Microlocal Analysis, Sharp Spectral Asymptotics and Applications I: Semiclassical Microlocal Analysis and Local and Microlocal Semiclassical Asymptotics [1st ed. 2019] 978-3-030-30556-7, 978-3-030-30557-4

Table of contents :
Front Matter ....Pages I-XLIX
Front Matter ....Pages 1-1
Introduction to Microlocal Analysis (Victor Ivrii)....Pages 2-125
Propagation of Singularities in the Interior of the Domain (Victor Ivrii)....Pages 126-195
Propagation of Singularities near the Boundary (Victor Ivrii)....Pages 196-284
Front Matter ....Pages 285-285
General Theory in the Interior of the Domain (Victor Ivrii)....Pages 286-406
Scalar Operators in the Interior of the Domain. Rescaling Technique (Victor Ivrii)....Pages 407-520
Operators in the Interior of Domain. Esoteric Theory (Victor Ivrii)....Pages 521-621
Front Matter ....Pages 622-622
Standard Local Semiclassical Spectral Asymptotics near the Boundary (Victor Ivrii)....Pages 623-741
Standard Local Semiclassical Spectral Asymptotics near the Boundary. Miscellaneous (Victor Ivrii)....Pages 742-800
Back Matter ....Pages 801-889

Citation preview

Victor Ivrii

Microlocal Analysis, Sharp Spectral Asymptotics and Applications I Semiclassical Microlocal Analysis and Local and Microlocal Semiclassical Asymptotics

Microlocal Analysis, Sharp Spectral Asymptotics and Applications I

Victor Ivrii

Microlocal Analysis, Sharp Spectral Asymptotics and Applications I Semiclassical Microlocal Analysis and Local and Microlocal Semiclassical Asymptotics

123

Victor Ivrii Department of Mathematics University of Toronto Toronto, ON, Canada

ISBN 978-3-030-30556-7 ISBN 978-3-030-30557-4 https://doi.org/10.1007/978-3-030-30557-4

(eBook)

Mathematics Subject Classification (2010): 35P20, 35S05, 35S30, 81V70 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speci cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on micro lms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speci c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional af liations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface The Problem of the Spectral Asymptotics, in particular the problem of the Asymptotic Distribution of the Eigenvalues, is one of the central problems in the Spectral Theory of Partial Differential Operators; moreover, it is very important for the General Theory of Partial Differential Operators. I started working in this domain in 1979 after R. Seeley [1] justified a remainder estimate of the same order as the then hypothetical second term for the Laplacian in domains with boundary, and M. Shubin and B. M. Levitan suggested me to try to prove Weyl’s conjecture. During the past almost 40 years I have not left the topic, although I had such intentions in 1985, when the methods I invented seemed to fail to provide the further progress and only a couple of not very exciting problems remained to be solved. However, at that time I made the step toward local semiclassical spectral asymptotics and rescaling, and new much wider horizons opened. So I can say that this book is the result of 40 years of work in the Theory of Spectral Asymptotics and related domains of Microlocal Analysis and Mathematical Physics (I started analysis of Propagation of singularities (which plays the crucial role in my approach to the spectral asymptotics) in 1975). This monograph consists of five volumes. This Volume I starts the general theory. It consists of three parts. The first one is devoted to the microlocal analysis and propagation of singularities, in the second and the third parts we derive local and microlocal spectral asymptotics inside domain, and near the boundary, correspondingly.

Victor Ivrii, Toronto, June 10, 2019.

V

Contents

Preface Introduction 1. Problems to Study . . . . . . . . . . . . . . . . . . . . . . . 2. Weyl’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Methods. II . . . . . . . . . . . . . . . . . . . . . . . . . . 5. What is in the Book . . . . . . . . . . . . . . . . . . . . . . Part I. Semiclassical Microlocal Analysis . . . . . Part II. Local and Microlocal Semiclassical Spectral Asymptotic in the Interior of the Domain Part II. Local and Microlocal Semiclassical Spectral Asymptotic near the Boundary . . . . . 6. How LSSA are Applied to Classical Asymptotics . . . . . . 7. Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. About the Author . . . . . . . . . . . . . . . . . . . . . . . Elements of Biography . . . . . . . . . . . . . . . . . . . My Way to Spectral Asymptotics and Around . . . . . . Weyl Conjecture (66–67 AW) . . . . . . . . . . . Aftermath (68–79 AW) . . . . . . . . . . . . . . . Multiparticle Quantum Theory (79 AW–88 AW) . . . . . New Dawn (88–93 AW) . . . . . . . . . . . . . . Etc (93 AW–113 AW) . . . . . . . . . . . . . . . Aftermath (113 AW–present time) . . . . . . . .

V XXII XXII XXIV XXV XXIX XXXI XXXI XXXV XXXIX XLI XLIII XLIV XLIV XLIV XLIV XLVII XLIX XLIX XLIX XLIX

VI

CONTENTS

I

Semiclassical Microlocal Analysis

1 Introduction to Microlocal Analysis 1.1 Pseudodifferential Operators . . . . . . . . . . . . . . . . 1.1.1 Calculus of Pseudodifferential Operators . . . . . h-Fourier Transform . . . . . . . . . . . . . . . . h-PDO Representation via h-Fourier Transform; Quantization of Different Flavors . . . . Symbols and h-Pseudodifferential Operators; qp, pq-, Weyl (Symmetric) Quantization. Double Symbols and Their Quantization Negligible Functions, Symbols, Operators. Going from One tType of Quantization to Another. Symbol and Operator Classes . . Algebra: Conjugate and Product of Operators. Symmetric Pseudodifferential Operators . . . Quantization: Abstract Operator Point of View . Matrix Pseudodifferential Operators . . . . . . . . Ellipticity, Parametrix, Resolvent. Functional Calculus . . . . . . . . . . . . . . . . . . . . Traces and Hilbert-Schmidt Norms . . . . . . . . Pseudodifferential Operators Applied to an Exponent 1.1.2 Analysis of Pseudodifferential Operators . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . Estimates . . . . . . . . . . . . . . . . . . . . . . Remainder Estimates . . . . . . . . . . . . . . . . Classes of Symbols and Pseudodifferential Operators Compactness . . . . . . . . . . . . . . . . . . . . Rescaling . . . . . . . . . . . . . . . . . . . . . . Ga ˚rding Inequalities . . . . . . . . . . . . . . . . Parametrix and iInverse . . . . . . . . . . . . . . Resolvent . . . . . . . . . . . . . . . . . . . . . . Analytic Functional Calculus of Pseudodifferential Operators . . . . . . . . . . . . . . . . . Smooth Functional Calculus of Pseudodifferential Operators . . . . . . . . . . . . . . . . . Families of Commuting Operators . . . . . . . . . Norms and Bounds . . . . . . . . . . . . . . . . .

VII

1 2 3 3 5 6

7

9 12 14 15 15 16 19 22 22 24 27 28 28 29 30 39 41 42 44 47 47

CONTENTS

VIII

Pseudodifferential Operators with Analytic Symbols and Logarithmic Uncertainty Principle . . . . . . Pseudodifferential Operators with Infinitely Smooth Symbols . . . . . . . . . . . . . . . . . . Pseudodifferential Operators with Analytic Symbols 1.1.4 Other Classes and Calculi of Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . Classical Pseudodifferential Operators . . . . . . . m Classes Sm σ,ς and Ψσ,ς . . . . . . . . . . . . . . . . Beals-Fefferman and H¨ormander Operators . . . . Global Operators . . . . . . . . . . . . . . . . . . h-Pseudodifferential Operators . . . . . . . . . . . Fourier Integral Operators . . . . . . . . . . . . . . . . . 1.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . Oscillatory Solutions to Evolution Equations . . . Blow-Up . . . . . . . . . . . . . . . . . . . . . . . Maslov Canonical Operator . . . . . . . . . . . . Propagator . . . . . . . . . . . . . . . . . . . . . 1.2.2 Lagrangian Distributions and Manifolds . . . . . Definition . . . . . . . . . . . . . . . . . . . . . . Main Properties . . . . . . . . . . . . . . . . . . . Maslov Index . . . . . . . . . . . . . . . . . . . . Other Properties . . . . . . . . . . . . . . . . . . Complex Phase . . . . . . . . . . . . . . . . . . . Coherent States . . . . . . . . . . . . . . . . . . . 1.2.3 Fourier Integral Operators . . . . . . . . . . . . . Final Remarks . . . . . . . . . . . . . . . . . . . 1.2.4 Metaplectic Operators . . . . . . . . . . . . . . . 1.2.5 Germs . . . . . . . . . . . . . . . . . . . . . . . . Wave Front Sets and Related Topics . . . . . . . . . . . 1.3.1 Principal nNotions . . . . . . . . . . . . . . . . . Philosophy . . . . . . . . . . . . . . . . . . . . . . Smooth (Standard) Theory . . . . . . . . . . . . Analytic Theory . . . . . . . . . . . . . . . . . . 1.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Propagator . . . . . . . . . . . . . . . . . . . . . 1.3.4 Alternative Theories . . . . . . . . . . . . . . . . Elliptic Boundary Value Problems . . . . . . . . . . . . . 1.1.3

1.2

1.3

1.4

48 50 51 53 53 56 60 60 61 62 62 62 65 65 68 68 68 70 75 76 77 78 79 87 88 90 93 93 93 94 96 96 100 101 101

CONTENTS Pseudodifferential Operators with Transmission Property . . . . . . . . . . . . . . . . . . . . . . . . . Causality . . . . . . . . . . . . . . . . . . . . . . Transmission Property . . . . . . . . . . . . . . . Pseudodifferential Operators in a Domain . . . . 1.4.2 Related Operators . . . . . . . . . . . . . . . . . Boundary (Poisson) Operators . . . . . . . . . . . Trace and Singular Green Operators . . . . . . . Boutet de Monvel Algebra . . . . . . . . . . . . . Ellipticity and Parametrices . . . . . . . . . . . . Resolvent and Functional Calculus . . . . . . . . 1.4.3 Applications to Elliptic Boundary Value Problems Parametrix Construction . . . . . . . . . . . . . . Dirichlet-to-Neumann and Similar Operators . . . Adjoint Operator; Self-Adjoint Operator . . . . . Projectors . . . . . . . . . . . . . . . . . . . . . . Semiclassical Theory . . . . . . . . . . . . . . . . 1.A Appendices . . . . . . . . . . . . . . . . . . . . . . . . . 1.A.1 Operator e itQ(D) . . . . . . . . . . . . . . . . . . . 1.A.2 Almost Analytic Compactly Supported Symbols . 1.A.3 Stationary Phase Method . . . . . . . . . . . . . 1.A.4 Function in the Box . . . . . . . . . . . . . . . .

IX

1.4.1

2 Propagation of Singularities in the Interior of the Domain 2.1 Energy Estimates Approach . . . . . . . . . . . . . . . . 2.1.1 Main Theorem . . . . . . . . . . . . . . . . . . . 2.1.2 Cauchy Problem. I . . . . . . . . . . . . . . . . . 2.1.3 Cauchy Problem. II . . . . . . . . . . . . . . . . . 2.1.4 Properties of Fundamental Solutions . . . . . . . 2.1.5 Overdetermined Systems . . . . . . . . . . . . . . Overdetermined Systems . . . . . . . . . . . . . . Multi-Time Systems . . . . . . . . . . . . . . . . 2.2 Geometric Interpretation . . . . . . . . . . . . . . . . . . 2.2.1 Discussion of the Microhyperbolicity Condition . Scalar Case . . . . . . . . . . . . . . . . . . . . . Cones of Directions of Microhyperbolicity . . . . Examples . . . . . . . . . . . . . . . . . . . . . .

101 101 102 104 105 105 107 110 110 112 113 113 116 117 119 120 121 121 122 123 124

126 127 127 133 136 138 143 143 143 145 145 145 145 148

CONTENTS

X

Microhyperbolicity: Algebraic Theory . . . . . . . Main Theorem . . . . . . . . . . . . . . . . . . . Main Theorem . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Examples of Propagation . . . . . . . . . . . . . . Constant Coefficients Case . . . . . . . . . . . . . Scalar Principal Type (Again) . . . . . . . . . . . Operators with Completely Factorizing Characteristic Symbol: Non-Involutive Case . . . Operators with Completely Factorizing Characteristic Symbol: Involutive Case . . . . . . Operators with not Factorizing Characteristic Symbol. I . . . . . . . . . . . . . . . . . . . Operators with not Factorizing Characteristic Symbol. II . . . . . . . . . . . . . . . . . . . Operators with not Factorizing Characteristic Symbol. III . . . . . . . . . . . . . . . . . . Operators with not Factorizing Characteristic Symbol. IV . . . . . . . . . . . . . . . . . . Operators with Rough Symbols . . . . . . . . . . . . . . 2.3.1 Main Theorem . . . . . . . . . . . . . . . . . . . Statement of the Main Theorem . . . . . . . . . . Proof of the Main Theorem . . . . . . . . . . . . 2.3.2 Corollaries . . . . . . . . . . . . . . . . . . . . . . Heisenberg Approach . . . . . . . . . . . . . . . . . . . . 2.4.1 Evolution of Operators . . . . . . . . . . . . . . . Heisenberg Approach . . . . . . . . . . . . . . . . Solving Heisenberg Equations. Standard Case . . Discussion . . . . . . . . . . . . . . . . . . . . . . Properties of Hamiltonian Flow . . . . . . . . . . Solving Heisenberg Equations for Long-Time Evolution . . . . . . . . . . . . . . . . . 2.4.2 Propagation of Singularities . . . . . . . . . . . . Scalar Principal Symbol Case . . . . . . . . . . . Diagonalization . . . . . . . . . . . . . . . . . . . Heisenberg Evolution for Matrix Principal Symbols Main Theorem . . . . . . . . . . . . . . . . . . . 2.2.2

2.3

2.4

149 150 150 153 154 154 155 155 159 160 161 163 167 167 168 168 169 175 176 177 177 180 181 182 183 187 187 188 191 193

CONTENTS 3 Propagation of Singularities near the Boundary 3.1 Energy Estimates Approach . . . . . . . . . . . . . . . . 3.1.1 Statement of the Main Theorem . . . . . . . . . . Preliminary Notes . . . . . . . . . . . . . . . . . . Finally, the Main Theorem . . . . . . . . . . . . . 3.1.2 Proof of the Main Theorem . . . . . . . . . . . . Reduction to a Special Case . . . . . . . . . . . . Elliptic Theory . . . . . . . . . . . . . . . . . . . Main Theorem Refined . . . . . . . . . . . . . . . Proof of Theorem 3.1.13. Step 1 . . . . . . . . . . Proof of Theorem 3.1.13. Step 2 . . . . . . . . . . Proof of Theorem 3.1.13. Step 3 . . . . . . . . . . Proof of Proposition 3.1.14 . . . . . . . . . . . . . Proof of Proposition 3.1.5 . . . . . . . . . . . . . 3.2 Geometric Interpretation . . . . . . . . . . . . . . . . . . 3.2.1 Main Theorem . . . . . . . . . . . . . . . . . . . Notations and Definitions . . . . . . . . . . . . . Finally, Main Theorem . . . . . . . . . . . . . . . 3.2.2 Discussion and Examples . . . . . . . . . . . . . . General Remarks . . . . . . . . . . . . . . . . . . Constant Coefficients Case . . . . . . . . . . . . . Transversal Points . . . . . . . . . . . . . . . . . Elliptic Points . . . . . . . . . . . . . . . . . . . . Bicharacteristics Tangent to the Boundary . . . . 3.3 Corollaries of Theorem 3.1.7 . . . . . . . . . . . . . . . . 3.3.1 Finite Speed of Propagation and Related Topics . Finite Speed of Propagation . . . . . . . . . . . . Decomposition of the Fundamental Solution . . . 3.3.2 Normal Singularity Property . . . . . . . . . . . . Preliminary Analysis . . . . . . . . . . . . . . . . Main Results . . . . . . . . . . . . . . . . . . . . 3.4 Energy Estimates Approach Revisited . . . . . . . . . . . 3.4.1 Theorem 3.1.7(ii) Revisited . . . . . . . . . . . . Energy Estimates . . . . . . . . . . . . . . . . . . Condition (3.4.13) . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . Theorems 3.1.13(ii), 3.1.7(ii) and 3.2.4(ii) Generalized . . . . . . . . . . . . . . . . . . . .

XI 196 197 197 197 203 204 204 210 215 216 218 222 223 226 226 226 226 229 231 231 231 231 233 234 239 240 240 244 245 245 248 253 253 254 256 257 258

CONTENTS

XII 3.4.2

Main Theorem . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . Symbol q: General Construction . . . . . . . . . . Energy Estimates. I . . . . . . . . . . . . . . . . Energy Estimates. II . . . . . . . . . . . . . . . . Energy Estimates. III . . . . . . . . . . . . . . . Results: Single Glancing Point . . . . . . . . . . . Further Discussion . . . . . . . . . . . . . . . . . 3.5 Propagation of Singularities along Long Bicharacteristics 3.5.1 Evolution of Operators . . . . . . . . . . . . . . . Standard Theory . . . . . . . . . . . . . . . . . . Reflections . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Long Time Evolution . . . . . . . . . . . . . . . . Main Theorem . . . . . . . . . . . . . . . . . . . Proof of the Main Theorem . . . . . . . . . . . . 3.5.3 Branching Billiards and Propagation . . . . . . . 3.A Appendices . . . . . . . . . . . . . . . . . . . . . . . . . 3.A.1 On Solutions to Linear Matrix Equations . . . . . 3.A.2 Structure of Gliding Points . . . . . . . . . . . . .

II

Local and Microlocal Semiclassical Spectral Asymptotics in the Interior of the Domain

4 General Theory in the Interior of the Domain 4.1 Introduction . . . . . . . . . . . . . . . . . . . . 4.1.1 Main Objects . . . . . . . . . . . . . . . 4.1.2 Kind of Assumptions to be Used . . . . 4.1.3 Plan of the Chapter . . . . . . . . . . . . 4.2 Preliminary Analysis and Tauberian Method . . 4.2.1 General Estimates . . . . . . . . . . . . 4.2.2 Semiclassical Spectral Gaps. I . . . . . . First Standard Set of Assumptions . . . Second Standard Set of Assumptions . . Estimates to Propagator . . . . . . . . . Estimates to Projector . . . . . . . . . . Estimates without Cut-offs . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

259 259 261 262 264 264 266 267 270 270 271 272 275 275 277 280 281 281 282

285 286 286 286 289 290 291 291 294 294 296 296 299 303

CONTENTS

4.3

4.4

4.5

Semiclassical Spectral Gaps under Reduced Smoothness Assumptions . . . . . . . . . . . . . 4.2.3 Main Tauberian Estimate . . . . . . . . . . . . . Main Tauberian Formula . . . . . . . . . . . . . . Asymptotics with Mollification . . . . . . . . . . Asymptotics with Averaging . . . . . . . . . . . . Operator Valued Functions . . . . . . . . . . . . . 4.2.4 Applications . . . . . . . . . . . . . . . . . . . . . Scheme of tThings . . . . . . . . . . . . . . . . . So, What We Get? . . . . . . . . . . . . . . . . . 4.2.5 Estimates in the Hilbert Scale . . . . . . . . . . . Hilbert sScale . . . . . . . . . . . . . . . . . . . . Estimates in the Hilbert Scale . . . . . . . . . . . The Method of Successive Approximations . . . . . . . . 4.3.1 The Method of Successive Approximations in the Standard Case . . . . . . . . . . . . . . . . . . . . Pilot-Construction . . . . . . . . . . . . . . . . . Transformation of the Successive Approximation Formula . . . . . . . . . . . . . . . . . . Minor Modifications . . . . . . . . . . . . . . . . 4.3.2 Successive Approximations in the Hilbert Scale . 4.3.3 Calculations . . . . . . . . . . . . . . . . . . . . . 4.3.4 Exploiting Microhyperbolicity and Increasing T . Exploiting Microhyperbolicity . . . . . . . . . . . General Basic Theorems . . . . . . . . . . . . . . . . . . 1 4.4.1 General Theory: T  h 2 −δ . . . . . . . . . . . . . 4.4.2 General Microhyperbolic Theory: T  1 . . . . . Asymptotics with Mollification . . . . . . . . . . Asymptotics without Mollification . . . . . . . . . Improved Asymptotics without Mollification . . . Useful Formulae . . . . . . . . . . . . . . . . . . . Sharper Asymptotics . . . . . . . . . . . . . . . . . . . . 4.5.1 Preliminary Discussion . . . . . . . . . . . . . . . Increasing T . . . . . . . . . . . . . . . . . . . . . Synthesis . . . . . . . . . . . . . . . . . . . . . . Sharp and Sharper . . . . . . . . . . . . . . . . . 4.5.2 Standard Sharp Theory . . . . . . . . . . . . . . Analysis of Wave Front Sets . . . . . . . . . . . .

XIII

306 310 310 311 312 314 314 314 317 318 318 320 321 321 321 323 325 327 332 338 338 341 342 346 346 347 349 352 354 354 354 355 356 357 357

XIV Periodic Trajectories and Loops . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . 4.5.3 Sharper Remainder Estimates . . . . . . . . Analysis as |t| ≤ T0 . . . . . . . . . . . . . . Successive Approximations . . . . . . . . . . Increasing T . . . . . . . . . . . . . . . . . . Tauberian eEstimates . . . . . . . . . . . . . Finally, Results . . . . . . . . . . . . . . . . 4.5.4 Discussion . . . . . . . . . . . . . . . . . . . Local Results . . . . . . . . . . . . . . . . . Simple Improvements . . . . . . . . . . . . . Sharper Remainder Estimates . . . . . . . . Matrix Case and Branching Bicharacteristics 4.6 Operators with Rough Coefficients . . . . . . . . . 4.6.1 Discussion: Rationale . . . . . . . . . . . . . 4.6.2 Estimates . . . . . . . . . . . . . . . . . . . 4.6.3 Successive Approximations and Calculations 4.6.4 Main Theorems . . . . . . . . . . . . . . . . 4.6.5 Applications: Bracketing Method . . . . . . 4.6.6 Final Remarks . . . . . . . . . . . . . . . . 4.7 Operators with Irregular Coefficients . . . . . . . . 4.7.1 Preliminary Analysis . . . . . . . . . . . . . 4.7.2 Tauberian Estimates . . . . . . . . . . . . . 4.7.3 Weyl Asymptotics . . . . . . . . . . . . . . 4.7.4 Sharper Asymptotics . . . . . . . . . . . . . 4.A Appendices . . . . . . . . . . . . . . . . . . . . . . 4.A.1 On the Traces of Almost Analytic Functions 4.A.2 Smoothing of Cϑm Functions . . . . . . . . .

CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Scalar Operators in the Interior of the Domain. Rescaling Technique 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Pilot: Schr¨odinger Operator . . . . . . . . . . . . 5.1.2 Plan of the Chapter . . . . . . . . . . . . . . . . . 5.2 Spectral Asymptotics for Scalar Operators . . . . . . . . 5.2.1 From Condition “No Critical Point” to “rank Hess a ≥ 2 at Critical Points”. I . . . . . . . . . . . . . . . Standard Asymptotics . . . . . . . . . . . . . . .

359 361 365 365 366 367 367 368 369 369 370 372 374 374 375 378 379 384 386 389 391 391 392 395 400 400 400 403

407 407 407 409 411 412 412

CONTENTS

5.3

5.4

Sharp and Sharper Asymptotics . . . . . . . . . . Near-Critical Zone . . . . . . . . . . . . . . . . . Sharp and Sharper Asymptotics (End) . . . . . . 5.2.2 From Condition “No Critical Point” to “rank Hess a ≥ 2 at Critical Points”. II . . . . . . . . . . . . . . 5.2.3 From Condition “rank Hess a ≥ 2 at Critical Points” to “rank Hess a ≥ 1 at Critical Points” . . . . . . 5.2.4 From Condition “rank Hess a ≥ 1 at Critical Points” to a Weak non-Degeneracy Condition . . . . . . . 5.2.5 From Microlocal to Local Spectral Asymptotics . Spectral Asymptotics for the Schr¨odinger and Dirac Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Schr¨odinger Operator . . . . . . . . . . . . . . . . Main Assumptions . . . . . . . . . . . . . . . . . Asymptotics with Spatial Mollification . . . . . . Asymptotics without Spatial Mollification . . . . Asymptotics without Spatial Mollification and Short Loops . . . . . . . . . . . . . . . . . . . 5.3.2 Dirac Operator . . . . . . . . . . . . . . . . . . . Assumptions . . . . . . . . . . . . . . . . . . . . . Calculations . . . . . . . . . . . . . . . . . . . . . Asymptotics for m ≥ c0 . . . . . . . . . . . . . . . Asymptotics for m ≤ c0 . . . . . . . . . . . . . . . Further Improvements . . . . . . . . . . . . . . . Sharp Remainder Estimates . . . . . . . . . . . . Operators with Irregular Coefficients . . . . . . . . . . . 5.4.1 Preliminary Remarks . . . . . . . . . . . . . . . . 5.4.2 Schr¨odinger Operator . . . . . . . . . . . . . . . . Reminder . . . . . . . . . . . . . . . . . . . . . . Getting Rid of Condition (5.4.2) . . . . . . . . . . From Condition (5.4.2) to Condition (5.4.3) . . . Case d ≤ s + 1 Reloaded . . . . . . . . . . . . . . Dropping Condition (5.4.3) . . . . . . . . . . . . Sharp Remainder Estimates . . . . . . . . . . . . Applying Aesults of Section 4.7 . . . . . . . . . . Applying Results of Section 4.7. II . . . . . . . . 5.4.3 Dirac Operator . . . . . . . . . . . . . . . . . . . Getting Rid of Condition (5.4.2) . . . . . . . . . .

XV 418 419 423 424 434 440 446 447 447 447 448 453 454 463 463 465 470 475 481 485 489 489 490 490 491 494 495 498 499 501 501 503 503

XVI

CONTENTS

From Condition (5.4.2) to Condition (5.4.3) . . . Dropping Condition (5.4.3) . . . . . . . . . . . . Sharp Remainder Estimate . . . . . . . . . . . . . 5.4.4 General Operators . . . . . . . . . . . . . . . . . Dropping Condition |∇ξ a| = 0 . . . . . . . . . . . Case ν(t) = o(t) . . . . . . . . . . . . . . . . . . Case ν(t) = o(t). II . . . . . . . . . . . . . . . . . Sharp Remainder Estimates . . . . . . . . . . . . 5.A Appendices . . . . . . . . . . . . . . . . . . . . . . . . . 5.A.1 Spectral Kernel Calculations for Model Operators Case d = 1 . . . . . . . . . . . . . . . . . . . . . Case d = 2 . . . . . . . . . . . . . . . . . . . . . 5.A.2 On Pauli Matrices . . . . . . . . . . . . . . . . .

505 505 506 506 506 509 511 512 513 513 513 515 516

6 Operators in the Interior of Domain. Esoteric Theory 6.1 Families of Commuting Operators . . . . . . . . . . . . . 6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.1.2 General Theory . . . . . . . . . . . . . . . . . . . Elliptic Arguments . . . . . . . . . . . . . . . . . Successive Approximations . . . . . . . . . . . . . Basic Results (General Case) . . . . . . . . . . . The Microhyperbolicity Condition . . . . . . . . . Exploiting Microhyperbolicity . . . . . . . . . . . Tauberian Arguments . . . . . . . . . . . . . . . Principal Results . . . . . . . . . . . . . . . . . . Possible Generalizations, Remarks . . . . . . . . . 6.1.3 Operators with Scalar Symbols . . . . . . . . . . Preliminary Remarks . . . . . . . . . . . . . . . . Asymptotics with Mollifications . . . . . . . . . . Principal Results . . . . . . . . . . . . . . . . . . 6.2 Operators with Periodic Hamiltonian Flow . . . . . . . . 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.2.2 Operator B . . . . . . . . . . . . . . . . . . . . . Scalar Case . . . . . . . . . . . . . . . . . . . . . Matrix Case . . . . . . . . . . . . . . . . . . . . . 6.2.3 Long Term Evolution . . . . . . . . . . . . . . . . Main Theorem . . . . . . . . . . . . . . . . . . . Corollaries . . . . . . . . . . . . . . . . . . . . . .

521 522 522 524 524 525 526 527 528 530 531 532 534 534 534 536 538 538 545 545 551 552 552 553

CONTENTS Spectral Gaps (the First Attempt) . . . . . . . . Exploiting Microhyperbolicity . . . . . . . . . . . Extending Time Interval . . . . . . . . . . . . . . Tauberian Arguments . . . . . . . . . . . . . . . Removing Cut-off ψ . . . . . . . . . . . . . . . . Operators with Periodic Hamiltonian Flow (End) . . . . 6.3.1 Scalar a and Subperiodic Trajectories . . . . . . . Structure of Subperiodic Trajectories . . . . . . . Propagation of Singularities near Π . . . . . . . . Successive Approximations . . . . . . . . . . . . . Exploiting Microhyperbolicity . . . . . . . . . . . Tauberian Estimates . . . . . . . . . . . . . . . . 6.3.2 The case of Degenerate Scalar Operator B . . . . Asymptotics with Mollification . . . . . . . . . . Estimates without Mollification . . . . . . . . . . 6.3.3 Some Remarks . . . . . . . . . . . . . . . . . . . 6.3.4 Spectral Gaps Reloaded . . . . . . . . . . . . . . 6.3.5 Families of Commuting Operators Approach . . . Operators with Completely Periodic Evolution . . Operators with the Global Decomposition . . . . General Operators . . . . . . . . . . . . . . . . . 6.3.6 Strong Perturbations . . . . . . . . . . . . . . . . 6.3.7 Operators with Massive Set of Periodic Trajectories 6.3.8 Examples . . . . . . . . . . . . . . . . . . . . . . Laplacian . . . . . . . . . . . . . . . . . . . . . . Harmonic Oscillator . . . . . . . . . . . . . . . . Coulomb pPotential . . . . . . . . . . . . . . . . . Asymptotics for Dirac Energy . . . . . . . . . . . . . . . 6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.4.2 Estimates . . . . . . . . . . . . . . . . . . . . . . Special Case . . . . . . . . . . . . . . . . . . . . . Smooth Case . . . . . . . . . . . . . . . . . . . . Singular Homogeneous Case . . . . . . . . . . . . Singular Homogeneous Case: Using ξ-Microhyperbolicity . . . . . . . . . . . 6.4.3 Calculations . . . . . . . . . . . . . . . . . . . . . Constant Coefficients Case . . . . . . . . . . . . . General ξ-Microhyperbolic Case . . . . . . . . . . 6.2.4

6.3

6.4

XVII 558 560 560 560 564 564 564 564 566 568 572 573 574 574 575 578 580 581 581 584 586 587 589 592 592 593 594 595 595 597 597 599 601 602 603 603 604

CONTENTS

XVIII 6.4.4

Scalar Case . . . . . . . . . . . . . . . . . . . . Smooth Case. I . . . . . . . . . . . . . . . . . . Smooth Case. II . . . . . . . . . . . . . . . . . Singular Homogeneous Case . . . . . . . . . . . 6.4.5 Rescaling Technique . . . . . . . . . . . . . . . Schr¨odinger operator . . . . . . . . . . . . . . . 6.5 Mollification with Respect to Non-Spectral Parameters 6.5.1 Linear Dependence . . . . . . . . . . . . . . . . 6.5.2 Microhyperbolic Case . . . . . . . . . . . . . . . 6.5.3 Averaging . . . . . . . . . . . . . . . . . . . . . 6.A Appendices . . . . . . . . . . . . . . . . . . . . . . . . 6.A.1 Hamiltonian Flows of Commuting Operators . . 6.A.2 One Property of Real Residues . . . . . . . . . 6.A.3 Eigenvalues of Some Model Operators . . . . . .

. . . . . . . . . . . . . .

606 606 608 608 611 611 614 614 616 618 618 618 619 620

Local and Microlocal Semiclassical Spectral Asymptotics near the Boundary

622

7 Standard Local Semiclassical Spectral Asymptotics near the Boundary 7.1 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . 7.1.1 The General Case . . . . . . . . . . . . . . . . . . 7.1.2 Elliptic Case . . . . . . . . . . . . . . . . . . . . . 7.1.3 Operators with Minimal Smoothness Conditions . 7.1.4 Estimates for the Schr¨odinger Operator . . . . . . 7.2 Method of Successive Approximations . . . . . . . . . . . 7.2.1 Approximating the Free Space Term . . . . . . . Standard Calculations Revised . . . . . . . . . . . Calculating Restriction to the Boundary . . . . . 7.2.2 Approximations for the Reflected Wave . . . . . . 7.2.3 Calculations . . . . . . . . . . . . . . . . . . . . . Starting Point . . . . . . . . . . . . . . . . . . . . 1-Dimensional Parametrices . . . . . . . . . . . . 1-Dimensional Parametrices (End) . . . . . . . . Calculations: What We Got . . . . . . . . . . . . Calculations: Taking Traces . . . . . . . . . . . . Calculations: Mollification with Respect to τ . . .

623 624 624 629 636 641 645 645 645 646 650 657 657 657 659 666 667 668

III

CONTENTS 7.2.4 Exploiting Microhyperbolicity . . . . . . . . . . . 7.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . 7.3 General Basic Theorems . . . . . . . . . . . . . . . . . . 7.3.1 Asymptotics with Mollification . . . . . . . . . . 7.3.2 Asymptotics under Microhyperbolicity Condition 7.3.3 Asymptotics in the Elliptic Zone . . . . . . . . . 7.3.4 Local Asymptotics . . . . . . . . . . . . . . . . . 7.4 Sharper Asymptotics . . . . . . . . . . . . . . . . . . . . 7.4.1 Discussion . . . . . . . . . . . . . . . . . . . . . . 7.4.2 General Theory . . . . . . . . . . . . . . . . . . . Sharp Asymptotics . . . . . . . . . . . . . . . . . Sharper Asymptotics . . . . . . . . . . . . . . . . 7.4.3 Billiard Flow for the Schr¨odinger Operator . . . . Definition of the Flow . . . . . . . . . . . . . . . Growth of the Billiard Flow. I . . . . . . . . . . . Growth of the Billiard Flow. II . . . . . . . . . . Non-Periodicity Condition and Results. I . . . . . Non-Periodicity Condition and Results. II . . . . Non-Periodicity Condition for Branching Billiards 7.5 Operators with Irregular Coefficients . . . . . . . . . . . 7.5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . Rescaling Technique . . . . . . . . . . . . . . . . Propagation . . . . . . . . . . . . . . . . . . . . . What Then? . . . . . . . . . . . . . . . . . . . . . Mollification . . . . . . . . . . . . . . . . . . . . . 7.5.2 O(h1−d ) Remainder Estimate for Scalar Operators Inner zZone . . . . . . . . . . . . . . . . . . . . . Moving Closer to the Boundary . . . . . . . . . . Boundary Strip . . . . . . . . . . . . . . . . . . . Synthesis . . . . . . . . . . . . . . . . . . . . . . 7.5.3 O(h1−d ) Remainder Estimate for Matrix Operators Covering . . . . . . . . . . . . . . . . . . . . . . . Estimates . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Sharp Remainder Estimate . . . . . . . . . . . . . Near Boundary Zone . . . . . . . . . . . . . . . . Inner Zone . . . . . . . . . . . . . . . . . . . . . . 7.5.5 Less Sharp Remainder Estimates . . . . . . . . . 7.A Appendices . . . . . . . . . . . . . . . . . . . . . . . . .

XIX 669 673 673 674 675 677 681 683 683 685 685 686 687 687 688 692 697 699 702 707 708 708 709 710 711 712 712 717 718 718 721 722 725 726 726 728 731 733

XX

CONTENTS 7.A.1 Sets Associated with the Microhyperbolicity Condition . . . . . . . . . . . . . . . . . . . . . .

733

8 Standard Local Semiclassical Spectral Asymptotics near the Boundary. Miscellaneous 8.1 Pointwise Spectral Asymptotics . . . . . . . . . . . . . . 8.1.1 Preliminary Analysis . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . Toy-Model . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Schr¨odinger Operator . . . . . . . . . . . . . . . . General Theory . . . . . . . . . . . . . . . . . . . Calculations and Main Theorem . . . . . . . . . . 8.1.3 Generalizations . . . . . . . . . . . . . . . . . . . 8.2 Scalar Operators and Rescaling . . . . . . . . . . . . . . 8.2.1 Schr¨odinger Operator. I . . . . . . . . . . . . . . General theory . . . . . . . . . . . . . . . . . . . Sharp Remainder Estimates . . . . . . . . . . . . Special Case d = 1 . . . . . . . . . . . . . . . . . Generalizations . . . . . . . . . . . . . . . . . . . 8.2.2 Schr¨odinger Operator. II . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . Microhyperbolicity Conditions . . . . . . . . . . . Analysis in the “Elliptic” Zone . . . . . . . . . . Asymptotics as V ≈ 0 . . . . . . . . . . . . . . . 8.2.3 Schr¨odinger Operator. III . . . . . . . . . . . . . 8.3 Operators with Periodic Hamiltonian Flows . . . . . . . 8.3.1 Discussion and Plan . . . . . . . . . . . . . . . . 8.3.2 Simple Hamiltonian Flow . . . . . . . . . . . . . Inner Asymptotics . . . . . . . . . . . . . . . . . Asymptotics near Boundary . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Branching Hamiltonian Flow with “Scattering” . Analysis in X2 . . . . . . . . . . . . . . . . . . . . Analysis in X1 . . . . . . . . . . . . . . . . . . . . 8.3.4 Two Periodic Flows . . . . . . . . . . . . . . . . . Examples and Discussion . . . . . . . . . . . . . . Reduction to the Boundary . . . . . . . . . . . . Analysis of the Evolution . . . . . . . . . . . . . .

742 743 743 743 745 747 747 750 751 753 753 753 757 757 760 760 760 761 762 764 766 767 767 769 769 772 774 776 776 778 783 783 784 786

CONTENTS 8.4

Spectral Asymptotics on Subspaces . . . . . . . . . . . 8.4.1 Examples and Discussion . . . . . . . . . . . . . 8.4.2 Analysis away from the Boubdary . . . . . . . . 8.4.3 Analysis near the Boundary. I . . . . . . . . . . 8.4.4 Analysis near the Boundary. II. Seeley’s Method

XXI . . . . .

789 789 793 794 798

Bibliography

801

Presentations

873

Index

875

Introduction 1. Problems to Study Finding the Spectral Asymptotics, in particular the Asymptotic Distribution of Eigenvalues, is one of the central problems of the Spectral Theory of Partial Differential Operators, and is very important for the General Theory of Partial Differential Operators. Apart from applications to Quantum Mechanics, Radio Physics, Continuum Media Mechanics (Elasticity, Acoustics, Hydrodynamics, Theory of Shells), etc., there are also applications to mathematics itself and deep though non-obvious links to Differential Geometry, Dynamical Systems Theory, Ergodic Theory and Number Theory; even the term Spectral Geometry was coined. All these circumstances make this topic very attractive for a mathematician. The original problem was formulated independently by A. Sommerfeld [1] and H. A. Lorentz [1] in 1910 who stated the Weyl’s Law as a conjecture based on the book of Lord Rayleigh “The Theory of Sound” (1887) (see in details in W. Arendt and others [1]). This conjecture was proven in 1911 by then young mathematician, a disciple of David Hilbert, specialist in partial differential and integral equations Hermann Weyl who published several papers [1–5] devoted to eigenvalue asymptotics for the Laplace operator (and also the elasticity operator) in a bounded domain with a regular boundary. After this article a huge number of various papers devoted to spectral asymptotics were published among the authors of which were numerous prominent mathematicians. Also, in [4] Weyl published his famous Weyl conjecture. Much later, in 1950, H. Weyl in his paper [7] returned to this topic. The theory was developed in two directions: on one hand, the theory was extended to consider more general operators, boundary conditions and

XXII

1. PROBLEMS TO STUDY

XXIII

geometrical domains; on the other hand, the asymptotics were improved and more accurate remainder estimates were derived. In the latter extensions the links with differential geometry, dynamical systems theory and ergodic theory appeared. Even the theory of eigenvalue asymptotics for the Laplace (or Laplace-Beltrami) operator has a long, dramatic and still unfinished history. Apart from asymptotics with respect to the spectral parameter , asymptotics with respect to other parameters appeared; the most important among them are (in my opinion) semiclassical asymptotics, i. e. asymptotics with respect to a small parameter h (the Planck constant in physics), tending to +0. For a long time these asymptotics were in the shadows: most attention was paid to the eigenvalue asymptotics for operators on compact manifolds (with or without boundary). The results which had been obtained in that case were then reproved for operators in Rd such as the Schr¨odinger operator h2 Δ + V (x) with fixed h > 0 and with V (x) → +∞ as |x| → ∞ where Δ is a (positive) Laplacian; less attention was paid to semiclassical asymptotics (i.e., asymptotics of the number of eigenvalues below some fixed level λ as h → +0). The asymptotics of small negative eigenvalues were considered in the case of fixed h and V (x) decreasing at infinity as |x|2m with m ∈ (−1, 0); under reasonable conditions in this case the discrete spectrum of the operator has accumulation point −0 and the essential spectrum coincides with [0, +∞). The result of the development of the theory was that at a certain time there existed four parallel (though not equally developed and not equally respected)1) theories, the statements in each of which had to be proven separately. This pluralism was over from ≈1985 because all other results are easily derived from the Local Semiclassical Spectral Asymptotics (LSSA in what follows)2) . 1)

While the achievements in the first theory were broadly known by mathematicians, the other results were familiar to the narrow circle of PDE-folk and mathematical physicists, many of whom (including myself, it is a real shame for me) even thought that only the first topic is of the prime interest. 2) I started to study semiclassical spectral asymptotics at the time when I foolishly assumed that the first theory was completely finished and I had nothing to do; so I decided to extend the results to semiclassical asymptotics before leaving the area. When I told my first results to M. Z. Solomyak, he asked if those results could be deduced from the results of the classical theory by the Birman-Schwinger principle. A quick check

INTRODUCTION

XXIV

One of the main tools here is a rescaling technique (see Section 6. How LSSA are Applied to Classical Asymptotics) I invented approximately in the same time. The semiclassical spectral asymptotics are the main object of this book, and all the other results are obtained as applications. We also consider applications to the multiparticle quantum theory the topic I began working on in 1990 with M.I.Sigal. Alternative introduction the reader can find in V. Ivrii [28]: “100 years of Weyl’s law”; see also my talk [9].

2. Weyl’s law Let us formulate here the Weyl’s law for a non-Magnetic Schr¨odinger operator H = h2 Δ + V (x),

(0.1)

where Δ = −∂x21 − ... − ∂x2d , h  1 and V (x) is the real valued function. Assuming that H is a self-adjoint operator, let E (λ) is its spectral projector and e(x, y , λ) its Schwartz kernel. Then Weyl’s law states (0.2) N(λ) ≈ (2πh)−d

 {|ξ|2 +V (x)≤λ}

dxdξ = (2πh)−d d





d/2

λ − V )+ dx,

where N(λ) := Tr(E (λ)) is an eigenvalue counting function i.e. a number of eigenvalues of H lesser than λ provided (−∞, λ) ∩ Specess (H) = ∅, otherwise N(λ) = +∞, d is a volume of the unit ball in Rd . There are also local Weyl’s law   d/2 −d (0.3) Tr(E (λ)ψ) ≈ (2πh) d λ − V )+ ψ(x) dx for ψ ∈ C0∞ and pointwise Weyl’s law (0.4)

 d/2 e(x, x, λ) ≈ (2πh)−d d λ − V )+ .

gave a very surprising answer: using “classical” asymptotics one can prove semiclassical asymptotics only under very unnatural restrictions; but using semiclassical asymptotics one can extend “classical” asymptotics to more general operators and problems.

3. METHODS

XXV

3. Methods In his papers, starting from the second one, H. Weyl applied the variational method (Dirichlet-Neumann bracketing) he invented; later this method was improved in various directions by many outstanding mathematicians and mathematical physicists (R. Courant, M. Sh. Birman, and others). This method is simple and works in the general situations (even for nonsemibounded operators) and seems to be the best method if one does not care about precise remainder estimates. Later Tauberian methods appeared (first due to T. Carleman). The idea is to consider some function F (H, t) of an operator in question H and some auxiliary parameter t. Then  (0.5) Tr(F (H, t)) = F (τ , t) dτ Tr(E (τ )), where E (τ ) is the spectral projector. One then tries to construct the lefthand expression by means of PDE theory and then recover Tr(E (τ )) with some error by means of an appropriate Tauberian theorem. The choice of the function F (H, t) is very important here. Taking F (H, t) = exp(−tA) (for a positive operator H), F (H, t) = (t − H)−1 and F (H, t) = H t one obtains respectively the heat equation, resolvent and ζ-function methods, which are usually simple in their PDE part but difficult in the Tauberian part and almost always fail to provide a decent remainder estimate. On the other hand, the hyperbolic operator method due to B. M. Levitan and V. G. Avakumovicˇ is difficult in its PDE part but is almost trivial in the Tauberian part and provides accurate remainder estimates. By this method all the asymptotics with the most accurate remainder estimates were obtained. This method is based on the function U(t) = F (H, t) = exp(itH) which satisfies (Dt − H)U = 0, U(0) = I , and therefore the Schwartz kernel u(x, y , t) of the operator is the fundamental solution to the Cauchy problem (or the initial-boundary value problem) for the operator Dt − H (where Dt = −i∂t , etc.). The Schwartz kernel is connected to the eigenvalue counting function of the operator H by the formula   (0.6) u(x, x, t) dx = exp(itλ) dλ N(λ). In the case of a matrix operator H, u(x, y , t) is a matrix-valued function and should be replaced by its trace in the left-hand expression. Then by

INTRODUCTION

XXVI

the means of the inverse Fourier transform we can recover N(λ) provided we have constructed u(x, y , t) by means of the methods of partial differential operator theory. In fact, we are never able (excluding some very special and rare cases) to construct u(x, y , t) precisely and for all the values t ∈ R. Usually (now we assume that H is an elliptic first-order pseudodifferential operator) the fundamental solution is constructed approximately (modulo smooth functions) for t belonging to some interval [−T , T ] with T > 0. As a consequence we obtain, modulo O(λ−s ) with any arbitrarily chosen exponent s, an expression for     (0.7) Ft→τ χT (t) u(x, x, t) dx = χˆT (τ − λ) dλ N(λ), where χ is a fixed smooth function supported in [−1, 1], χT (t) = χ(t/T ) and the hat as well as Ft→τ mean the Fourier transform. Then, if we know the left-hand expression, we are able, by the Tauberian theorem due to B. M. Levitan, to recover N(λ) approximately by the formula  λ     Ft→τ χT (t)σ(t) (τ ) dτ + O λd−1 , (0.8) N(λ) = −∞

where d is the dimension of the domain and  (0.9) σ(t) = u(x, x, t) dx. The explicit construction of u(x, x, t) in this situation leads us to the formula   N(λ) = κ0 λd + O λd−1 (0.10) with the leading coefficient (0.11)

κ0 = (2π)

−d

 dxdξ, H(x,ξ) 0. For the h-pseudodifferential operators, considered in this book, this approach is essentially simpler and more transparent because there is a selected parameter h; this case will be discussed below. This method allows us to prove the asymptotics (0.8) for an arbitrary self-adjoint mth order elliptic operator with m > 0 and the spectral parameter λm on a compact manifold with or without boundary (in the former case the boundary conditions are also supposed to be elliptic), scalar or matrix, semibounded from below or not semibounded at all (in this case N(λ) is replaced by N± (λ) which is the number of eigenvalues lying between 0 and ±λm ); the formula for κ0 should be changed if necessary. The two-term asymptotics   (0.12) N(λ) = κ0 λd + κ1 λd−1 + o λd−1 , suggested by H. Weyl [4] (who also gave a formula for κ1 ) fails to be true without some additional condition. Surely this asymptotic expansion is wrong for d = 1 and for the Laplace-Beltrami operator on the sphere Sd ; this fact is associated with the high multiplicities of its eigenvalues. Moreover, it remains wrong in the case when this Laplace-Beltrami operator is perturbed by a potential or even by a symmetric first-order operator with small coefficients; in these cases all the eigenvalues of high multiplicities will be split into narrow eigenvalue clusters separated by spectral gaps. On the other hand, under conditions of a global nature the asymptotics (0.12) is valid. For a scalar operator on a compact manifold without boundary the relevant condition is:

XXVIII

INTRODUCTION

The measure of the {set of all the points of the cotangent bundle periodic with respect to the Hamiltonian flow generated by the principal symbol} equals 0 (see J. J. Duistermaat and V. Guillemin [1]). The condition is more complicated for matrix operators. For a scalar second-order operator on a compact manifold with the boundary one needs only to consider trajectories which are transversal to the boundary and which reflect according the laws of geometrical optics. There are some points of the cotangent bundle through which infinitely long trajectories do not pass but the measure of these dead-end points vanishes and we need not take them into account. However, for higher-order operators as well as for matrix operators the trajectories reflected from the boundary can branch and in this case it is necessary to follow every branch. This circumstance guides us to a situation which is essentially more complicated: the measure of the set of all these dead-end points does not necessarily vanish, so the following additional condition appears: the measure of {the set of all dead-end points} equals 0; this condition is not automatically fulfilled now. Let us clarify, for scalar first-order operators on a manifold without boundary, the link between the asymptotics (0.12) and periodic Hamiltonian trajectories. It is well known that singularities of the solutions of hyperbolic equations propagate along Hamiltonian trajectories. This fact guides us to the conclusion that the singular support of σ(t) is contained in the set of all the periods of the Hamiltonian trajectories including t = 0 (this is the so-called Poisson relation); in particular, t = 0 is an isolated point of the singular support (this fact remains true also in the much more general situation). Hence, if there is no periodic trajectory with the period not exceeding T , then we know that in the interval [−T , T ] the distribution σ(t) is singular only at 0 and hence we know σ(t) in this interval modulo a smooth function. The Tauberian theorem impliesthat the remainder in asymptotics (0.12) does not exceed C λd−1 T −1 + O λd−2 with a constant C , which does not depend on T ; however “O” here is not necessarily uniform with respect to T . Therefore, if t = 0 were the only period (surely I know that it is impossible!), then  we could choose T arbitrarily large and obtain the remainder estimate o λd−1 .

4. METHODS. II

XXIX

In the general (realistic) case one should consider a partition of unity by means of two pseudodifferential operators Qj (j = 1, 2) for every chosen T , such that the support of the first operator contains no periodic point with a period not exceeding T , and the measure of the support of the symbol of the second operator is less than ε with arbitrarily chosen ε > 0. Applying the Tauberian theorem to each term  (0.13) Nj (λ) = (Qj e)(x, x, λ) dx,  d−2  d−1 −1 we obtain the remainder estimate C λ T + O λ for j = 1 and   C ελd−1 + O λd−2 for j = 2. Therefore the total remainder estimate for   N(λ) = N1 (λ) + N2 (λ) is C λd−1 (ε + T −1 ) + O λd−2 with arbitrarily large T and arbitrarily small ε > 0; this estimate again implies (0.12). Moreover, under certain more restrictive conditions to the   Hamiltonian d−1 −1 flow, one can improve  d−1−δ  the remainder estimate in (0.12) up to O λ (log λ) or even O λ with a small exponent δ > 0. It appeared later that the presence of periodic trajectories and, moreover, the presence of eigenvalue clusters allow two-term asymptotics of the form   (0.14) N(λ) = κ0 λd + f (λ)λd−1 + o λd−1 with an explicitly calculable function f (λ) which is bounded and oscillating as λ → +∞ with the characteristic “period” of oscillations  1. In particular, this fact permits us to obtain the asymptotic distribution of eigenvalues inside clusters. In addition, under some assumptions including the assumption that all the trajectories are periodic, one can obtain the asymptotics (0.14)  with the remainder estimate O λd−2 !

4. Methods. II The theory presented above has major exceptions. First, it could happen that in some part of our domain we should not apply the semiclassical spectral asymptotics (rescaled or not) but instead we simply need to estimate the main part of the asymptotics using methods different from those explained (usually those are variational methods). So, our domain is split into two parts: a semiclassical zone (aka a regular zone) and a singular zone; this partition depends on the parameters. F. e., if we consider the Schr¨odinger operator h2 Δ − |x|−1 , then the semiclassical zone is {x : |x|  h−2 } and the

XXX

INTRODUCTION

singular zone is {x : |x|  h−2 }. Definitely joining results in these two zones requires some work. Second, it may happen that (in some part of the domain) we need to split variables x and ξ to two parts: x = (x  ; x  ), ξ = (ξ  ; ξ  ), and only consider (x  , ξ  ) as Weyl variables, which means that one should consider the operator in question as a (partial) differential operator with respect to x  with operator-valued coefficients (acting in some auxiliary Hilbert space) and apply the Weyl procedure to this operator. F.e., if we consider Laplacian in the domain has a cusp: {x = (x  , x  ) : x  ∈ f (x  )Ω}, where Ω  is a bounded domain in Rd with a smooth boundary, and f (x  ) decays as |x  | → ∞, then for |x  | ≥ g (λ) we can consider x  as Weyl variables and an auxiliary space is K = L2 (Ω). In the more general case one should divide the phase space into a few parts, one of which should be removed from consideration altogether while in the other parts the “Weylization” (preceded by a certain transform) should be made only with respect to certain variables. The previous remarks are not a survey, even an incomplete one. Their purpose is only to motivate this book in general and the following part of this introduction in particular. Nor are the comments in each chapter complete. As the best surveys I recommend the books of M. S. Birman and M. Z. Solomyak [5] and G. Rozenblioum, M. Z. Solomyak and M. Shubin [1], despite being outdated. It also should be noted that though this book demands from the reader only a good and detailed knowledge of the theory of pseudodifferential operators and a familiarity with the theory of Fourier integral operators (see, e.g., the books of L. H¨ormander [1], F. Treves [1], M. Taylor [7] and M. Shubin [3]); an elementary familiarity with the theory of spectral asymptotics (see the same books) is also very desirable. As an elementary introduction to Semiclassical analysis I recommend M. Zworski [1], A. Martinez [1], B. Helffer [2]. It should be noted that this book has few intersections with the books of S. Levendorskii [6] and Yu. Safarov and D. Vassiliev [3] which are also devoted to spectral asymptotics. I recommend both of those books, especially the second one. Finally, I would like to mention that there are some problems in this book. There are problems the solutions of which are not known to me. There are problems the solutions of which are known to me but which cannot be published here or elsewhere by me because of lack of time due to my

5. WHAT IS IN THE BOOK

XXXI

efforts on this book. I welcome readers to solve these problems either by the method suggested or by some other method and to publish their solutions. Finally, there are problems I consider rather simple exercises. The structure of this book and the numbering of statements and formulas is clear from the table of contents and from the Section below.

5. What is in the Book Let us now discuss the contents and principal ideas of this book. It consists of five volumes: Volume I covers the general microlocal and local theory. Volume II covers the general functional-analytic theory which in combination with the microlocal theory of Volume I leads us to spectral asymptotics in domains with singularities. Then we apply these results to more specific problems. We also consider here some problems not covered by the general theory. Volume III is devoted to Schr¨odinger and Dirac operators with the strong magnetic field. Volume IV is devoted to Schr¨odinger and Dirac operators with the strong magnetic field. Volume V is devoted to the asymptotics of the ground state energy of heavy atoms and molecules and related topics. It also contains some latest articles, which are not included into the main body of the book. Let us discuss in details the current volume, consisting of two parts and eight chapters. Part I. Semiclassical Microlocal Analysis There are no spectral asymptotics yet in Part I, “Semiclassical Microlocal Analysis”; we develop the apparatus (both basic notions and technique) of the semiclassical microlocal analysis here.

INTRODUCTION

XXXII

Chapter 1. Introduction to Microlocal Analysis. This chapter contains the necessary survey of the theory of the h-pseudodifferential operators (in Section 1.1), the functional calculus of these operators (in Section 1.4), the theory of the h-Fourier integral operators (in Section 1.2) and the theory of singularities (wave front sets and related topics, in Section 1.3). There are no new non-trivial results in this chapter but there is a rather obvious but very important new Definition 0.1. A family of functions u = uh , depending on a small parameter h ∈ (0, h0 ] (and maybe depending on some other parameters of an arbitrary nature), tempered (i.e., with u ≤ Ch−n with some constant C and exponent n where here and below . means the L2 -norm) is negligible in the box Π = {(x, ξ) : |xj − x¯j | ≤ γj , |ξj − ξ¯j | ≤ ρj ∀j = 1, ... , d}, if there exists a function φ ∈ C0∞ (R2d ), equal to 1 in the unit cube {(x, ξ) : |xj | ≤ 1, |ξj | ≤ 1 ∀j = 1, ... , d} and such that φwΠ (x, hD)u ≤ Chs , ¯ note that the exponent s enters where φΠ (x, ξ) = φ(γ −1 (x − x¯), ρ−1 (ξ − ξ)); into the definition of negligibility and here and below we use a notation γ −1 x = (γ1−1 x1 , ... , γd−1 xd ) etc. In the standard definition, leading to the notion of an wave front set, one ¯ and (γ1 , ... , γd ; ρ1 , ... , ρd )); this takes a fixed box Π (i.e., with constant (¯ x , ξ) definition is too crude for our needs and we admit variable (i.e., depending ¯ and (γ1 , ... , γd ; ρ1 , ... , ρd ), satisfying the microlocal on parameters) (¯ x , ξ) uncertainty principle (0.15)

γj ρj ≥ h1−δ

∀j = 1, ... , d

with an arbitrarily small exponent δ > 0. An important special case is 1 γ1 = ... = ρd ≥ Ch 2 (1−δ) 3) 3)

We can weaken (0.15) to the logarithmic uncertainty principle

(0.16)

γj ρj ≥ Cs h| log h|

∀j = 1, ... , d 1

with important special case γ1 = ... = ρd ≥ C (h| log h|) 2 albeit not to the standard uncertainty principle (0.17)

γj ρj ≥ h

∀j = 1, ... , d.

5. WHAT IS IN THE BOOK

XXXIII

Chapter 2. Propagation of Singularities in the Interior of the Domain. This chapter is devoted to the propagation of singularities (or, more precisely, of the negligibility) of solutions in the interior of the domain. The key definition of the microhyperbolicity of an operator P at the given point z of the cotangent bundle in the direction T ∈ Tz T ∗ X appears in Section 2.1, and the main theorem concerning propagation of singularities is proven there; the theorem is formulated in terms of real-valued functions φk , such that the operator P under consideration is microhyperbolic at every point z ∈ Ω in the directions of the Hamiltonian fields ∇# φk , generated by φk ; here and in what follows Ω ⊂ T ∗ X is a phase domain under consideration. This theorem is proven by means of the energy estimates. We then apply this theorem to the operator P = hDt − Aw (x, hDx , h) and to the analysis of the fundamental solution u(x, y , t) of the Cauchy problem for this operator, establishing the finite speed of propagation of singularities (in the domain Ω ⊂ T ∗ X ). In particular, under the microhyperbolicity condition for the operator A on the energy level λ we prove that the singularities of u(x, y , t) leave the diagonal {x = y , ξ = η}; this statement yields the negligibility of  (0.18) σQ (t) = (Qx u)(x, x, t)dx in boxes of the form {(t, τ ) : Ch1−δ ≤ ±t ≤ T0 , |τ − λ| ≤ 0 } with small enough positive constants T0 and 0 and with arbitrarily small exponent δ > 0; here and in what follows Q is a cutting h-pseudodifferential operator supported in Ω. The results of Section 2.1 are reformulated in Section 2.2 in the terms of the generalized bicharacteristics of P, which, in the case of constant multiplicities of the eigenvalues αj (x, ξ) of the symbol a(x, ξ) = A(x, ξ, 0), are the standard bicharacteristics of the symbols τ − αj (x, ξ). Also results of Section 2.1 are extended in Section 2.3 to operators with rough symbols. In Section 2.4 the classical theorem on the propagation of singularities along bicharacteristics for operators with constant multiplicities of the characteristics is proven in a more general and refined form. Namely, for the operator P = hDt − Aw (x, hDx , h) with self-adjoint operator Aw (x, hDx , h) and with constant multiplicities of the eigenvalues αj (x, ξ) of the symbol a(x, ξ) we replace the usually used negligibility in a small fixed neighborhood by negligibility in the γ-neighborhood of a Hamiltonian trajectory with

XXXIV

INTRODUCTION 

γ ≥ h 2 (1−δ) ; it is assumed that this trajectory is not too long (i.e., T ≤ ch−δ with a small enough exponent δ  > 0); the symbol A(x, hDx , h) and its derivatives up to large enough order K and the elements of the Jacobi matrix of the corresponding Hamiltonian flow are also assumed to be not  too large (i.e., their absolute values or norms are also less than ch−δ ). The proof is based on the Heisenberg equation. In what follows we sometimes return to the investigation of the propagation of singularities. 1

Chapter 3. Propagation of Singularities near the Boundary. Here we realize approximately the same program for operators near the boundary on which boundary conditions are given. In Section 3.1 we give the key definition of the microhyperbolicity of the problem (P, B) at the point z ∈ T ∗ ∂X in the multidirection T = (T  , ζ1 , ... , ζn ) ∈ Tz T ∗ ∂X × Rn , where n is the number of the different characteristic points zk of the operator P, lying over z. Then we prove the main theorem concerning the propagation of singularities; this theorem is formulated in terms of real-valued functions φk (x, ξ  ), coinciding at {x1 = 0} and such that the problem (P, B) is microhyperbolic in the domain Ω ⊂ T ∗ ∂X under consideration in the multidirection (∇# φ, ∂x1 φ1 , ... , ∂x1 φn ), where we assume that X = {x1 = 0} locally and x  = (x2 , ... , xd ), φ = φk (0, x, ξ  ). Then in Section 3.2 we discuss some geometrical aspects of the propagation of singularities; however the general results are not very precise. Next, in Section 3.3 we deduce the same corollaries as in Section 2.1 (concerning the finite speed of the propagation of singularities for the operator P = hDt − Aw (x, hDx , h) and the negligibility of σQ (t) in the same boxes as before provided (A, B) is microhyperbolic at the level λ) from this theorem. Further, in Section 3.4 we develop a more refined energy method and prove that there is no propagation along creeping rays. Finally, Section 3.5 is devoted to the propagation of singularities along “long trajectories”; here we consider only h-differential second-order operators and assume that at the point where a trajectory meets the boundary and reflects the trajectory is transversal enough to the boundary (with angle  greater than hδ ); moreover, we assume that a certain non-degeneracy condition on the boundary operator is also fulfilled at such reflection points. It should be noted that the number of reflection points is not too large (less

5. WHAT IS IN THE BOOK

XXXV



than h−δ with exponent δ  = δ  (δ  ) > 0, δ  (δ  ) → +0 as δ  → +0). Part II. Local and Microlocal Semiclassical Spectral Asymptotic in the Interior of the Domain The main goal of Part II is to obtain asymptotics as h → +0 for  EQ (λ1 , λ2 ) = (Qx e)(x, x, λ1 , λ2 ) dx (0.19) with (0.20)

λ=λ e(x, y , λ1 , λ2 ) = e(x, y , λ)λ=λ21 ,

where e(x, y , λ) is a Schwartz kernel of E (λ) (which is a spectral projector of A) and either Q = Q w (x, hDx ) is an h-pseudodifferential operator with compactly supported symbol or Q = ψ(x) is a compactly supported function; we speak about microlocal spectral asymptotics and local spectral asymptotics respectively. Here and below the subscript “x” means that the corresponding operator acts with respect to x, etc. We also consider the same expressions mollified with respect to the spectral parameter. Chapter 4. General Theory in the Interior of the Domain. Here we consider the case when supp(Q) is disjoint from the boundary of the domain X . Section 4.2 contains a preliminary analysis and the results obtained there are rather trivial. First of all, microlocal estimates EQ (λ1 , λ2 ) ≤ Ch−d are derived in the general case and estimates EQ (λ1 , λ2 ) ≤ Chs are derived in the case when the interval (λ1 , λ2 ) is classically forbidden in a neighborhood of supp(Q) (i.e., the interval is disjoint from the spectrum of the symbol A(x, ξ, h) for the indicated (x, ξ)); one can weaken this condition provided the multiplicities of the eigenvalues of the symbol a(x, ξ) are constant. We obtain similar local estimates for operators elliptic in the standard sense. Then in Section 4.3 we apply the method of successive approximations to u(x, y , t); we take an unperturbed operator A¯ = A(y , hDx , 0) with the symbol depending only on ξ. This method turns out to be applicable in the 1 rather short time interval {|t| ≤ h 2 +δ } with arbitrary small exponent δ > 0. This immediately yields in Section 4.4 complete semiclassical asymptotics for the spectral means   1 ¯ dλ EQ (λ1 , λ) (0.21) φ (λ − λ) L

INTRODUCTION

XXXVI

with any L ≥ h 2 −δ and with φ ∈ C0K (R); further, one can take L ≥ h1−δ provided the eigenvalues of a(x, ξ) have constant multiplicities. This yields complete asymptotics for EQ (λ1 , λ2 ) in the case when both levels λ1 and λ2 are classically forbidden but the whole interval (λ1 , λ2 ) is not; in all these ¯ is cases the leading term is κ0 h−d . Then we consider the main case when λ not a classically forbidden level and A is microhyperbolic at this level. Combining the results of Section 4.3 with the statement from Section 2.1 concerning the complete asymptotics of σQ (t)  negligibility of σQ (t), we obtain ¯ ≤ 0 with small positive constants T0 in the zone (t, τ ) : |t| ≤ T0 , |τ − λ| ¯ ≤ 0 we get asymptotics in the interval [−h 12 +δ , h 12 +δ ]. It and 0 . For |τ − λ| was proven in Section 2.1 that σ(t) is negligible in [−T0 , −h1−δ ] ∪ [h1−δ , T0 ], and these intervals overlap! We immediately get asymptotics of (0.21) for L ≥ h1−δ . Moreover, the Tauberian method yields the estimate 1

(0.22)

||EQ (λ1 , λ2 ) − κ0Q (λ1 , λ2 )h−d || ≤ Ch1−d

¯ − 1 0 , λ ¯ + 1 0 ) and miscellaneous related asymptotics in which for λ1 , λ2 ∈ (λ 2 2 the possible smallness of supp(Q) is taken into account. The same method also yields asymptotics of the Riesz means of order ϑ > 0 with remainder estimate Ch1−d+ϑ . However, all these asymptotics are due to the simple Tauberian estimate (0.23)

||EQ (λ1 , λ2 )|| ≤ Ch1−d

¯ − 0 , λ ¯ + 0 ] : |λ1 − λ2 | ≤ h ∀λ1 , λ2 ∈ [λ  s and asymptotics of (0.21) with L ≥ h and remainder estimate Ch1−d h/L with an arbitrarily large exponent s. Actually estimates of this type seem to be the most important for our internal needs. In Section 4.5 we derive sharper remainder estimates under assumptions of the global nature. Applying the results of Section 2.3 under a condition forbidding periodic trajectories of length not exceeding T , we obtain complete asymptotics of σQ (t) in the zone indicated above with T0 replaced by T ; then the same arguments yield the estimate (0.24) ||EQ (λ1 , λ2 ) − κ0Q (λ1 , λ2 )h−d − κ1Q (λ1 , λ2 )h1−d || ≤ C εh1−d + Ch1−d+δ



with  = T −1 and small enough exponent δ  > 0; under some more general and realistic conditions we obtain an estimate with a more complicated

5. WHAT IS IN THE BOOK

XXXVII

expression for ε. Those adjustments are also achieved for Riesz means (the first term in the remainder estimate is Ch1−d+ϑ T −1−ϑ ) and for expression (0.21) with L ≥ hT −1 (the first term of the remainder estimate is s Ch1−d T −1 h/LT ). Furthermore, in Section 4.6 we generalize these results to operators with rough coefficients. Then for operators with irregular coefficients we construct two operators with with rough coefficients, bracketing them, and close enough to provide the same principal parts of the asymptotics. Considering a(x, ξ) as the symbol of an m-th order differential operator and assuming that it is elliptic in the standard sense, we switch from microlocal to local asymptotics. It allows us in certain cases to consider operators with non-smooth ( Cl ) coefficients by the bracketing method. Finally, in Section 4.7 we consider operators with non-smooth coefficients directly by the perturbation method. We discuss Chapters 4–6 so comprehensively because its results and methods are highly important for the parts of the book which follow. The scheme of the arguments in the main stream of these chapters is the following: Construction (by means of the method of successive approximations) of u(x, y , t) and hence σQ (t) in the short time-interval {|t| ≤ T1 }—step-by-step extension of this interval by different methods—application of the Tauberian method; in what follows this scheme is used repeatedly. An important general idea is that In the theory of spectral asymptotics the step-by-step approach with improvement is much better than more bold attempts to do everything in one shot. It should be mentioned that we do not use the results of Chapter 3 in Chapters 4–6 and that the results of Sections 4.2–4.4 are obtained for standard operators as well as for operators with operator-valued symbols; however, the results for operators with operator-valued symbols are applied only in Chapter 12 (except Section 12.6).

XXXVIII

INTRODUCTION

Chapter 5. Scalar Operators in the Interior of the Domain. Rescaling Technique. Here we consider scalar operators (and sometimes operators with the similar properties). In Section 5.2 we analyze operators with constant multiplicities of the eigenvalues of the principal symbol. By means of an admissible partition of unity and rescaling with respect to x with a change of h on every element of this partition we replace the microhyperbolicity condition by another much less restrictive condition (which is void for Riesz means). The method applied here is very important and will be used very often in this book. We call it the rescaling method or partition-rescaling method in what follows. To get the best possible results one should combine this method with a refined analysis of the propagation of singularities along trajectories which are long after rescaling but are usually of bounded length or even short before: time and energy are rescaled automatically. Section 5.3 may be summarized simply: for the Schr¨odinger, Schr¨odingerPauli and Dirac operators in dimension d ≥ 2 one can simply drop the microhyperbolicity condition. Here we apply the same rescaling method as before with some simplifications and modifications. For these operators we also consider pointwise asymptotics, i.e. asymptotics of e(x, x; λ1 , λ2 ) or (Qx,y e)(x, x; λ1 , λ2 ) without mollification with respect to the spatial variables. In this case not only periodic trajectories but also loops play important role: namely, trajectory returns to the same point but from the different direction. In Section 5.4 we generalize previous results to operators with rough and irregular symbols. Chapter 6. Operators in the Interior of Domain. Esoteric Theory. In this chapter we consider more special questions. First, in Section 6.1 we extend the above results to the families of commuting operators. Further, Sections 6.2 and 6.3 have a more special character. Here we consider scalar operators with periodic Hamiltonian flow of the principal symbol, or some weak perturbation of an operator of this type; there automatically appears an additional parameter η ∈ [hn , hδ ] with arbitrarily large n and arbitrarily small δ > 0 (this parameter equals h in the “normal” situation); for η ≤ h with a small constant  > 0 there arise spectral zones of length ≤ C η separated by lacunary zones of length  h in the semiclassical

5. WHAT IS IN THE BOOK

XXXIX

approximation to the spectrum; it is proven that EQ (λ1 , λ2 ) = O(hs ) with arbitrarily large s provided the interval (λ1 , λ2 ) lies in a lacunary zone; in the case when λ1 and λ2 belong to different lacunary zones we obtain complete non-Weyl asymptotics for EQ (λ1 , λ2 ). Turning to the long-time propagation of singularities (with |t| ≤ η −1 ) we prove that singularities propagate along Hamiltonian trajectories of the symbol a(x, ξ), drifting in the direction of the Hamiltonian field generated by an auxiliary symbol b(x, ξ) with drift speed  η. Now turning to EQ (λ1 , λ2 ) we finally derive, under certain conditions, the estimate   (0.25) EQ (λ1 , λ2 ) − κ0Q (λ1 , λ2 )h−d − κ1Q (λ1 , λ2 ) − fQ (λ1 , λ2 )h1−d  ≤ C (η + h)h1−d , where the quickly oscillating function f is explicitly calculated. It arises from singularities of σQ (t) different from t = 0. Nevertheless the last term in the asymptotics is of the order of the second one and its variation on the interval of length  h is of the same order  h1−d as the variation of the first term4) . For η ≥ h1−δ we obtain asymptotics (0.24) with ε = η under certain assumptions. In Section 6.4 we consider asymptotics of expression  (0.26) I := ω(x, y )|e(x, y , τ )|2 dxdy with ω(x, y ) := Ω(x, y ; x − y ), where function Ω(x, y , z) is smooth with respect to x and y and smooth positively homogeneous of degree κ with respect to z =  0. This expression generalize the Dirac energy used in Chapters 25–28 (in which case ω(x, y ) = |x − y |−1 ). Finally, in Section 6.5 we consider improved asymptotics with mollifications with respect to some parameter, which is different from a spectral parameter. Part II. Local and Microlocal Semiclassical Spectral Asymptotic near the Boundary Chapter 7. Standard Local Semiclassical Spectral Asymptotics near the Boundary. This chapter is essentially parallel to Chapter 4: 4)

As it should be because the decrease of the last term along the spectral gap should compensate for the increase of the first term; along the cluster both terms grow and if η  h then the last term grows more quickly.

XL

INTRODUCTION

the content of each of Sections 7.1–7.5 is very similar to the content of its counterpart from 4.2–4.6. We underline only the following “shortcomings”: (a) We now consider only standard operators, not operators with operatorvalued symbols. (b) We do not weaken the microhyperbolicity condition for operators with constant multiplicities of the eigenvalues of the principal symbol; we do it only for the Schr¨odinger operator in Section 7.4. (c) In Section 7.4 boundary is also irregular and we cannot smooth it up by a change of coordinates because then coefficients become too irregular. Instead we use (generalized) Seeley’s method. It also should be noted that if the interval [λ1 , λ2 ] is classically forbidden for the symbol a(x,  ξ) but is not forbidden  for the 1D-problem (on the semiaxis R+  t) a(x  , Dt , ξ  ); b(x  , Dt , ξ  ) (as above we assume that X = {x1 > 0} locally), then in the above estimates and asymptotics one should replace d by (d −1); sometimes we are even able to get two-term asymptotics. Chapter 8. Standard Local Semiclassical Spectral Asymptotics near the Boundary. Miscellaneous. This chapter contains results similar to those of Chapters 5 and 6. In Section 8.1 we consider pointwise asymptotics (i.e. asymptotics of e(x, x; λ1 , λ2 )) near the boundary. Because there are short loops (when a trajectory, reflected from the boundary, goes in the exactly opposite direction and returns to the same point x) such asymptotics with a decent remainder estimate has an additional term of the boundary layer type of the width h, i.e. the term of the magnitude h−d if dist(x, ∂X )  d and decaying away from ∂X . In Section 8.2 we consider scalar operators and develop a rescaling technique for them. In Section 8.3 we consider operators with periodic Hamiltonian flows but with a twist. First, as we know from Section 6.2 a quantum periodicity may be violated by the phase shift; now however there may be an additional phase shift at the reflections from the boundary. Second, in the case of branching billiards the periodicity could be partial: only some branches return to the original point (x, ξ) while other branches do not, so there is an energy decay along periods.

6. HOW LSSA ARE APPLIED TO CLASSICAL ASYMPTOTICS

XLI

Finally, in Section 8.4 we consider spectral asymptotics on subspaces (when the main Hilbert space is not L2 (X , CD ) but it subspace which is defined by underdetermined elliptical boundary value problem (f.e. a space of solenoid vector fields)). Such generalization in the case without boundary is rather trivial.

6. How LSSA are Applied to Classical Asymptotics Finally, let us discuss three examples which illustrate Chapter 11 and show how LSSA yield asymptotics with respect to the spectral parameter. This basically explains Example 0.2. Let X be a compact Riemannian manifold without boundary and let Δ be a (positive) Laplace-Beltrami operator. Let us consider the Schr¨odinger operator Ah = h2 Δ + V (x); it is well known that   (0.27) N− (Ah ) = κ0 h−d + O h1−d with Weyl constant κ0 provided 0 is not a critical value of V (x) 5) . The Birman-Schwinger principle implies that N− (Ah ) = N(h−2) where N(λ) is the number of eigenvalues μ ∈ [0, λ) of the spectral problem Δ + μV (x) u = 0 (counting their eigenvalues). These two equalities immediately yield the asymptotics (0.28)

 1  d N(λ) = κ0 λ 2 + O λ 2 (d−1)

provided 0 is not a critical value of V (x) (the condition which was claimed not to be necessary). Try to obtain this result directly in the case when V (x) vanishes at some point! It is striking that nobody had observed this non-trivial new result which trivially follows from two well known facts. Example 0.3. Let us consider the Schr¨odinger operator A = Δ + V (x) in Rd where Δ is the (positive) Laplacian and the real-valued potential V (x) 5) It is shown in this book that one can skip this condition for d ≥ 2 and for manifolds with boundary.

INTRODUCTION

XLII satisfies the following conditions:  α  D V  ≤ x2m−α (0.29) and V ≥ 0 |x|2m (0.30)

∀α : |α| ≤ K , ∀x : |x| ≥ c,

 1 where x = |x|2 + 1 2 here and in what follows. Let us take a γ-admissible 1 partition  of unity with γ(x) = 4 x. If we make−1the rescaling transforming B x¯, γ(¯ x ) to B(0, 1) and multiply A − λ by λ , we obtain a Schr¨odinger 1 operator for |x| ≤ C λ 2m in B(0, 1) with the standard restrictions to the 1 potential and with h = γ(¯ x )−1 λ− 2 . Let us apply corresponding results of Chapter 4 6) . Then we obtain Weyl asymptotics for spatial means of e(x, x, λ) in B x¯, γ(¯ x ) with the  1−d   1 (d−1)  d−1 2 remainder estimate O h =O λ γ(¯ x) ; there is no additional factor because e(x, x, λ) is a density rather than a function. Summing 1/2m with respect to the partition }, we obtain the  (d−1)l of unity in {x : |x| ≤ C λ remainder estimate O λ with l = (1 + m)/2m.   1 On the other hand, let us consider the ball B x¯, γ(¯ x ) with |x| ≥ C λ 2m . After rescaling and multiplication of A − λ by ρ−2 (¯ x ) with ρ(¯ x ) = xm we obtain a Schr¨odinger operator with the standard restrictions to the potential v (x) ≥  and with h = γ(¯ x )−1 ρ(¯ x )−1 ; then we derive the estimate s −d |e(x, x, λ)| ≤ Ch γ(¯ x ) with an arbitrary exponent s and hence the con1 tribution of the domain {x : |x| ≥ C λ 2m } to the remainder is O(λ−s ) with another, also arbitrarily large, exponent s. Combining with the contribution of the previous zone, we obtain the asymptotics   N(λ) = N(λ) + O λ(d−1)l (0.31) with   d N(λ) = κ0 λ − V (x) +2 dx (0.32) as λ → +∞ and N(λ)  λdl . Example 0.4. Let us consider the Schr¨odinger operator in Rd with potential V (x) satisfying (0.29) with m ∈ (−1, 0) and now let λ < 0 be asmall param eter. Then for the same ρ and γ as before the rescaling of B x¯, γ(¯ x ) into 6) In which case we are able to skip assumption that V is non-degenerate, i.e., |∇V | ≥ 0 |x|2m−1 for |x| ≥ c.

7. PATHS

XLIII

x )−2 produce a Schr¨odinger operator B(0, 1) and multiplication of A − λ by ρ(¯ with standard restrictions to the potential and with h = γ(¯ x )−1 x )−1pro ρ(¯ 1 vided |x| ≤ C |λ| 2m . Hence for the spatial mean of e(x, x, λ)in B x¯, γ(¯ x ) the  Weyl formula holds with remainder estimate O h1−d = O ρ(¯ x )d−1 γ(¯ x )d−1 . 1 On the other hand, for |x| ≥ C |λ| 2m rescaling and multiplication by |λ|−1 produce a Schr¨odinger operator with standard restrictions to the potential 1 v (x) ≥  and with h = γ(¯ x )−1 |λ|− 2 . Hence the estimate |e(x, x, λ)| ≤ Chs γ(¯ x )−d again holds. Summing with respect to a partition of unity we obtain asymptotics (0.31)–(0.32) as λ → −0 with the same l = (1 + m)/2m as in Example 0.3; moreover N(λ)  |λ|dl provided V ≤ −|x|2m

(0.33)

for x : |x| ≥ c

in some non-empty open cone in Rd . V (x)

|x| ≤ C0 |λ|l

|x| ≥ C0 |λ|l x

V (x) λ

λ

x |x| ≤ C0 λ

l

|x| ≥ C0 λl

(a) Illustration to Example 0.3

(b) Illustration to Example 0.4

7. Paths For those interested in the basics of the theory I recommend the following sequence of chapters: 1 −→ 2 −→ 4 −→ 11 and then 1 −→ 2 −→ 4 −→ 9 −→ 11; in addition, I suggest considering only the Schr¨odinger operator in Chapter 11 and only the standard (not the operator-valued) case in Chapters 2 and 4.

INTRODUCTION

XLIV

Those interested in domains with boundaries (still on a rather elementary level) should read also Chapter 3 and then Chapter 7. Those interested in magnetic field theory should start reading Chapters 13 (and maybe 17) and 23 (but Chapters 3 and 7 are not necessary for this). Those interested in the results of Chapter 12 should consider references to every its section. Those interested in the results of other chapters should look at the map and read the references to each chapter they are interested in.

8. About the Author Elements of Biography Elements of biography: http://weyl.math.toronto.edu/victor/victor ivrii-bio.html , http://weyl.math.toronto.edu/victor/victor ivrii-vita.pdf and http://en.wikipedia.org/wiki/Victor Ivrii (the latter is authored by someone and, while it looks accurate, it may contain some errors).

My Way to Spectral Asymptotics and Around Weyl Conjecture (66–67 AW) Background. It was Winter ’78-’79 when Michael Shubin and B. M. Levitan suggested me to prove Weyl conjecture. R. Seeley just proved remainder estimate O(λ(d−1)/2 ). He considered (as anyone would at this time) 1 σ(t) = Tr(cos(tΔ 2 )) and respectively u(x, y , t) was the Schwartz kernel of 1 cos(tΔ 2 ), satisfying initial-boundary value problem (0.34) (0.35) (0.36)

(Dt2 − Δx )u = 0, u|t=0 = δ(x − y ), u|x∈∂X = 0,

Dt u|t=0 = 0,

where for simplicity we discuss the Dirichlet Laplacian; the Neumann Laplacian is considered in the same way. So it was the wave equation rather than the Schr¨odinger equation (which came later, with semiclassics).

8. ABOUT THE AUTHOR

XLV

I did not messed up with spectral asymptotics before but I was an established expert in the propagation of singularities, and here I used the energy estimates which is much more robust and general method than the explicit construction of the parametrix. My idea was to invent a new approach because (as I thought) if Seeley’s method worked for Weyl conjecture then Seeley would prove it! I thought wrong! Couple of years later D. Vassiliev who was a secret physicist at that time gave the proof of Weyl conjecture using Seeley’s method. And much later I combined Seeley’s approach with my own. But it was the best mistake I ever made! Because the method I invented worked in many situations Seeley’ method did not, f.e. for general systems. Normal Singularity. First, I conjectured that the singularity of σ(t) at 0 is normal i.e. (tDt )n σ(t) at 0 have the same order of singularity for any n exactly as for manifolds without boundary. I proved this conjecture by implicit method of propagation of singularities in Spring ’79. At this moment I had a powerful method of the energy estimates with the theorem stated in the form: If Ω is a domain in the phase space and φ satisfies microhyperbolicity condition and Pu is smooth in Ω ∩ {φ < 0}, and u is smooth in ∂Ω ∩ {φ < 0}, then u is smooth in Ω ∩ {φ < 0}. Sure, it looks similar to Holmgren uniqueness theorem! Using this technique plus rescaling arguments I proved that singularity of σ(t) at t = 0 is normal. Successive Approximations. My next idea was rather crazy: to calculate u(x, y , t), using successive approximations, first going to the coordinate system where boundary was planar and getting operator with variable coefficients even if it originally was not, then to the problem (0.34)–(0.35) and apply successive approximation method, freezing coefficients of Δ at point y . This looks insane because such perturbation decreases smoothness by 2 while parametrix to the problem (Dt2 − Δ)u = f , u|t=0 = ut |t=0 = 0, u|∂X = 0

XLVI

INTRODUCTION

increases it only by 1, so each next term in this approximation approach is more singular than the previous one! However, the perturbation contains factors (xj − yj ) which are of magnitude t due to the finite propagation speed and each parametrix contains factor t due to the Duhamel integral, so in fact, each next term in successive approximations acquires an extra factor t 2 Dt . Big deal, so what? But then the same is true for σ(t) as well, but for σ(t) I knew already that near 0 each multiplication by t compensates one differentiation and this allowed me to justify the successive approximation method for σ(t) near 0 without justification it for u. So, a complete asymptotics (with respect to smoothness) of σ(t) near 0 was done. This would imply Seeley’s result. Other Singularities. It was early August ’79 and I wrote to Michael Shubin about my progress. His answer came two weeks later (no email at that time!): “So what? Other singularities (at t = 0) are much more difficult!” But at this moment I already had a solution! Actually other singularities were easier. I have proven that if the set of periodic geodesic billiards has measure 0 then two-term (Weyl) asymptotics holds. To tackle “other” singularities of σ(t) I analyzed Duistermaat-Guillemin method, purging all irrelevant FIO stuff. As a result arguments became very simple: Fix arbitrary T > 0. A set ΠT of geodesic billiards, periodic with periods ≤ T , is a closed set. Then due to our assumption it is a closed nowhere dense set of measure 0. And the set ΛT of all dead-end billiards of the length ≤ T is also of this type. Recall that dead-end billiards are those which become tangent to the boundary or behaving badly. Let I = Q1 + Q2 , where Q1 is a pdo with symbol in ε-vicinity of ΠT ∪ ΛT and the small vicinity of the boundary and Q2 has a symbol vanishing in the 1 vicinity of this set. Then σ(t) = σQ1 (t)+σQ2 (t) with σQ (t) = Tr(cos(tΔ 2 )Q). Here σQ2 (t) has no “other” singularities on [−T , T ] and Tauberian methods let me to recover asymptotics of Tr(E (λ)Q2 ) with the remainder d−1 estimate CT −1 λ 2 + CT with C which does not depend on T or ε. On the other hand, I recovered asymptotics of Tr(E (λ)Q1 ) with the d−1 remainder estimate C ελ 2 + CT ,ε where ε = mes(supp(Q1 )) is arbitrarily small. So, I recovered asymptotics of N(λ) = Tr(E (λ)) with the remainder

8. ABOUT THE AUTHOR

XLVII

estimate

 1  d−1 λ 2 + CT ,ε C ε+ T and the rest was a second-year calculus exercise! Aftermath (68–79 AW) Using this method I instantly proved asymptotics with the remainder estid−1 mate O(λ m ) for m-th order elliptic systems on manifolds without boundaries, and later on manifolds with the boundaries. When I was peacefully exploiting my method and harvesting results and even published a book two events happened: Corners, edges etc. I got from RZh Matematika (a Russian version of Math. Reviews, but better) a some weird paper (K. Otsuka [1]), deriving two-term Weyl asymptotics for the Euclidean Laplacian in polygons. The author was spending a lot of efforts to consider wave equation near vertices and I realized that this was a completely unnecessary job. So I decided to do a proper job and I with my student Svetlana Fedorova proved (’84) Weyl formula for Laplace-Beltrami operators in domains with edges, vertices, conical points, cuts etc. But what is more important: rescaling technique was invented! Going Semiclassic. I decided to go cheap and prove some semiclassic asymptotics. I said “cheap” because at this time I believed as many did that really great mathematicians like H. Weyl, R. Courant, L. H¨ormander, J. J. Duistermaat, V. Guillemin, R. Seeley and myself (smile!) study the classical N(λ) on compact manifolds while less great mathematicians study N(λ) for the Schr¨odinger operator with growing potential, semiclassics, etc. So I proved some semiclassical results and told M. Z. Solomyak about them. He asked: “Why you just do not deduct it from classical asymptotics by a cheap trick 7) ?” I tried to follow this advice with a rather surprising result. Semiclassical results derived this way were less general than I had already; however, working the opposite way I derived more general classical results than I had. It was an eye-opener: Semiclassical Asymptotics are the most important! 7)

It was a colloquial name for the Birman-Schwinger principle.

XLVIII

INTRODUCTION

So, I began to study semiclassical asymptotics as a prime target. And in this framework my method became not only more general, but also more natural and transparent, and it worked better with the rescaling technique. Going Ballistic. First, I discovered that the rescaling applied to the semiclassic produces a bunch of new results. In particular, it allowed to eliminate |V | + |∇V | = 0 (if d ≥ 2) as a precondition for the sharp semiclassical asymptotics for the Schr¨odinger operator, which using cheap trick got instantly new results for the classical asymptotics. Further, I studied degenerations and singularities of different kinds, horns and cusps, other degenerations and singularities. I derived classical asymptotics, asymptotics of eigenvalues for operators, generalizing the Schr¨odinger operator with the potential growing at infinity, and for operators, generalizing the Schr¨odinger operator with potential slowly decaying at infinity, and these operators had their degenerations and singularities too. It was a complete Results Explosion! And it was so damn easy! I was feeling like a prospector who found a place with native ores of gold lying just on the surface, ready to be picked. These results were subject of my ICM-1986 talk8) . But often remainder estimates were not as good as in the non-degenerate or non-singular cases and I felt that the reason was not my lack of skill but a more profound one. Going Deeper. I modified my method to treat operators with symbols which were operators in auxiliary spaces and treated only some variables as semiclassical variables and others as non-semiclassical variables, and I reexamined some singularities and degenerations and cusps and derived more sharp asymptotics than before. But these asymptotics either contained non-Weyl correction terms of the magnitude of the remainder estimates derived on the previous stage or even were non-Weyl in their main parts. I also considered the Schr¨odinger and Dirac operators with the strong magnetic field. This was longer, slower and much more difficult process than before and its results appeared in my research monograph of ’98. 8)

My ICM-1978 was devoted to propagation of singularities. By Soviet bureaucracy I was not allowed to go to ICMs, further, the permission to send a talk for ICM-1978 came too late and, while this talk was published, it was not read. 8 years later I was smarter–and sent the talk without any permission. L. H¨ormander read it. I visited Berkeley many times since and in 2014 I visited Helsinki the first time–36 years too late!

8. ABOUT THE AUTHOR

XLIX

Multiparticle Quantum Theory (79 AW–88 AW) In ’90 Michael Sigal suggested me to work together to justify the Scott correction term for or asymptotics of the ground state energy for molecules consisting of very large atoms (for atoms even more precise results were already justified by C. Fefferman and L. Seco, using harmonic analysis). My part was to derive sharp asymptotics for the trace of the Schr¨odinger operator with a Coulomb type singularity. My first attempt was a (partial) success: the remainder estimate was sufficiently sharp to justify the Scott correction, but not sharp enough to justify Dirac and Schwinger corrections or even to approach to them. A bit later I was able to find more subtle arguments and succeeded to derive estimates, sufficiently sharp to justify Dirac and Schwinger corrections. Further, I considered this problem in the presence a strong external magnetic field. New Dawn (88–93 AW) Next few years I have been working on sharp spectral asymptotics for operators with not very regular coefficients. The major tool is the same kind of the analysis as before, applied to the mollified operator. Mollification scale depends on h and equals to Ch| log h| in the simplest case. This is related to the Logarithmic Uncertainty Principle. These arguments were applied also to the Schr¨odinger operator with the strong magnetics field. Etc (93 AW–113 AW) I did many other things later (apart of improving my old results and writing this monograph, which I started in 96 AW). The whole Parts VI (except Chapter 13), VII and VIII, and also Chapters 27 and 28 are brand new. Further, many sections of the “old” chapters are brand new, so chapters became much larger and the material was rearranged. Aftermath (113 AW–present time) As I mentioned, I decided not to change the main body of the monograph, but to add new results as separate articles (actually, I even moved two results from the main body to Part XII Articles). See also the List of my presentations with links to them.

Part I Semiclassical Microlocal Analysis

Chapter 1 Introduction to Microlocal Analysis In this chapter we give a survey of the theory of h-pseudodifferential operators (in Section 1.1), h-Fourier integral operators (in Section 1.2), the notion of wave front sets and related topics (in Section 1.3). Proofs are mostly sketchy since all the results of this chapter are trivial consequences of the results proven in L. H¨ormander’s monograph [1] or they can be proven by trivial modifications of the arguments used in that book. Further, (in Section 1.4) we give a concise introduction to L. Boutet de Monvel calculus and its application to elliptic boundary value problems.

What is Microlocal Analysis? Microlocal Analysis is a theory of Pseudo-Differential Operators and Fourier Integral Operators (PDOs and FIOs). It draws on many rich traditions in mathematics including the links between analysis and geometry and has a powerful application to partial differential equations. For example, all the significant progress in linear PDEs for the last 50 years has been based on microlocal analysis and many important advances in non-linear PDEs over this period were also based on microlocal analysis. Pseudodifferential operators are generalization of (partial) differential operators (DOs) and also appear as parametrices (almost inverse) of elliptic differential operators (and parabolic differential operators unless we stick with “classical” pseudodifferential operators). 0-order PDOs are singular in-

© Springer Nature Switzerland AG 2019 V. Ivrii, Microlocal Analysis, Sharp Spectral Asymptotics and Applications I, https://doi.org/10.1007/978-3-030-30557-4_1

2

1.1. PSEUDODIFFERENTIAL OPERATORS

3

tegral operators. There is a very nice calculus of pseudodifferential operators which make them extremely useful tool. Fourier integral operators appear as propagators of hyperbolic and similar equations. While from the Quantum Mechanics point of view pseudodifferential operators are observables, Fourier integral operators are canonical transformations. Microlocal analysis comes in few flavors. One of the possible classifications is: semi-classical with explicit asymptotics with respect to a small parameter h (Plank constant or wavelength) and classical (standard or more general) where there is no explicit parameter and asymptotics is with respect to the smoothness (or something like this). However, there is still implicit small parameter and microlocal analysis is asymptotical by its nature. We will consider semi-classical pseudodifferential operators and Fourier integral operators: one can always get from there to standard pseudodifferential operators and Fourier integral operators but semi-classical theory has more applications. Less formal description of (semi-classical) microlocal analysis is “a semiclassical limit of quantum mechanics” or “high-frequency limit of electrodynamics” (etc). This is really difficult and exciting topic to understand how quantum mechanics yields classical mechanics as Plank constant h goes to 0 (in contrast to trivial “Special relativity yields Newton mechanics as light speed c → ∞”; the fundamental difference is that quantum and corresponding classical objects are mathematically completely different while special relativity and Newton mechanics objects are exactly the same). So, Microlocal means local both in coordinates and momenta. There are direct and deep connections with Global Analysis.

1.1 1.1.1

Pseudodifferential Operators Calculus of Pseudodifferential Operators

Officially Microlocal Analysis was born in 1965 after two articles of L. H¨ormander [2] and J. J. Kohn and L. Nirenberg [1] were published where pseudodifferential operators were introduced by formula  (1.1.1)

(Au)(x) =

a(x, ξ)e ix,ξ u(ξ) ˆ dξ

4

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

or by alternative formula (1.1.2)

(Au)(ξ) =



a(x, ξ)e −ix,ξ u(x) dx

where (1.1.3)

u(ξ) ˆ = Fx →ξ u = (2π)

−d



e −ix,ξ u(x) dx

is a Fourier transform. Formulae (1.1.2) are not exactly equivalent. Really, if  (1.1.1) and α a(x, ξ) = a (x)ξ is a polynomial then (1.1.1) defines a differen|α|≤m α  tial operator A(x, D) = |α|≤m aα (x)D α while (1.1.2) defines differential  operator A(x, D) = |α|≤m D α aα (x), D = −i∂x . In the first case we differentiate first and multiply by coefficients after while in the second case we reverse this order. Following terminology originated from physics we call (1.1.1) by qp-quantization and (1.1.2) by pq-quantization. In these articles symbol a(x, ξ) was supposed to be either positive homogeneous of degree m or the asymptotic sum of such (1.1.4)

a∼

∞

am−j (x, ξ)

j=0

with am−j positively homogeneous of degree m −j. Later more general classes were introduced. However one can notice that essentially the same operators were introduced by A. Calderon-A. Zygmund and S. G. Mihlin as singular integral operators. Also quantization procedure was introduced first by H. Weyl, who used yet another formula  x +y a( , ξ)e ix−y ,ξ u(y ) dydξ (1.1.5) (Au)(x) = (2π)−d 2 which is called symmetric or Weyl quantization. The procedure symbol → operator is called quantization. Even if there are different methods of quantization we get the same classes of pseudodifferential operators. So, each pseudodifferential operator has pq-, qp- and Weyl-symbols and the navigation between different symbols will be given.

1.1. PSEUDODIFFERENTIAL OPERATORS

5

Moving from differential operators to pseudodifferential we lose property supp(Au) ⊂ supp(u) but retain sing supp(Au) ⊂ sing supp(u). Microlocal Analysis of this type is not exact in the sense that we consider infinitely smooth functions and infinitely smoothing operators as negligible. Microlocal Analysis is asymptotic in the sense that we will consider (1.1.4)type decomposition of symbols and decomposition on functions which are more and more smooth. The most important properties of such calculus are: Remark 1.1.1. The most important properties of such calculus are: (i) Product of operators of orders m1 and m2 has degree m1 + m2 and its principal symbol is equal to the product of principal symbols. (ii) Commutator of operators of orders m1 and m2 has degree m1 + m2 − 1 and its principal symbol is equal to the Poisson bracket of principal symbols. (iii) Elliptic operator (i.e. operator with non-vanishing principal symbol) has parametrix (or almost inverse): AB ≡ BA ≡ I modulo infinitely smoothing operators. (iv) There is a functional calculus of 0-order pseudodifferential operators. This section is devoted to the calculus of pseudodifferential operators and to major inequalities. However we will deal mainly with h-pseudodifferential operators a(x, hD) with small parameter h → 0 (in this case homogeneity is not assumed). This makes asymptotic nature of Microlocal Analysis more explicit. The price to pay: we have not functions or operators but families of them depending on h. Temperate function or operator (or rather the family!) is with the norm O(h−l ) for some l. Negligible function or operator (or rather the family!) is with the norm O(hs ) for any s. h-Fourier Transform Important role in our constructions plays h-Fourier transform given by  −1 −d (1.1.6) Fh : u(x) → (Fh u)(ξ) = (2πh) e −ih x,ξ u(x) dx and its inverse is (1.1.7)

Fh−1 : v (ξ) → (Fh−1 v )(x) =

 e ih

−1 x,ξ

v (ξ) dξ.

6

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

We could put factor (2πh)− 2 in both expressions (instead of (2πh)−d and 1, to get unitary operators Fh and Fh∗ in L2 (Rd ). Physicists call u(x) a coordinate representation and Fh u a momentum representation because F(hDxj )u = ξj Fh , Dxj = −i∂xj and hDxj is momentum operator. In physics q and p is reserved for coordinates and momenta. −1 The only operators commuting with multiplication by e ih x,ξ for every ξ are operators of multiplication u(x) by a function ν(x). Operator norm of such operator ν(x) in L2 (Rd ) is ν∞ = ess sup |ν(x)|. Respectively, the only operators commuting with all shifts are multiplicators : d

(1.1.8)

μ(hD) = Fh−1 μ(ξ)Fh .

Operator norm of such operator μ(hD) in L2 (Rd ) is μ∞ = ess sup |μ(ξ)|. h-PDO Representation via h-Fourier Transform; Quantization of Different Flavors Now we will construct much broader algebra including both operators of multiplication and multiplicators (with admissible symbols a(x), b(ξ). The trouble is that these operators do not commute! So, how to quantize a(x, ξ) = ν(x)μ(ξ)? The first way is ν(x)μ(hD) which would be given by formula   2 1  −1 −d (1.1.9) a(x, hD)u (x) = (2πh) e ih x−y ,ξ a(x, ξ)u(y ) dydξ; we can do it in the reverse order ν(x)μ(hD):   1 2  −1 −d e ih x−y ,ξ a(y , ξ)u(y ) dydξ. (1.1.10) a(x, hD)u (x) = (2πh) One can see easily that  α (1.1.11) α (x)ξ formulae (1.1.9) and (1.1.10) define oper As a(x, ξ) =α α a ators α aα (x)(hD) and α (hD)α aα (x) respectively; (1.1.12) If a(x, hD) is defined by (1.1.9) or (1.1.10) then Fh a(x, hD)Fh−1 = a(−hDξ , ξ) where a(−hDξ , ξ) is defined by method (1.1.10) or (1.1.9) respectively.

1.1. PSEUDODIFFERENTIAL OPERATORS

7

Formulae (1.1.9), (1.1.10) are called qp- and pq-quantization respectively to reflect order in which x (aka q) and hDx (aka p) are applied. more symmetric? The first guess  Should we look for something  1 ν(x)μ(hD) + μ(hD)ν(x) is not that bad but 2 (1.1.13)





w

−d

 e ih

a (x, hD)u (x) = (2πh)

−1 x−y ,ξ

a(

x +y , ξ)u(y )dydξ 2

is way better. One can prove easily that (1.1.14) For a(x, ξ) =



aα (x)ξ α formula (1.1.13) gives h-DO;

aw (x, hD) =

(1.1.15)



2−|α|−|β|

α,β

(α + β)! (hD)α aα+β (x)(hD)β ; α!β!

(1.1.16) If a(x, hD) is defined by (1.1.13) then Fh a(x, hD)Fh−1 = a(−hDξ , ξ) where a(−hDξ , ξ) is defined by the same method (1.1.13). Formula (1.1.13) is called Weyl or symmetric quantization. Apart of p and q equality it has yet another advantage: if a(x, ξ) is real valued then a(x, hD) is symmetric and often self-adjoint operator. Symbols and h-Pseudodifferential Operators; qp-, pq-, Weyl (Symmetric) Quantization. Double Symbols and Their Quantization Now we can define using formulae (1.1.9), (1.1.10) or (1.1.13) pseudodifferential operators for more general classes of symbols than it was before: we simply plug an (admissible) general symbol a(x, ξ) in such formula. For a sake of generality and simplicity in some arguments we, however, introduce quantization double symbols a(x, ξ, y ) by formula (1.1.17)

 3 2 1  a(x, hD, x)u (x) = (2πh)−d

 e ih

−1 x−y ,ξ

a(x, ξ, y )u(y ) dydξ

which for a(x, ξ, y ) = a(x, ξ), a(x, ξ, y ) = a(y , ξ), a(x, ξ, y ) = a( x+y , ξ) gives 2 2

1

1

2

a(x, hD), a(x, hD) and aw (x, hD) respectively.

8

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

Surely there are two shortcomings of this way: completely lost symmetry between x and hD and very non-unique symbol ↔ operator correspondence. However such symbols are technically useful. 3

2

1

Operator A = a(x, hD, x) defined by (1.1.17) has Schwartz kernel  −1 −d e ih x−y ,ξ a(x, ξ, y ) dξ (1.1.18) KA (x, y ) = (2πh) Recall that each linear operator A : C0∞ (Y ) → D (X ) where D (Y ) is the space of distributions over D(X ) = C0∞ (X ) has a (unique) Schwartz kernel KA ∈ D (X × Y ) such that Aφ, ψ = KA , φ ⊗ ψ where (φ ⊗ ψ)(x, y ) = φ(x)ψ(y ). For an infinitely smooth and fast decaying with respect to ξ symbol this Schwartz kernel belongs to C∞ (Rd × Rd ) and operator is infinitely smoothing and can be extended to operator E (Rd ) → C∞ (Rd ) where E the space of distributions over E = C∞ . However it is not uniform with respect to h and in contrast to classical theory (to follow) smoothness of the function (distribution) is not the subject of our prime interest. Let us consider operator defined by (1.1.18) in L2 (Rd ) (our usual playground). The following very weak statements are nevertheless useful: Proposition 1.1.2. If (1.1.19)

|∂ξα a| ≤ M(1 + |ξ|)−s

∀α : |α| ≤ K

then (1.1.20)

 −K |KA (x, y )| ≤ Cs,K Mh−d 1 + h−1 |x − y |

Proof. Obviously statement holds as K = 0. −1 Let us multiply KA by (xj − yj ), observe that e ih x−y ,ξ (xj − yj ) = −1 i∂ξj e ih x−y ,ξ (xj − yj ) and integrate by parts with respect to ξj . We will get that (1.1.21) KA (x, y )(xj − yj ) has the Schwartz kernel defined by (1.1.18) with symbol a(x, ξ, y ) replaced by −ih∂ξj a(x, ξ, y ). Then (1.1.20) holds with K = 1. Repeating the same argument K times we will get (1.1.20) for any K .

1.1. PSEUDODIFFERENTIAL OPERATORS

9

Obviously, KA (x, y )(xj − yj ) is the Schwartz kernel of commutator [xj , A]. Theorem 1.1.3. If (1.1.19) (or (1.1.20)) holds for K > d then operator norm of A in L2 (Rd ) does not exceed CM. Proof. There is a theorem in Real Analysis that the operator norm in Lp (Rd ) does not exceed    p1   p−1  p sup |K(x, y )|dx . (1.1.22) sup |K(x, y )|dy x

y

The proof of it based on the trivial proofs for p = 1, ∞ and then interpolation. It follows from (1.1.20) with K > d that both integrals in (1.1.22) do not exceed CM. Later we will replace assumption that a decays in ξ by a more natural one. Negligible Functions, Symbols, Operators. Going from One tType of Quantization to Another. Symbol and Operator Classes Now when we noticed that factor (xj −yj ) in the double symbol leads to extra h in the operator norm estimate, we can assuming some smoothness reduce 3

2

1

2

1

1

2

a(x, hD, x) to b  (x, hD) or to b  (x, hD) or to b w (x, hD) (modulo operators with small norms). 2

1

Namely, to go to b  (x, hD) we need to get rid of y . The simplest way to do this is to use the Taylor expansion (1.1.23) a(x, ξ, y ) =

α:|α| d. Then a(x, hD, x) ≡ b w (x, hD) modulo operator with operator norm not exceeding CN,s MhN where b = bN is defined by (1.1.28). We refer to b  , b  , b as to pq-, qp- and Weyl symbols. . Definition 1.1.5. (i) We call operator negligible if its operator norm does not exceed ChN with some constant C . (ii) We call double symbol negligible if it satisfies (1.1.25) or (1.1.27) or (1.1.29) and leads to bN = 0 or cN = 0 or eN = 0 respectively. (iii) We call a family of functions uh depending on parameter h negligible if uh  ≤ ChN . (iv) We call a family of functions uh depending on parameter h admissible if uh  ≤ C . Sometimes we admit the family with uh  ≤ Ch−l . Surely these definitions depend on N; we will just assume that N is large enough. Note that if a satisfies one of the conditions (1.1.25),(1.1.27),(1.1.29), it is not generally true for b  , b  , b. More robust formulae will be provided later. Definition 1.1.6. Fix s and N ≥ 1. (i) Denote by SsN (Rd × Rd × Rd ) the class of double symbols (1.1.30)

a(x, ξ, y ) =

N−1

aj (x, y , ξ)hj

j=0

with aj satisfying condition (1.1.29) with N replaced by (N −j) and with some constant M. Denote by SsN (Rd × Rd ) the class of corresponding ordinary symbols.

12

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

(ii) Denote by ΨsN (Rd ) the class of operators obtained (modulo negligible) from double symbols a ∈ SsN (Rd × Rd × Rd ) by quantization procedure (1.1.17). (iii) According to Theorem 1.1.3 we get the same class if we start from ordinary symbols b ∈ SsN (Rd × Rd ) and apply quantization procedures (1.1.9),(1.1.10),(1.1.13); in this case we refer to b as qp-symbol, pq-symbol and Weyl symbol respectively. (iv) We call a0 (x, ξ)|h=0 or a0 (x, ξ, x)|h=0 the principal symbol. It does not depend on method of quantization. Remark 1.1.7. Several questions arise immediately: Can two different symbols generate the same operator (modulo negligible)? We have seen this non-uniqueness for double symbols but how about ordinary symbols? We will see later that they are uniquely determined. Note that if we allow dependence aj on h decomposition of the symbol is not unique for sure but we are talking about symbol itself. In particular, the principal symbol is defined uniquely. Algebra: Conjugate and Product of Operators. Symmetric Pseudodifferential Operators There are two operations which are not completely trivial: multiplication and finding adjoint (finding resolvent and functional calculus will follow later). Since we study now bounded operators, we do not need to study the domain of the adjoint operator: (1.1.31)

(Au, v ) = (u, A∗ v )

∀u, v ∈ L2 (Rd )

where (.,.) is an inner product in L2 (Rd ). Adjoint is rather easy: (1.1.32)



3

2

1

a(x, hD, x)

∗

1

2

3

= a† (x, hD, x)

where a† means a complex or Hermitian conjugate to a. In particular (1.1.33)1 (1.1.33)2 (1.1.33)3

2 1  1 2 ∗ a(x, hD) = a† (x, hD), 1 2  2 1 ∗ a(x, hD) = a† (x, hD), ∗  w a (x, hD) = a†w (x, hD),

1.1. PSEUDODIFFERENTIAL OPERATORS

13

(1.1.34) a† = a implies that aw (x, hD) is symmetric. 2

1

1

2

Product is more tricky but we can restrict ourselves by a(x, hD)b(x, hD). One can see easily that 2

(1.1.35)

1

1

2

3

2

1

a(x, hD)b(x, hD) = c(x, hD, x),

c(x, ξ, y ) = a(x, ξ)b(y , ξ).

Further, using (1.1.35) and (1.1.24), (1.1.26) and (1.1.28) one can see easily that (1.1.36)

a(x, hD)b(x, hD) ≡ (a • b)(x, hD)

with the same quantization method where (with indicated quantization method) (−ih)α a •qp,L b = (∂ξα a)(∂xα b), (1.1.37)1 α! α:|α| d/2. functions Φ(ξ)m Proof. The conditions on γ¯j , ρj and ρ¯j imply that the weight  and φ(x)l are tempered with respect to the metrics j dxj2 + h2 dξj2 . Then Theorem 18.5.5 of [1] implies that the operator Φ(hD)m A∗ AΦ(hD)m is a scalar h-pseudodifferential operator with a symbol satisfying (1.1.27) with M replaced by CM 2 ; hence it is a bounded operator from L2 (Rd ) to L2 (Rd ) and its norm does not exceed CM 2 . Then Sobolev embedding theorem implies that the norm of the operator 1 1 d ∗ A AΦ(hD)m from L2 (Rd ) to L∞ (Rd , H) does not exceed CM 2 ρ¯12 · · · ρ¯d2 h− 2 and therefore  ∀x ∈ Rd . |KA∗ AΦ(hD)m (x, y )|2 dy ≤ CM 4 ρ¯1 · · · ρ¯d h−d

18

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

But the left-hand expression is equal to  |Φ(hDy )m KA∗ A (x, y )|2 dy and the same arguments imply that |KA∗ A (x, y )| ≤ CM 2 ρ¯1 · · · ρ¯d h−d

(1.1.56)

∀x, y ∈ Rd .

On the other hand,  (1.1.57)

K

A∗ A

(x, x) =

KA† (x, y )KA (x, y ) dy

 =

||KA (x, y )||2 dy

since the range of KA (x, y ) is a matrix-column. This equality and (1.1.56) imply (1.1.54). Moreover, (1.1.56) holds with A replaced by Aφ(x)l and hence the inequality for A remains true if we integrate with respect to x ∈ Rd provided l < d/2. This inequality and (1.1.57) imply (1.1.55). Theorem 1.1.12. (i) Let A operator be an operator with a symbol a (simple or double) satisfying (1.1.29) with Hilbert-Schmidt rather than operator norm. Let m > d/2, l > d/2. Then A is Hilbert-Schmidt operator and its HilbertSchmidt norm does not exceed CM(¯ ρ1 · · · ρ¯d γ¯1 · · · γ¯d )1/2 h−d/2 ; (ii) Let A operator be an operator with a symbol a (simple or double) satisfying (1.1.53) with trace norm rather than operator norm. Let m > d, l > d. Then A is trace class operator and its trace norm does not exceed CM(¯ ρ1 · · · ρ¯d γ¯1 · · · γ¯d )h−d . Proof. Assume first that min(dim H, dim H ) = 1. Then there is no difference between Hilbert-Schmidt, trace or operator norms for L(H , H) operators. Without any loss of the generality one can assume that m = l. Obviously (i) and (ii) hold as m is large enough. Further, both (i) and (ii) hold for Λm = (1 + |x|2 + h2 D 2 )−m/2 . One can see easily that A = BΛm + C where B is an operator with the symbol a(1 + |x|2 + |ξ|2 )m/2 and C is an operator with the symbol satisfying (1.1.53) with m, K replaced by m + 1, K − 1. Then “induction” with respect to m proves the Theorem in this restricted case. The general case can be forced to the restricted one considering operators Aw with symbols a(x, ξ)w with w ∈ H, ||w || = 1.

1.1. PSEUDODIFFERENTIAL OPERATORS

19

Remark 1.1.13. All the conclusions of Proposition 1.1.11 and this Theorem remain true with constants not depending on x¯, ξ¯ if we replace Φ(ξ) and ¯ and φ(x − x¯) respectively, with arbitrary (¯ ¯ ∈ R2d . φ(x) by Φ(ξ − ξ) x , ξ) Note that the shift x → x−¯ x , ξ → ξ−ξ¯ is equivalent to the transformation ¯ ∗ A ∈ T AT with the operator T : (Tu)(x) = e −ix,ξ u(x + x¯) which preserves all the Lp -norms. Pseudodifferential Operators Applied to an Exponent We need to study pseudodifferential operators applied to an exponent u = −1 f (x)e ih φ(x) where currently (1.1.58) phase φ(x) is a real-valued function, φ(x) and amplitude f (x) satisfy |∂ α φ| ≤ c

(1.1.59)

∀α : |α| ≤ N + 1,

|∂ α f | ≤ M1

∀α : |α| ≤ N

where for f it is a norm in the auxiliary space H. Let us consider 2 1   −1 (1.1.60) a(x, hD) f (x)e ih φ(x) =  −1 −d a(x, ξ)f (y )e ih [x−y ,ξ +φ(y )] dξdy = (2πh)  −1 −d ih−1 φ(x) a(x, ξ)f (y )e ih [x−y ,ξ +φ(y )−φ(x)] dξdy (2πh) e

which after substitution ξ → ∇φ(x) + ξ becomes (2πh)

−d ih−1 φ(x)

e

 a(x, ∇φ(x) + ξ)f (y )e ih e ih

−1 φ(x)

−1 [x−y ,ξ +ϕ(x,y )]

2

1

a(x, ∇φ(x) + hDy )e ih

with (1.1.61)

dξdy =

ϕ(y , x) = φ(y ) − φ(x) − y − x, ∇φ(x).

−1 ϕ(x,y )

  

y =x

20

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

If an original operator  was h-differential operator of order m the last expres−1 −1 sion would be e ih φ(x) n≤m bn (x)hn where differentiating e ih ϕ(x,y ) ones we   would need to differentiate factor ∇φ(y ) − ∇φ(x) at least ones to prevent getting 0 as y = x; so, at least one factor h would not be compensated by h−1 arising from the differentiation of exponent. We can apply stationary phase method . We will learn it later in full but now we need its poor man’s approach. Step 1 . Let us replace ϕ(y , x) by (1.1.62) ϕN (x, y ) = α:2≤|α|≤N

1 (y − x)α ∂xα φ(x). α!

Note that (1.1.63) e ih

−1 ϕ(x,y )

− e ih

−1 ϕ

N (x,y )

=

−1



ih [ϕ(y ) − ϕN (x, y )]

1

e ih

−1 [tϕ(y )+(1−t)ϕ

N (x,y )]

0

and one can decompose φ(y ) − φN (x, y ) into the sum (y − x)α bα (x, y ) with −1 |α| = N + 1. Then replacing (y − x)α by (ih∂ξ )α applied to e ih x−y ,ξ and integrating N + 1 times by parts we will obviously get that this error is O(hN−d ). Similarly, one can replace f by (1.1.64) fN−1 (x, y ) = α:|α|≤N−1

1 (y − x)α ∂xα f (x). α!

One can skip Step 1 assuming extra smoothness of φ and f . Step 2 . Now let us assume that φ and f are smooth enough (or we plug φN −1 and fN−1 instead) and let us replace f (y )e ih ϕ(a,y ) by its Taylor decomposition 1   −1 (y − x)α ∂yα f (y )e ih ϕ(x,y )  α! y =x α:|α| 2d; it is enough to assume that the corresponding estimate holds as |α| ≤ K , |β| ≤ K , |ν| ≤ K with K > d. Then in (1.1.89) we get CM(1 + |ξμ − ξν |)−K (1 + |ξν − ξλ |)−K . This remark holds for all statements of this section. Remainder Estimates Theorem 1.1.19 and Proposition 1.1.18(ii) immediately imply Theorem 1.1.21. Let be fulfilled with K > 2d. Then for h ∈(0, h0 ]  (1.1.76) 2 d  and L ≤ N − 1 the L L (R , H ), L2 (Rd , H) -norm of operator Op a(j) −  a(j,L) with a(j,L) defined by (1.1.24), (1.1.26) and (1.1.28) does not exceed CMhL . Theorem 1.1.22. Let a and b be L(H , H )- and L(H , H)-valued ordinary symbols satisfying (1.1.77) with M = Ma and M = Mb respectively. Let a(x, hD), b(x, hD) and (a •L b)(x, hD) be quantizations of symbols a, b and a •L b where L ≤ N − 1 and here and in what follows a •pq,L b, a •qp,L b, a •w,L b are defined by (1.1.37)1 –(1.1.37)3 for qp-, pq- and Weyl quantization respectively.   Then the L L2 (Rd , H ), L2 (Rd , H) -norm of the operator a(x, hD)b(x, hD) − (a •L b)(x, hD) does not exceed ChL Ma Mb . Proof. Proof follows immediately from Proposition 1.1.18(ii) and Theorem 1.1.16. Remark 1.1.23. Again there are sharp but non-local formulae   a •qp b = Γx,ξ e ihDy ,Dξ a(x, ξ)b(y , η) , (1.1.92)1   a •pq b = Γx,ξ e −ihDx ,Dη a(x, ξ)b(y , η) , (1.1.92)2   ih ih (1.1.92)3 a •w b = Γx,ξ e 2 Dy ,Dξ − 2 Dx ,Dη a(x, ξ)b(y , η) , with Γx,ξ is the operator of restriction to the diagonal {x = y , ξ = η}.

28

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

Classes of Symbols and Pseudodifferential Operators From now on we in use Definition 1.1.24. (i) Denote by Sh,N (Rd × Rd × Rd , H , H) and Sh,N (Rd × Rd , H , H) classes of double and ordinary symbols in the form (1.1.30) N−1 a(x, ξ, y ) = aj (x, y , ξ)hj j=0

with aj satisfying condition (1.1.76) with N replaced by (N − j) and with some constant M. (ii) Denote by Ψh,N (Rd , H , H) the class of operators obtained from double symbols a ∈ Sh,N (Rd × Rd × Rd ) by quantization; operators of this class are defined modulo negligible i.e. with the operator norm not exceeding MhN . (iii) The smallest M such that (1.1.76) with N replaced by (N − j) holds for all j = 0, ... , N and (ii) also holds, the norm of A: |||A||| = AΨ . We preserve the notions of the principal and subprincipal symbol. Remark 1.1.25. Discussing trace or Hilbert-Schmidt operators we need to use trace or Hilbert-Schmidt norms both in (1.1.76) and for negligible operator estimate as well. Discussing compact operators (see below) we need to assume that both symbols and negligible operator are compact. Compactness Theorem 1.1.26. Let A operator be an operator with a symbol a (simple or double) satisfying (1.1.76) and (1.1.93) ||Dxα Dyβ Dξν a(x, ξ, y )|| ≤ (R) ∀(x, ξ, y ) ∈ Rd × Rd × Rd : |x| + |y | + |ξ| ≥ R ∀α, β, ν : |α| ≤ K , |β| ≤ K , |ν| ≤ K (R) → 0 as R → ∞, N ≥ 0. Further, let us assume that a(x, ξ, y ) is a compact operator for each (x, y , ξ). Then A is compact operator.

1.1. PSEUDODIFFERENTIAL OPERATORS

29

Proof. We need to prove that one can approximate A by compact operators (in operator norm). Let AR be an operator with the symbol aϕR with  1  ϕR = χ 2 (|x|2 + |y |2 + |ξ|2 ) , R χ ∈ C0∞ (R) equal 1 on (−1, 1). Note that (A − AR ) is a quantization of the symbol a(1 − ϕR ) which satisfies (1.1.76) with M replaced by C (R). Therefore its operator norm does not exceed C (R). Let Pn be a sequence of finite-dimensional projectors in H converging to I strongly (i.e. Pn w → w for each w ∈ H. Then a(x, y , ξ)Pn → a(x, y , ξ) in operator norm for each (x, y , ξ) and due to (1.1.76) ||Dxα Dyβ Dξν a(x, ξ, y )(I − Pn )|| ≤ R (n) ∀(x, ξ, y ) ∈ Rd × Rd × Rd : |x| + |y | + |ξ| ≤ R ∀α, β, ν : |α| ≤ K /2, |β| ≤ K /2, |ν| ≤ K /2 with R (n) → 0 as n → ∞. Similar inequalities then hold for aϕR . Therefore operator norm of AR (I − Pn ) does not exceed C R (n). On AR Pn is Hilbert-Schmidt (and even the trace class) operator due to Theorem 1.1.12. Therefore operator A can be approximated in operator norm by compact operators AR Pn and therefore is compact as well. Rescaling We will need to consider operators with symbols which could be reduced to those considered before by rescaling x → xγ −1 , ξ → ξρ−1 with ρ = (ρ1 , ... , ρd ) and γ = (γ1 , ... , γd ) which means xj → xj γj−1 , ξj → ξj ρ−1 and j 4) h → h = (h1 , ... , hd ) with hj = h/(ρj γj ) . To reach after this rescaling (1.1.76) or (1.1.77) with h ≤ 1 we need to assume that (1.1.94) ||Dxα Dyβ Dξν a|| ≤ Mρ−ν γ −α−β

∀(x, y , ξ) ∈ Rd × Rd × Rd

∀α, β, ν : |α| + |β| ≤ N + K , |ν| ≤ N + K or (1.1.95) ||∂xα ∂ξν a(x, y , ξ)|| ≤ Mρ−ν γ −α 4)

I.e. xj,new =

xj,old γj−1

etc.

∀α, ν : |α| ≤ N + K , |ν| ≤ N + K

30

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

respectively with (1.1.96)

 = max h/(ρj γj ) ≤ 1 j

( ⇐⇒ ρj γj ≥ h).

Remark 1.1.27. (i) Under these assumptions Theorems 1.1.21, 1.1.22 remain true with the remainder estimate CML . Condition (1.1.96) is known as an uncertainty principle. (ii) To get negligible remainder estimate O(hs ) one should assume that  = O(hδ ) with an arbitrarily small exponent δ > 0 i.e. ρj γj ≥ h1−δ .

(1.1.97)

Condition (1.1.97) is known as a (standard) microlocal uncertainty principle. Later we will introduce a logarithmic uncertainty principle. Remark 1.1.28. Using the same methods one can generalize Theorem 1.1.22 to the case when a, b satisfy (1.1.95) with Ma , ρa , γ a and Mb , ρb , γ b respectively such that ρa,j γa,j ≥ h, ρb,j γb,j ≥ h ∀j and with exactly the same methods of quantizations of all symbols; in this case an operator norm of the remainder does not exceed C L Ma Mb with (1.1.98)1 (1.1.98)2 (1.1.98)3

1 , ρa,j γb,j 1  = h max , 1≤j≤d ρb,j γa,j  1 1   = h max + 1≤j≤d ρa,j γb,j ρb,j γa,j

 = h max

1≤j≤d

for qp-, pq- and Weyl quantizations respectively. Ga ˚rding Inequalities By Ga ˚rding inequalities we call statements of the type “if symbol is positive then operator is (almost) positive”. The following theorem will be our workhorse: Theorem 1.1.29. Let a be an L(H, H)-valued Hermitian symbol satisfying (1.1.77) with N = 0 and such that (1.1.99)

a(x, ξ)w , w  ≥ 0

∀(x, ξ) ∈ Rd × Rd

∀w ∈ H

1.1. PSEUDODIFFERENTIAL OPERATORS

31

where here ., . means the inner product in H. Then for h ∈ (0, h0 ] (1.1.100)

(aw (x, hD)u, u) ≥ −CMhu2

∀u ∈ L2 (Rd , H)

where here and below (., .) and . mean the inner product and the norm in L2 (Rd , H) and h0 = min1≤j≤d ρj γj . Proof. This is an adapted proof of Theorem 18.6.4 L. H¨ormander [1]. Without any loss of the generality one can assume that M = 1. First of all we need a really fine partitions of unity. Consider (1.1.101) 1= φ2ν (x), 1 = ψμ2 (ξ) ν

μ

where all functions are real, supported in B(xν , 1) or B(ξμ , 1) which are balls of radius 1 with the centers at the lattice points xν , ξμ and such that |∂xα φν | ≤ cα ,

(1.1.102)

|∂ α ψμ | ≤ cα

∀α, ν, μ

uniformly with respect to x, ξ, ν, μ. 1 1 Let us introduce φˆν = φν (h− 2 x), ψˆμ = ψμ (h 2 D), and consider equality (1.1.103) (Au, u) =



 φˆ2ν Au, u =

ν



     φˆν Au, φˆν u = Aφˆν u, φˆν u − φˆν [A, φˆν ]u, u .

ν

ν

ν

Note that operator [A, φˆν ] is skew-symmetric, and therefore  ∗  1 φˆν [A, φˆν ] + φˆν [A, φˆν ] = − [A, φˆν ], φˆν 2

(1.1.104)

and (1.1.103) becomes (1.1.105)

(Au, u) =

 ν

   1  Aφˆν u, φˆν u + [A, φˆν ], φˆν u, u . 2 ν

We claim that (1.1.106) Operator norm of

 ν

 [A, φˆν ], φˆν does not exceed Ch.

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

32

On the heuristic level, calculating commutators formally, one can notice 1 that every time a factor h 2 appears, so the second commutator has an extra factor h. To justify (1.1.106) if a(x, ξ) is fast-decaying with respect to ξ one can notice that the Schwartz kernel of the second commutator is  2 1 1 (1.1.107) K[[A,φˆν ],φˆν ] (x, y ) = KA (x, y ) φν (h− 2 x) − φν (h− 2 y ) and therefore for K ≥ 2 |K[[A,φν ],φν ] (x, y )| ≤ ν

C (1 + h−1 |x − y |)−d−3 × h−1 |x − y |2 ≤ Ch(1 + h−1 |x − y |)−d−1 . The proof in the general situation repeats one of Theorem 1.1.9. After (1.1.106) is proven, (1.1.105) implies (1.1.108) (Au, u) ≥ (Aφˆν u, φˆν u) − Chu2 ν

and therefore it is sufficient to prove (1.1.100) for u replaced by φˆν u. The same arguments work for the second partition. Therefore it is sufficient to prove (1.1.100) for u replaced by ψˆμ φˆν u. Remark 1.1.30. We used the finest x- and ξ- partitions for which (1.1.106) is still valid. Furthermore, now one can replace a(x, hD) by its approximation on support of ψˆμ φˆν : (1.1.109) a¯(x, hD) =     ¯ + ¯ j − x¯j ) + ∂ξ a (¯ ¯ ¯ a(¯ x , ξ) ∂xj a (¯ x , ξ)(x x , ξ)(hD j − ξj )+ j j

¯2 C0 |x − x¯|2 + C0 |hD − ξ| where overbar replaces index μν . Really, the quadratic term of the Taylor decomposition is O(h) on the support of φˆν , ψˆμ . To justify it we can freeze x = xν first (before introduction ψμ ), and hD = ξμ after it. Proofs repeat one of Theorem 1.1.19.

1.1. PSEUDODIFFERENTIAL OPERATORS

33

Consider first the case of the scalar operator a. Then one can rewrite quadratic operator a¯(x, hD) as ¯ 2 + C0 |x − y − x¯|2 + b a¯(x, hD) = C0 |hD − η − ξ| with η=−

1 ∇ξ a, 2C0

y =−

1 ∇x a, 2C0

b =a−

1 (|∇x a|2 + |∇x a|2 ) 4C0

¯ calculated at x = x¯, ξ = ξ. Now without any loss of the generality one can assume that x¯ + y = ξ¯ + η = 0 (otherwise we can make a shift). So, we have a harmonic oscillator C0 (|x|2 + |hD|2 ) plus b. Here b ≥ 0 (provided we picked up a large enough constant C0 ) due to the known fact that if the second derivatives of f are bounded and f is nonnegative then f ≥ C0 |∇f |2 . On the other hand, harmonic oscillator is non-negative operator; in fact its lower bound is Cdh; we will analyze it later. So, our reduced operator a¯(x, hD) satisfies (1.1.100) and thus our operator a satisfies it as well. Later we reexamine scalar case to improve sharp Ga ˚rding inequality. Consider general (matrix) case now . Then (1.1.110)    ¯ + L(¯ ¯ y , η) v , v + c(|y |2 + |η|2 )||v ||2 ≥ 0 a(¯ x , ξ) x , ξ;

∀v ∈ H ∀y , η ∈ Rd

where ¯ y , η) = (∇x a)(¯ ¯ y  + (∇ξ a)(¯ ¯ η. L(¯ x , ξ; x , ξ), x , ξ), We can plug 2y , 0 instead of y , ξ. Then we L becomes an operator of multiplication and then    ¯ + 2L(¯ ¯ x − x¯, 0) u, u + 4cxu2 ≥ 0; a(¯ x , ξ) x , ξ; here u is an arbitrary H-valued function. Plugging in (1.1.110) 0, 2η instead of y , η and making h-Fourier transform (recall that such transform does not break Weyl quantization) we will get    ¯ + 2L(¯ ¯ 0, hD − ξ) ¯ u, u + 4chDu2 ≥ 0. a(¯ x , ξ) x , ξ; Adding two last inequalities we will arrive to (¯ a(x, hD)u, u) ≥ 0 as long as C ≥ 4c. It concludes the proof.

34

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS In the scalar case (i.e., for H = C) one can prove a better estimate.

Theorem 1.1.31. Let a be a scalar real valued symbol satisfying (1.1.77) with N = 0 and such that (1.1.111) Then (1.1.112)

a(x, ξ) ≥ 0. (aw (x, hD)u, u) ≥ −CMh2 u2

∀u ∈ L2 (Rd ).

Proof. This is very precise result and it looks like no tight partition of unity can save us because if we take scale hδ the double commutator has the norm  h2−2δ which will be too large for any δ > 0. However, we will run rescaling method. The proof is really difficult and idea rich. We will follow the proof of Theorem 18.6.8 of L. H¨ormander [1]. We will run induction with respect to d. z , ) of radius  Step 1. Non-degenerate case. First, let us consider ball B(¯ with the center 0 in R2d where  > 0 is a small enough constant. If a(¯ z) ≥ κ then a(z) ≥ κ/2 in B(¯ z , ) where κ > 0 is an arbitrarily small constant and  = (κ) > 0 and then (1.1.113) provided (1.1.114)

(aw (x, hD)φ(x, hD)u, φ(x, hD)u) ≥ −C0 h2 u2 supp(φ) ⊂ B(¯ z , r ),

|∇α φ| ≤ cα r −|α|

∀α

with r = . Assume now that |∇2 a|(¯ z ) = 1 and a(¯ z ) < κ with a small enough constant κ > 0. Then without any loss of the generality one can assume that (1.1.115)

z ) ≥ κ. ∂ξ21 a(¯

Really, if one of the second derivatives with respect to ξj is greater than κ by absolute value we can reach our inequality by rotation (and multiplication of a by some constant); if it is true for one of the second derivatives with respect to xj can reach the previous case by rotation and h-Fourier transform; if one of the mixed x, ξ-derivatives we can reach the previous case by gauge transformation: aw (x, hD) → e − 2 h i

−1 Qx,x

i

aw (x, hD)e 2 h

−1 Qx,x

1.1. PSEUDODIFFERENTIAL OPERATORS

35

with symmetric real matrix Q, transforming the symbol a(x, ξ) to a(x, ξ+Qx). One can prove easily that (1.1.116) e − 2 h i

−1 Qx,x

i

aw (x, hD)e 2 h

−1 Qx,x

= b w (x, hD), b(x, ξ) = a(x, ξ + Qx).

Then under assumption (1.1.115) in B(¯ z , ) (1.1.117) a(x, ξ) = b(x, ξ)2 + a1 (x, ξ  ),

x  = (x2 , ... , xd ), ξ  = (ξ2 , ... , ξd )

where a1 = a(x, t(x, ξ  ), ξ  ) ≥ 0 and ξ1 = t(x, ξ  ) is a solution to equation ∂ξ1 a = 0 and a, b are real-valued. Further, one can prove easily that derivatives of a1 , b up to order k are estimated by derivatives of a up to order k + 2 (multiplied by a some constant). Then (1.1.118)

aw (x, hD) ≡ bw (x, hD)∗ b w (x, hD) + a1w (x, hD  ) + h2 Ψ

where Ψ denotes the class of bounded h-pseudodifferential operators. Therefore inequality (1.1.112) for aw (x, hD) is reduced to the same inequality for a1w (x, hD  ); this is an operator-valued function and one needs to prove (1.1.112) for each x1 with an inner product in L2 (Rd−1 ). Reduction is done. Step 2. Rescaling. Assume now that (1.1.119)

|∇α a| ≤ cα r 4−|α|

∀α : |α| ≤ K

and that 1

z )|  r 2 with r ≥ Ch 2 . (1.1.120) Either a(¯ z )  r 4 or |∇2 a(¯ In this case we can rescale: x → xnew = r −1 x, ξ → ξnew = r −1 ξ and thus B(¯ z , r ) → B(r −1 z¯, )), h → hnew = r −2 h and a(x, ξ) → anew (x, ξ) = 2 r −4 a(rx, r ξ). Then we get −Cr 4 hnew = −Ch2 in the right-hand side of our target inequality. Step 3. Choice of scale. Let us introduce scaling function  1 1 (1.1.121) r (z) = 0 a(z) + |∇2 a(z)|2 4 + Ch 2 in R2d . Furthermore

36

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

(1.1.122) If f (z) ≥ 0 and |∇4 f | ≤ c in Rn then in each point 1

1

3

(1.1.123)

|∇f (z)| ≤ Cf (z) 2 |∇2 f | 2 + Cf 4 ,

(1.1.124)

|∇3 f (z)| ≤ C |∇2 f | 2 + Cf

1

1 4

with an appropriate constant C . One can prove (1.1.123), (1.1.124) basing on Taylor decomposition up to fourth order terms. Further, one can prove then that for an appropriate constant 0 > 0 1 |∇z r | ≤ , 2

(1.1.125)

1

(1.1.119) holds and either (1.1.120) holds or r ≤ 2Ch 2 . Then we conclude that (1.1.126) Inequality (1.1.112) holds for symbol a replaced by aφ = aφ where φ satisfies (1.1.120) for some z¯ and r = r (¯ z ). 1

1

Indeed, if r (¯ z ) ≥ 2Ch 2 it follows from Step 2. If r (¯ z ) ≤ 2Ch 2 then hnew  1 and our inequality is trivial (after rescaling). Step 4. Partition. Now we need an r -admissible partition. Namely, as scaling  2 function r (z) satisfies (1.1.125), there is a partition of unity 1 = ν ψν , such that (1.1.127) Each ψν satisfies (1.1.114), is supported in the ball B(zν , 12 r (zν )) and balls B(zν , r (zν )) cover Rn with multiplicity not exceeding C (d). n To construct such partition one needs to construct first  covering of R 1 1 by balls B(zν , 3 r (zν )) with the distance |zν − zμ | ≥ 4 max r (zν ), r (zμ ) ; then we have multiplicity condition fulfilled. Let ψν (z) = χ((z − zν )/r (zν )) with smooth χ, supported in B(0, 12 ) and equal 1 in B(0, 13 ). Then  function φ = ν ψν (z)2 is a smooth function, satisfying (1.1.114) in B(zν , 23 r (zν )) for 1 each ν and disjoint from 0. Finally ψν = φ− 2 ψν satisfy (1.1.127).

1.1. PSEUDODIFFERENTIAL OPERATORS

37

More general scaling should reflect the different scales with respect to x, ξ and even more general scaling can be introduced by temperate metric (see Definition 18.5.1 in L. H¨ormander [1], Vol. III). Now one can quantize ψν and construct instead operators ψˆν = ψν w ,    ˆ ˆ ˆ ˆ ˆ− 12 . One can see easily that ψˆν and φˆ are φˆ = ν ψν ψν and ψν = ψν φ Hermitian operators, hν−1 (ψˆν∗ − ψˆν ) satisfies (1.1.114) with hν = h/r (zν )2 and I =

(1.1.128)



ψˆν∗ ψˆν .

Then we arrive to decomposition (Au, u) = with

1 (ψˆν∗ ψˆν Au, Au) = (Aν ψˆν u, ψˆν u) − (Tu, u) + (Ru, u) 2 ν ν

T = ψˆν∗ [Aν , ψˆn ] − [Aν , ψˆν∗ ]ψˆν ,

R=



ψν∗ ψν (A − Aν )

ν

where Aν = (Aφν ) (x, hD), φν satisfies (1.1.114), is supported in B(zν , 34 r (zν )) and equal 1 in B(zν , 32 r (zν )). w

One can see easily that operator norms of each term in the sum defining T , R does not exceed r (zν )4 hν2 = h2 . Then using arguments of the proofs of Theorems 1.1.9, 1.1.22 one can prove easily that both (Tu, u) and (Ru, u) do not exceed Ch2 u2 (surely we have a more tricky partition but it is not a grave problem. We are skipping this part). 2 ˆ 2 ˆ ˆ  Onˆ the2 other 2hand, due to Step 2 (Aν ψν u, ψν u) ≥ −Ch ψν u and ν ψν u = u due to (1.1.128). Combining these inequalities we arrive to (1.1.112). It looks like Theorem 1.1.31 is very sharp. Surely it is. But in many cases it gives a wrong lower bound! Namely, let us look in d = 1 at Harmonic oscillator H = x 2 + (hD)2 . Its spectrum is {h, 3h, 5h, ...} and (Hu, u) ≥ hu2 while Theorem 1.1.31 claims that (Hu, u) ≥ −Ch2 u2 . Actually formally this theorem is not applicable since H has unbounded symbol and is unbounded operator but one can overcome this easily. Let us discuss another inequality for scalar operators which covers this gap even in general it is less sharp than Theorem 1.1.31. It also involves some symplectic notions.

38

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

Theorem 1.1.32. Let a(x, ξ) be scalar, real-valued and satisfy (1.1.77). Let a(x, ξ) + h Tr+ (Hess# a)(x, ξ) ≥ 0

(1.1.129) where

 (1.1.130)

Hess a =

 axx axξ , aξx aξξ

 #

Hess a =

aξx aξξ −axx −axξ



 0 I Hess a −I 0

 =

are 2d × 2d-matrices called Hessian and skew-Hessian of a respectively, Tr+ (Hess# )a is a half-sum of absolute values of eigenvalues of Hess# a. Then for any  > 0 (1.1.131)

(aw (x, hD)u, u) ≥ −(h − C h2 )u2

for all u, h ≤ 1. Proof. Let sketch the proof. First of all we can always make ε-partition of unity in R2d with arbitrarily small constant  > 0 (actually since our 1 remainder estimate is o(h) we can take even L h 2 -partition with large enough L ). Let us look at Step 1 of the proof of Theorem 1.1.31. Repeating it as long as |∇2 a| or |∇2 a1 | etc are disjoint from 0 we will get that we can reduce symbol to b12 + ... + bl2 + al where |∇2 al | is small enough and then it would be sufficient to prove (1.1.131) for symbol a = g − h Tr+ Hessσ g , g = b12 + · · · + bl2 . Furthermore, in this construction ∇b1 , ... ∇bl are linearly independent. Now it would be sufficient to check (1.1.129) on manifold Σ = {(x, ξ) : b1 = ... = bl = 0} and one can check easily that on Σ non-zero eigenvalues of Hess# g are those of skew-symmetric matrix ({bj , bk })j,k=1,...,l (and thus these eigenvalues are purely imaginary).  Further, one can replace system b1 , ... , bl by b1 , ... , bl , bj = k αjk bk with orthogonal real matrix (αjk ) in the way that at some point z¯ ∈ Σ matrix ({bj , bk }) is in the canonical form: {bj , bk } = ρm as j = 2m − 1, k = 2m, m = 1, ... , r , {bj , bk } = −ρm as j = 2m, k = 2m − 1, m = 1, ... , r and {bj , bk } = 0 otherwise where ±iρm (m = 1, ... , r ) are non-zero eigenvalues of

1.1. PSEUDODIFFERENTIAL OPERATORS

39

({bj , bk }) and ρm > 0. Therefore it is sufficient to prove (1.1.131) as l = 2. Now it is reduced to (1.1.132)

B1 u2 + B2 u2 + i([B1 , B2 ]u) ≥ −Ch2 u2

because replacing Bj by Bj + h2 μj we can always get rid of terms (hμj Bj u, u) with pseudodifferential operators μ1 , μ2 . However the left-hand expression of (1.1.132) is exactly (B1 + iB2 )u2 as long as Bj are symmetric. Inequality (1.1.131) (Melin’s inequality) gives a correct lower bound of the spectrum of a(x, hD). Really, for quadratic symbol a(x, ξ) = cαβ x α ξ β α,β:|α|+|β|=2

which is non-negative, the bottom of the spectrum of aw (x, hD) is exactly h Tr+ (a). There is more limited but more precise statement which we will not prove: Theorem 1.1.33. Let b1 , ... , bk be real valued symbols with ∇b1 , ... , ∇bl linearly independent at Σ = {b1 = ... = bl = 0}. Furthermore, let matrix β = ({bj , bk }) have a constant rank on Σ and let a = b12 + · · · + bl2 + hb0 with b0 = −h Tr+ (β) on Σ. Then (1.1.131) holds. Remark 1.1.34. (i) Theorems 1.1.31–1.1.33 remain true even if only the principal symbol a0 is scalar. (ii) A generalization of Melin’s inequality to 2 × 2-matrix case is known. See f.e. R. Brummelhuis [2]. Parametrix and iInverse Now let us consider parametrix construction. Theorem 1.1.35. Let a ∈ Sh,N (Rd × Rd , H , H). Assume that |||a||| ≤ c and its principal symbol a0 is left invertible or right invertible at every point (x, ξ) ∈ Ω ⊂ Rd × Rd : (1.1.133)

b0 (x, ξ)a0 (x, ξ) = I

or

a0 (x, ξ)b0 (x, ξ) = I

40

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

with (1.1.134)

|||b0 ||| ≤ c.

Then there exists a symbol b ∈ Sh,N (Rd × Rd , H, H ) with |||b||| ≤ C = C (d, N, c) such that respectively (1.1.135)

b •N a = I

or

Proof. One can define bn recursively by   b m h m • a b0 , (1.1.136) bn = − m 0) f (A) ∈ Ψh,N and coincides modulo O(hN ) with  1 • (1.1.145) f (a) = f (λ)r(N) (λ, h) dλ 2πi  with r(N) (λ, h) defined in Theorem 1.1.38(ii). Furthermore, the principal symbol of f (a) is f (a0 ). Formula (1.1.139) implies that if a0 is a scalar symbol then  s (1.1.146) f (A) = −as f  (a0 ). Remark 1.1.41. (i) Assume instead that Σ(a) ⊂ G ∪ G  where G ∩ G = ∅. Then (1.1.144) defines f (A)ΠG where ΠG = χG (A) is also defined by (1.1.144) with f = 1 is a projector, χG is a characteristic function of G. In particular, as a0 is Hermitian elliptic symbol we conclude that |A|t and θ(±A)|A|t are pseudodifferential operators where θ is a Heaviside function and t ∈ C.

1.1. PSEUDODIFFERENTIAL OPERATORS

43

(ii) One can define Σ(a, Ω) and run the whole construction on the (full) symbol rather than operator level. We need however to consider a more delicate case and to be more precise. Namely let us now assume that (1.1.147) ||(τ − a0 (x, ξ))−1 || ≤ cdist(τ , Σ)−p ∀(x, ξ) ∈ Rd × Rd ∀λ ∈ C : |λ| ≤ C0 with an exponent p ≥ 1 where Σ(a) ⊂ Σ ⊂ {|λ| < C0 }. Remark 1.1.42. Condition (1.1.147) is satisfied with Σ = Σ(a) provided either (1.1.148) or (1.1.149)

H = Cr ⊗ H 0 ,

a0 (x, ξ) = b(x, ξ) ⊗ I ,

a0 (x, ξ)† = a0 (x, ξ) ∀(x, ξ),

p=r

p = 1.

Note that in the resolvent construction each new term norm gains factor O(h||(τ − a0 (x, ξ))−1 ||2 ). Therefore as dist(x, Σ) ≥ η 1/p with (1.1.150)

η ≥ h 2 −δ 1

with an arbitrarily small exponent δ > 0 the resolvent construction is justified and the error is O(hN η −(2N+1) ). Further, after rescaling x → x/η, ξ → ξ/η, h →  = h/η 2p j-th term satisfies (1.1.77) with Mj = hj η −2j−1 . In particular   (1.1.151) (τ − A)−1  ≤ C η −1 , (τ − A)−1 − Op (τ − a0 )−1  ≤ Chη −3 . Also we conclude that (1.1.152) Spec(A) is contained in Ch1/2p vicinity of Σ. Combining with (1.1.144) we arrive to Theorem 1.1.43. Let (1.1.47) be fulfilled. Let f be analytic in G and  be a closed contour in G going around spectrum of A exactly once in counterclock-wise direction on the distance no less than 0 η 1/p from Σ. Then   (1.1.153) f (A) − Op f • (a)  ≤ ChN η −2N−1 f L1 ()

44

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

where symbol f • is defined by (1.1.145) and each term in its decomposition does not exceed Chj η −2j−1 f L1 () . Further,   (1.1.154) f (A) ≤ C η −1 f L1 () , f (A) − Op f (a0 )  ≤ CMhη −3 . Smooth Functional Calculus of Pseudodifferential Operators As operator A is Hermitian (or unbounded self-adjoint) one can construct f (A) with smooth function f . Let us look for a smooth function of a Hermitian pseudodifferential operator. Since one can define f (A) through propagator e itA let us consider such operators first but we need to consider them for large t ∈ R as well. Let us apply analytic theory to ϕt (τ ) = e itτ under condition (1.1.49) (we do not need a stronger condition at this moment). Then p = 1. To avoid exponential growth with respect to t we must bound |t Im τ |; so let  be a boundary of the rectangle {| Re τ | ≤ c, | Im τ | ≤ |t|−1 }, where we assume that Σ(a) ⊂ [− 12 c, 1 c]. Then ϕt L1 ()  1 and therefore (1.1.155) and (1.1.156)

e itA − Op(ϕ•t (a)) ≤ Chs  Op(ϕ•t (a)) ≤ C |t|,

 Op(ϕ•t (a)) − Op(e ita0 ) ≤ Ch|t|3

provided t ∈ R, 1 ≤ |t| ≤ h(δ−1)/2 . Moreover, note that the terms rn (τ ) in the decomposition of the symbol r (τ ) are sums of terms of the form (τ − a0 )−1 b1 (τ − a0 )−1 · · · (τ − a0 )−1 bj−1 (τ − a0 )−1 hn with j ≤ 2n + 1, bk = bnjk ∈ Sh,N−(2n+1)+j (Rd × Rd , H, H) do not depend on τ and the number of these terms does not exceed K0 (n). Then we conclude that the terms ϕ•tn in the decomposition of the symbol ϕ•t are sums of terms of the form g (t) ∗ b1 g (t) ∗ · · · ∗ bj−1 g (t)hn where “∗” means convolution and  1 g (t) = (τ − a0 )−1 e itτ dτ = e ita0 ; 2πi  hence ϕ•tn is a sum of terms of the form   j−1 n (1.1.157) t h ··· e itz1 a0 b1 e itz2 a0 · · · bj−1 e itzj a0 dz  Δj−1

1.1. PSEUDODIFFERENTIAL OPERATORS

45

with zj = 1 − z1 − ... − zj−1 , dz  = dz1 · · · dzj−1 and Δj−1 = {z  ∈ Rj−1 : zk > 0 ∀k, z1 + ... zj−1 < 1}. Recall that j ≤ 2n + 1. Note that for any first-order differentiation ∂ the following formula holds:  t   (1.1.158) ∂e ita0 = e it a0 (i∂a0 )e i(t−t )a0 dt  . 0

Applying this formula to (1.1.157) we conclude easily that (1.1.159)

 Op(ϕ•t (a)) ≤ C ,

 Op(ϕ•t (a)) − Op(e ita0 ) ≤ Chγ −2

provided |t| ≤ h(δ−1)/2 . So we have improved estimates (1.1.156). Therefore we arrive to Theorem 1.1.44. Under condition (1.1.149) e itA is a pseudodifferential operator with the symbol satisfying (1.1.95) with ρ = γ = |t|−1 1 provided |t| ≤ ch(δ−1)/2 , 1 = (1, 1, ... , 1). Remark 1.1.45. (i) One can see easily that symbol e ita0 satisfies (1.1.95) with ρ = γ = |t|−1 1 provided |t| ≤ ch(δ−1)/2 and one cannot improve this statement; (ii) Note that for a scalar symbol a0 the symbol ϕ•tn is a symbol of the form (1.1.160) e ita0 bnj hn t j . 0≤n≤N, j≤2n

Now we can consider smooth functions of Hermitian operator. Assume that A∗ = A and f ∈ CN (R) satisfies (1.1.161)

τ 2 |Dτj f (τ )| ≤ Mη −j

∀j ≤ N

1

for η ∈ [h 2 (1−δ) , 1], δ > 0

with τ  = (1 + |τ |2 )1/2 . Then f (A) is given by the formula  fˆ(t)e itA dt (1.1.162) f (A) = R

where (1.1.161) implies that (1.1.163)

|fˆ(t)| ≤ CM(1 + η|t|)−N .

46

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS 



Consider integral (1.1.162) separately on [−h(δ −1)/2 , h(δ −1)/2 ] and on the remaining part of R where δ  > 0 is a small enough exponent; then (1.1.163) implies that the operator norm of the second integral does not exceed   1−N −1 C h(δ −1)/2 η η = O(hs ) while in the first integral we can replace e itA by Op(ϕ•t (a)). One can consider  fˆ(t)ϕ•t (a)) for |t| ≥ h(δ −1)/2 as a small and thus quantizable (generating a negligible operator); so one can take the integral on R again; the operator norm of the new error also is O(hs ). We obtain then the sum of terms of the form  n Jnj (τ )f (τ ) dτ (1.1.164) h R

where

 1  Jnj (τ + i0) − Jnj (τ − i0) 2πi and Jnj (τ ) are operator-valued functions of the form (1.1.157).  Therefore  Jnj (τ ) are distributions supported in C = (τ , x, ξ) : τ ∈ Spec a0 (x, ξ) ; so supp(Jnj ) ⊂ {|τ | ≤ C0 }. Moreover, |Jnj (τ )| ≤ C | Im τ |−j ∀τ ∈ C and hence  (1.1.165) | Jnj (τ ± i0)f (τ ) dτ | ≤ C f  Cj Jnj (τ ) = ResR Jnj =

for every f ∈ C0j ([−2c, 2c]). Therefore (1.1.158) implies that the term α (1.1.164) does not exceed Chn η −j . Moreover, applying Dx,ξ to (1.1.157) we obtain a function of the same type with j + |α| instead of j. So we obtain the first assertion of the following statement: Theorem 1.1.46. (i) Let conditions (1.1.149) and (1.1.161) be satisfied. Then the described procedure leads to a construction of quantizable symbol f • (a) such that  Op(f • (a)) − Op(f (a0 )) ≤ CMhη −3 . In particular as A is Hermitian (f.e. a† = a and we use Weyl quantization) (1.1.153) and (1.1.154) hold. (ii) Moreover, for scalar symbol a0   (1.1.166) f (A) − Op f (a0 )  ≤ CMhη −2 .

1.1. PSEUDODIFFERENTIAL OPERATORS

47

The proof of the second assertion is based on the fact that in this case  2c |Jnj (τ ± iη)| dτ ≤ C η 1−j −2c

provided j > 1 and hence in the right-hand expression in (1.1.165) one can replace j by (j − 1). Finally, for a scalar symbol a0 formula (1.1.146) remains true. Remark 1.1.47. Instead of (1.1.161) we can assume that f1 = τ −2l f ∈ CbK and derive Theorem 1.1.46 with M = f1  CbK where sup |D α g | < ∞}; CbK = {g : g K = |α|≤K

surely in this case K and C depend on l also. In fact, f (A) = (A2 + I )l f1 (A) where f1 satisfies (1.1.161) with η = 1. Families of Commuting Operators One can easily derive results similar to those obtained in two previous sections for a family A1 , ... , An of commuting operators; these operators have commuting principal symbols; instead of spectrum one should consider a joint spectrum and introduce Σ(a1 , ... , an ) ⊂ Cn ; respectively one should consider f (A1 , ... , An ) where either f is analytic in the vicinity of Σ(a1 , ... , an ) or all operators A1 , ... , An are Hermitian. We leave details to the reader. Norms and Bounds Ga ˚rding inequalities imply that for Hermitian operator its lower bound is not less than inf Σ(a0 ) − CMh and its upper bound is not greater than sup Σ(a0 ) + CMh; this is a better result than Ch1/2 error due to (1.1.152). On the other hand, one can easily generalize Theorem 1.1.14 to operators of the type considered here: usingarguments of the proof of Theorem 1.1.19 −1 one can prove easily that a(x, hD) e ih φ f ) is negligible as supp(a) is disjoint from the set {x, ∇φ(x)}. −1 ¯ + i|x − x¯|2 /2 and Let us consider u = (2πh)−d/4 e ih φ w with φ = x, ξ w ∈ H; one can prove easily that ¯ , w  + O(h) x , ξ)w (aw (x, hD)u, u) = a(¯

48

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

and therefore we arrive to Theorem 1.1.48. For Hermitian operator operator (1.1.167) and (1.1.168)

| inf Spec(A) − inf Σ(a)| ≤ CMh | sup Spec(A) − sup Σ(a)| ≤ CMh.

Remark 1.1.49. (i) In the general case one can apply Theorem 1.1.48 to operator B = 12 (A + A∗ ); (ii) If dim H ≥ 2 it may happen that Σ(a) is not connected. Let G be an open set in C such that Σ(a) ∪ G = ∅ and dist(G, Σ(a) \ G) ≥ 0 . Then one can apply (i) to AΠG with projector ΠG introduced in Remark 1.1.41(i). This generalization makes sense for Hermitian operators as well as general ones. (iii) Further, we can apply Theorem 1.1.46 to A∗ A resulting in (1.1.169)

| A − sup ||a0 (x, ξ)|| | ≤ CMh. (x,ξ)

1.1.3

Pseudodifferential Operators with Analytic Symbols and Logarithmic Uncertainty Principle

So far we considered pseudodifferential operators with finitely smooth symbols N < ∞. How one needs to modify results as N = ∞? Surely it concerns only results with the remainder estimates depending on N (unlike Ga ˚rding inequalities). The first answer is rather easy: all sums became asymptotic and all such remainder estimates are “LN hN for all N < ∞”. The next question is “how LN depend on N?” The answer we got already f.e. for the product of aw (x, hD) and b w (x, hD) is LN ≤ CN ||Dxα Dξβ a||L∞ × ||Dxα Dξβ b||L∞ |α|+|β|≤N

|α|+|β|≤N

1 and if we look at the regular terms containing α!β! etc we can think that CN = N −1 C0 (N!) . Inequality (1.A.5) and formulae (1.1.92)1 –(1.1.92)3 immediately prove this remainder estimate for all types of quantization.

1.1. PSEUDODIFFERENTIAL OPERATORS

49

To exploit this observation we need to consider operators with ultrasmooth symbols: those whose derivatives derivatives grow with the controlled rate: f.e. with (1.1.170)

||Dxα Dξβ a||L∞ ≤ M|α|+|β| (|α|!)σ (|β|!)ς

with σ ≥ 1, ς ≥ 1 and  independent on N. To be consistent we should assume that junior terms satisfy (1.1.171)

||Dxα Dξβ an ||L∞ ≤ M|α|+|β|+2n (|α|!)σ (|β|!)ς (n!)σ+ς−1

Then the remainder estimate is c N hN (N!)σ+ς−1 . One can see easily that there is a minimal value N = Nh = 0 h−1/(σ+ς−1) when estimate hits the minimum (1.1.172)

exp(−h−1/(σ+ς−1) )

and this is “supersmall” remainder estimate. We will reconstruct the theory of two previous subsections for such symbols and remainder estimates and with this notion of negligible. Obviously the best possible case is σ = ς = 1 which corresponds to analytic symbols. One can object that there are no compactly supported symbols and therefore there is no microlocalization. However M. Bronstein observed that we need to satisfy estimates only as |α| + n ≤ Nh , |β| + n ≤ Nh and there are very regular families of symbols satisfying a bit weaker but still allowing quantization condition (see Proposition 1.A.2). Further, this theory brings fruits even for the old notion of negligible O(hs ). Namely, if estimate (1.1.171) holds only after rescaling xj → xj /γj , ξj → ξj /ρj , then the remainder estimate would be O(exp(−1/(σ+ς−1) )) = O(hs ) as  ≤ s −1 | log h|−(σ+ς−1) . Recall that  = supj h/(ρj γj ); so the previous condition becomes (1.1.173)

ρj γj ≥ Cs h| log h|σ+ς−1 .

As σ = ς = 1 we arrive to the most important case (1.1.174)

ρj γj ≥ Cs h| log h|

which we call logarithmic uncertainty principle.

50

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

Pseudodifferential Operators with Infinitely Smooth Symbols Definition 1.1.50. Let Sh,∞ be a class of the symbols satisfying (1.1.77) for all N (with constant depending on N). Denote by Ψh,∞ the corresponding class of pseudodifferential operators. We immediately arrive to Theorem 1.1.51. Let aj ∈ Sh,∞ . Then there exist a ∈ Sh,∞ such that a∼

(1.1.175)



h j aj

j

which means that for any N a−

(1.1.176)



hj aj ∈ hN Sh,∞ .

0≤j≤N−1

Proof. Proof is the standard asymptotic exercise (very similar to the proof that for any sequence of numbers ck there exists a function f such that f (k) (0) = ck ). Namely let φ be a smooth function, supported in (−2, 2) and equal 1 in (−1, 1). Let us consider a=



hj aj φ(hlj )

j

with lj → ∞ as j → ∞. Then a is defined properly for each h > 0 (the sum is finite). We leave to the reader to prove that a is a proper symbol in the following steps: prove that the first of inequalities holds (1.1.177) hj aj φ(hlj )| ≤ CN hN ∀α : |α| ≤ N |∂ α j≥N

(1.1.178)

|∂

α



hj aj (φ(hlj ) − 1)| ≤ CN hN

∀α : |α| ≤ N

j0 Sh,an () where a ∈ Sh,an () if the formal norm defined by a formal power series (1.1.80) converges for |ζ| < : (1.1.181)

P(a, ζ) =

α,n

−n −1 1 0 ζ 2n+|α| sup ||∇αz a(z)|| (2n + |α|)! z

where z = (x, ξ) and we select 0 , 1 > 0 later. Proposition 1.1.53. For small enough 0 , 1 norm P(a, ζ) has following properties: (1.1.182)

P(a + b, ζ) ≪ P(a, ζ) + P(b, ζ),

(1.1.183)

P(a • b, ζ) ≪ P(a, ζ) · P(b, ζ),

where• is calculated for N = ∞ and  we consider here formal symbols i.e. sum an hn etc and n An ζ n ≪ n Bn ζ n means that An ≤ Bn for all n. Proof. We prove the second estimate. From obvious ||∇α (a • b)n || ≤

l+m+|γ|=n, γ, α ≥γ, α ≥γ, α +α =α+2γ

α!   ||∇α al || · ||∇α bm || (α − γ)!(α − γ)!γ!

52

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

we conclude that P(a • b, ζ) ≪

l,m,γ,α ,α

(α + α − 2γ)! × (α − γ)!(α − γ)!(2m + 2l + |α | + |α |)!γ! 



l+m+|γ| 2l+2m+|α |+|α |

||∇α al || · ||∇α bm || × −1 1 0

ζ

which equals

|γ|

l,m,γ,α ,α

(α + α − 2γ)! (2l + |α |)!(2m + |α |)! 1 0 · · × (α − γ)!(α − γ)! (2m + 2l + |α | + |α |)! γ! l m −1 −1   1 0 1 0 α 2l+|α | ||∇ ||∇α bm ||ζ 2m+|α | a ||ζ · l   (2l + |α |)! (2m + |α |)!

where in the first line the first and the second factor decrease if  do not |γ| we replace γ by 0 and l, m by 0 as well; meanwhile γ γ!1 0 ≤ e 2d0 and Proposition 1.1.53 follows from estimate5) (|α | + |α |)! |α |! |α |! |α + α |! (α + α )! ≤ ⇐= · ≤ α !α ! |α |!(|α |)! α ! α ! (α + α )! with the trivial from the combinatorial point of view last inequality. Then we arrive immediately to Theorem 1.1.54. Let a, b ∈ Sh,an (). Then a • b ∈ Sh,an () and with N = N(−1 , h)) (1.1.184)

¯ )P(b, ¯ ) exp(−2 h−1 )  Op(a) Op(b) − Op(a •N b) ≤ C0 P(a,

¯ ) is an actual sum of P(a, ). where C0 depends only on d and P(a, Finally, consider two-sided parametrix. Theorem 1.1.55. Let a ∈ Sh,an satisfy (1.1.137). Then there exist b ∈ Sh,an such that a • b = b • a = I . Proof. Note first that the following statement holds: 5) In [1] more general inequality is considered to be “easy”. Still We prove this one (1.A.5).

1.1. PSEUDODIFFERENTIAL OPERATORS

53

(1.1.185) Let a ∈ Sh,an () with  > 0. Assume that sup(x,ξ) ||a0 (x, ξ)|| ≤ c and sup(x,ξ) ||a0 (x, ξ)−1 || ≤ c. Then a0−1 ∈ Sh,an ( ) with some  > 0. Then a • a0−1 = I − hr with r ∈ Sh,an ( ) and b = r  • a with r ∼ I +

n≥1

hn r • r •· · · • r n factors

and one can see easily that (1.1.186)

P(r  , ζ) ≪ 1 +



ζP(r , ζ)

n

n≥1

¯ ,  ) < 1. which converges as |ζ| <  with  P(r

1.1.4

Other Classes and Calculi of Pseudodifferential Operators

Let us describe now more traditional classes and calculi of pseudodifferential operators without visible parameter h. Instead of it a hidden parameter  = heff appears. Now the notion of order appears; in semiclassical theory it was masked: instead of m-th order operator we used 0-th order one multiplied by h−m . Another new property which comes with certain classes is invariance with respect to change of coordinates. Classical Pseudodifferential Operators Classical pseudodifferential operators were the first to be introduced historically: Definition 1.1.56. (i) Symbol a ∈ C∞ (Rd ×Rd ) belongs to Sm cl with m ∈ R if (1.1.187)

|Dxα Dξβ a(x, ξ)| ≤ Cαβ ξm−|β|

and (1.1.188)

a∼

j≥0

aj

∀α, β

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

54

  with symbols aj ∈ C∞ Rd × (Rd \ 0) positively homogeneous of degrees (m − j) in the sense that   (1.1.189) |Dxα Dξβ a(x, ξ) − aj (x, ξ) | ≤ Cnαβ ξm−n=|β| ∀n, α, β. j≤n

(ii) Meanwhile S−∞ is a class of symbols satisfying (1.1.86) for all m and such symbols are negligible 6) . (iii) Class Ψm cl consists of pseudodifferential operators of the form Op(a) with h = 1 and a ∈ Sm cl . Such operators are defined modulo negligible operators where negligible means infinitely smoothing operator i.e. continuous from H t to H r for all t, r ∈ R. Here a0 = a0 is principal symbol and with Weyl quantization as = a1 is subprincipal symbol . Here and below H r are equipped with the standard norms uHr = (I + |D|2 )r /2 uL2 . One can consider m ∈ C; then one needs to replace m by Re m in the right-hand expressions of (1.1.187) and (1.1.189). We do not list any properties of them except that m2 m1 +m2 1 ; if one Theorem 1.1.57. (i) Let A ∈ Ψm cl , B ∈ Ψcl . Then AB ∈ Ψcl m1 +m2 −1 of operators is scalar then [A, B] ∈ Ψcl . −1 (ii) Let A ∈ Ψm cl with principal symbol a0 and |a0 (x, ξ)| ≤ c for all (x, ξ). −m Then there exists a parametrix B ∈ Ψcl : AB ≡ BA ≡ I (modulo negligible); similar statements hold for left and right parametrices.

(iii) Let m = 0 and either f be analytic in the vicinity of Spec(A) or f be smooth and A be Hermitian. Then f (A) ∈ Ψ0cl . Moreover let m < 0 be an integer; then f (A) − f (0) ∈ Ψm cl (as m not being integer powers ml − k with l, k ∈ Z+ will appear in decomposition (1.1.188). (iv) Ψm cl is invariant with respect to change of coordinates ( see Teorem 1.1.62 below for more details). Remark 1.1.58. Since “negligible” now means “infinitely smoothing” then parametrix does not mean exact invertibility and Spec(A) does not shrink to Σ(a) defined by (1.1.40) as m = 0 or to 0 as m < 0. 6)

m So actually symbol classes Sclm and Sσ,ς defined below are considered modulo S−∞ .

1.1. PSEUDODIFFERENTIAL OPERATORS

55

d Theorem 1.1.59. (i) Let A ∈ Ψm cl (R , H, H) be self-adjoint and moreover

A − (I + |D|2 )m/2 ≥ 0

(1.1.190)

d in operator sense. Then Al ∈∈ Ψml cl (R , H, H) for any l ∈ R (and even l l ∈ C); further, its principal symbol is a0 and if a0 is scalar, its subprincipal symbol is las a0l−1 . d (ii) Let A ∈ Ψm cl (R , H, H) be self-adjoint and moreover

Au ≥ (I + |D|2 )m/2 u

(1.1.191)

∀u ∈ L(Rd , H);

Consider orthogonal (in L2 (Rd , H)) projectors Π± on positive and negative subspaces of A. Then Π± ∈∈ Ψ0cl ; their principal symbols are π ± (x, ξ) (orthogonal projectors to positive and negative subspaces of a0 (x, ξ). Proof. Note first that assumptions (1.1.190) and (1.1.191) imply respectively that a0 (x, ξ) − |ξ|m ≥ 0 and ||a0−1 (x, ξ)|| ≤ c|ξ|−m . (i) Without any loss of the generality one can assume that Re m ≥ 0 (otherwise we substitute A by A−1 and l by −l) and −1 < Re l < 0 (otherwise l = l  n + n with l  satisfying this assumption and n, n ∈ Z). Then 1 A =− 2πi



+∞i

l

(1.1.192)

−∞i

ζ l (ζ − A)−1 dζ

along iR. One can see easily that operator (|ζ| + 1)(ζ − A)−1 belongs to Ψ01,0 uniformly with respect to ζ ∈ iR (see discussion of Sm σ,ς in the next two subsubsections) and therefore Al ∈ Ψ01,0 . Decomposing symbol of (ζ − A)−1 into asymptotic series r (x, ξ, ζ) ∼



rn (x, ξ, ζ)

n≥0

 −1 with rn (x, λξ, λm ζ) = λ−m−1 rn (x, ξ, ζ), r0 = ζ − a0 (x, ξ) and rn a finite sum terms in the form 

ζ − a0 (x, ξ)

−1

 −1  −1 b1 (x, ξ) ζ − a0 (x, ξ) b2 (x, ξ) · · · bj (x, ξ) ζ − a0 (x, ξ)

56

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

 −1 with (j + 1) factors ζ − a0 (x, ξ) we calculate easily that symbol of Al ml belongs to Scl ; furthermore its principal and subprincipal symbols equal (1.1.193)

0 a(l)

(1.1.194)

s a(l)

 +∞i 1 =− ζ l (ζ − a0 )−1 dζ, 2πi −∞i  +∞i 1 =− ζ l (ζ − a0 )−1 as ζ l (ζ − a0 )−1 dζ, 2πi −∞i

which implies the rest of Statement (i). (ii) Observe that orthogonal projectors Π± for A and A(A2 )−1/2 coincide and A(A2 )−1/2 ∈ Ψ0cl due to (i). Then Statement (ii) follows from the functional calculus (Theorem 1.1.57(iii)).

m Classes Sm σ,ς and Ψσ,ς

If we consider symbols satisfying only (1.1.187) we get classes of symbols m and operators denoted by Sm 1,0 and Ψ1,0 ; all other properties remain. Next generalization is given by Definition 1.1.60. (i) Symbol a ∈ C∞ (Rd × Rd ) belongs to Sm σ,ς with m ∈ R, 0 ≤ ς < σ ≤ 1 if (1.1.195)

|Dxα Dξβ a(x, ξ)| ≤ Cαβ ξm−σ|β|+ς|α|

∀α, β.

(ii) Class Ψm σ,ς consists of pseudodifferential operators of the form Op(a) with h = 1 and a ∈ Sm cl . Such operators are defined modulo negligible operators where negligible means infinitely smoothing operator. If we consider |ξ|  ρ  1 and rescale x → xρς , ξ → ξρσ we get pseudodifferential operator with  = ρσ−ς and to have it small for ρ  1 one needs to set ς < σ. We state only few most important results here: m2 m1 +m2 1 Theorem 1.1.61. (i) Let a ∈ Sm and σ,ς , b ∈ Sσ,ς . Then a • b ∈ Sσ,ς

(1.1.196)

a • b ≡ ab

m1 +m2 −σ+ς . mod Sσ,ς

1.1. PSEUDODIFFERENTIAL OPERATORS

57

m1 +m2 −σ+ς and (ii) If one of symbols a, b is scalar then (a • b − b • a) ∈ Sσ,ς

(1.1.197)

m1 +m2 −σ+ς mod Sσ,ς .

a • b − b • a ≡ {a, b}

(iii) Further, if one of symbols a, b is scalar then (1.1.198)

a •w b + b •w a ≡ 2ab

1 +m2 −2σ+2ς mod Sm , σ,ς

(1.1.199)

a •w b − b •w a ≡ −i{a, b}

1 +m2 −3σ+3ς mod Sm . σ,ς

−1 (iv) Let A ∈ Ψm σ,ς with principal symbol a0 and |a (x, ξ)| ≤ c for all (x, ξ) : |ξ| ≥ c. Then there exists a parametrix B ∈ Ψ−m σ,ς : AB ≡ BA ≡ I (modulo negligible); similar statements hold for left and right parametrices.

(v) Let m = 0 and either f be analytic in the vicinity of Spec(A) or f be smooth and A be Hermitian. Then f (A) ∈ Ψ0σ,ς . Moreover, let m < 0. Then f (A) − f (0) ∈ Ψm σ,ς . Until now we did not discuss change of variables with a good reason: semiclassical pseudodifferential operators of subsection 1.1.2 did not possess this property (but those of Subsection 1.1.1 did). Theorem 1.1.62. (i) Let a ∈ Sm σ,ς with 0 ≤ 1 − σ ≤ ς < σ ≤ 1.

(1.1.200)

Let G and G  be open domains in Rd and let κ : G → G  be a diffeomorphism (onto G  ). Then there exists a symbol a(κ) ∈ Sm σ,ς such that for every pair of functions f , g ∈ C0∞ (G  ) (1.1.201)

κ ∗ fa(x, hD)g κ ∗−1 ≡ f  a(κ) (x, D)g 

modulo negligible where κ ∗ is the induced map: κ ∗ u = u◦κ, f  = f ◦κ −1 , g  = g ◦ κ. (ii) Moreover, in T ∗ G  the symbol a(κ) is defined uniquely modulo negligible: in particular for qp-quantization (1.1.202) a(κ) (x, ξ) ∼ Γx

1   a(α) κ(x), tκ −1 (x)ξ × α! α   (hDy )α exp iκ(y ) − κ(x) − κ  (x)(y − x), ξ

where κ  (x) is the Jacobi matrix of κ at x and here, a(α) (x, ξ) = ∂ξα a(x, ξ).

t

means matrix transposition

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

58

Proof. Note that we need 1−σ ≤ ς in order to have a(ϕ(x), tκ  (x)ξ) belonging to Sm σ,ς . Really, differentiating this symbol with respect to x through ξ we gain factor ξ−σ because of definition of the class but also tκ  (x)ξ i.e. we get factor ξ1−σ which should not exceed ξς . Consider in qp-quantization      ∗ ∗−1 −d a κ(x), ξ e iκ(x)−y ,ξ u κ −1 (y ) dydξ κ (Aκ u)(x) = (2π) where we accommodated f and g  in a(x, ξ) and u respectively. To be closer to the semiclassical arguments let us consider integral over zone {|ξ|  h−1 } only and change ξ → ξh arriving to    −1 (2πh)−d a κ(x), ξ e ih κ(x)−κ(y ),ξ | det κ  (y )|u(y ) dydξ where we also changed variables of integration yold = κ(y ). Plugging  κ(x)−κ(y ) = κ  (x)(x −y )−ϕ(x, y ) we get exp ih−1 x −y −ϕ1 (x, y ), tκ(x)ξ with ϕ1 (x, y ) = κ −1 ϕ(x, y ) and changing variables ξold = κ  (x)ξ we get (2πh)−d



  −1 a κ(x), tκ −1 (x)ξ e ih x−y +ϕ1 (x,y ),ξ × | det κ  (y )| · | det κ  (x)|−1 u(y ) dydξ.

Note that e ih

−1 ϕ

1 (x,y ),ξ

∼

1 h−n ϕ1 (x, y ), ξn n! n

with each term O(|x − y |2n h−n ) and the similar error estimate and then integrating 2n times by parts with respect to ξ we get also factor h2n ; however these 2n derivatives could be applied to a symbol which leads to the factor h2n(σ−1) in the estimates7) ; so the total factor in estimates is hn(2σ−1) . Noting that σ > 1/2 we prove that what we got is an asymptotic expansion. Further, these arguments imply that it is sufficient to prove (1.1.202) for operators D β only; however for them this formula is obvious. Corollary 1.1.63. Let a ∈ Sm cl . Then 7)

Differentiating power of ϕ1 (x, y ), ξ causes no extra powers of h.

1.1. PSEUDODIFFERENTIAL OPERATORS (i) For pq-quantization (1.1.203)

a(κ)n =



59

(Ln−k ak ) ◦ Ψ

0≤k≤n

where Ψ : T ∗ G → T ∗ G  is a symplectomorphism, induced by κ: (1.1.204)

Ψ(x, ξ) = (κ(x),tκ −1 (x)ξ),

Lk are linear 2k-th degree differential operators with respect to ξ only, depending on κ; in particular (1.1.205)

L0 = I ,

Lk 1 = 0

∀k ≥ 1.

(ii) For any method of quantization L0 = I and (1.1.206) Lk = lkαβ ∂ξα ∂xβ

∀k ≥ 1.

|β| 0;

further, if we want negligible operators to act from S (Rd ) to S(Rd ) we need a condition (Φφ)(x, ξ) ≥ (1 + |ξ|2 + |x|2 )δ/2 ,

(1.1.212)

δ > 0.

Taking μ(x, ξ) = (Φφ)(x, ξ)m we get operators of order m; then principles “ord(ab) = ord(a) + ord(b)” and “ord({a, b}) = ord(a) + ord(b) − 1” hold. Without any loss of the generality we can assume that each of functions a = Φ, a = φ and a = μ satisfies (1.1.209) with a weight function in the righthand expression Φ, φ and μ respectively; otherwise one can replace them with equivalent functions satisfying this condition. Then one can introduce corres sponding spaces H s = HΦφ,μ with norms uHs =  Op(Φφ)s Op(μ)uL2 . The most general classes due to L. H¨ormander use temperate metrics instead of temperate scaling functions L. H¨ormander [1]. Global Operators In particular one can select Φ(x, ξ) = φ(x, ξ) = (1 + |x|2 + |ξ|2 )1/2 and consider global symbols of order m: (1.1.213)

||Dxα Dξβ a(x, ξ)| ≤ Cαβ (1 + |x|2 + |ξ|2 )m−(|α|+|β|)/2

∀α, β.

1.1. PSEUDODIFFERENTIAL OPERATORS

61

Operators of this class are invariant with respect to Fourier transform. One can consider classical version of such symbols: they should admit decomposition (1.1.188) with aj positively homogeneous of degrees 2(m − j) with respect to (x, ξ); operators of this class are invariant with respect to Fourier transform. The simplest example is harmonic oscillator |D|2 + |x|2 ; in the framework of this calculus it has an order 1. Slightly more general class of operators is described by scaling functions Φ(x, ξ) = (1 + |ξ|2 + |x|2ν )1/2 , φ(x, ξ) = (1 + |ξ|2 + |x|2ν )1/2ν : (1.1.214) ||Dxα Dξβ a(x, ξ)| ≤ Cαβ (1 + |x|2ν + |ξ|2 )m(1+1/ν)−|α|/(2ν)−|β|/2

∀α, β.

One can consider classical version of such symbols: the should admit decomposition (1.1.188) with aj positively (1, ν)-quasihomogeneous of degrees (m − j)(ν + 1) with respect to (x, ξ): aj (λx, λν ξ) = λ(m−j)(ν+1) aj (x, ξ) as λ > 0. The simplest example is an anharmonic oscillator |D|2 + |x|2ν ; in the framework of this calculus it has an order 2ν/(ν + 1). In this case one can prove statements similar to Theorems 1.1.57 and 1.1.59. Both propagation of singularities and spectral asymptotics demonstrate that discussed operators behave like operators of prescribed orders on compact manifolds. Furthermore, one can consider symbols and operators “global” with respect to some variables and ordinary with respect to others or with vector ν ∈ R+d .

h-Pseudodifferential Operators One can consider classes of operators described in this subsection and introduce semiclassical parameter h: i.e. replace operator a(x, D) by a(x, hD). Surely it affects the spaces as well: f.e. standard H s should be replaced by spaces with the norm us = (1 + h2 |D|2 )s/2 uL2 . Then effective semiclassical parameter would be  = h/(Φφ)(x, ξ) (for Beals-Fefferman operators). h-pseudodifferential operators with Sm σ,ς symbols admit a smooth change of coordinates provided 0 ≤ 1 − σ ≤ ς < σ ≤ 1; h-pseudodifferential operators with classical symbols also admit a smooth change of coordinates.

62

1.2 1.2.1

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

Lagrangian Distributions and Fourier Integral Operators Motivation

Oscillatory Solutions to Evolution Equations Let us consider evolution equation (1.2.1)

hDt u + a(x, t, hDx )u = 0

with the scalar symbol a and a real-valued a0 and let us find oscillatory solutions (1.2.2)

u(x, t) = e ih

−1 φ(x,t)

f (x, t).

Plugging (1.2.2) into (1.2.1) and applying Theorem 1.1.14 and Remark 1.1.16 we find that   (1.2.3) hDt + a(x, t, hDx ) u ∼   n  −1 h Ln f × e ih φ(x,t) a0 (x, ∇φ) + φt + n≥1

where Ln are differential operators of order n; in particular, (1.2.4) L1 = −i∂t − i (∂ξj a0 )(x, ∇φ)∂xj + K , j

where for qp-quantization (1.2.5) K =

i 2 (∂ a0 )(x, ∇φ)∂x2j xk φ + a1 (x, ∇φ) = 2 j,k ξj ξk  i (j) ∂xj a0 (x, ∇φ) + as (x, ∇φ) − 2 j

where only in the last sum we first plug ξ = ∇φ(x, t) and differentiate after; (j) recall that a0 = ∂ξj a0 . To eliminate the first term in the right hand expression of (1.2.3) we need to impose condition to φ: (1.2.6)

φt + a0 (x, ∇φ) = 0

1.2. FOURIER INTEGRAL OPERATORS

63

which is called eikonal equation as φ is called eikonal . For a simplicity of the terminology we assume that (1.2.7) a does not depend on t. Note that this is Hamilton-Jacobi equation for classical Hamiltonian dynamics ⎧ dx ⎪ ⎨ = ∇ξ a0 (x, ξ), d dt # (x, ξ) = ∇ a0 (x, ξ) ⇐⇒ (1.2.8) ⎪ dt ⎩ dξ = −∇x a0 (x, ξ), dt where ∇# b = (∇ξ b, −∇x b) is a Hamiltonian field generated by b. We refer to V. I. Arnold [2] for details. For such equations d  φ = ξ, a0ξ (x, ξ), dt

(1.2.9)

∇φ = ξ

along trajectories of (1.2.8). Equation (1.2.8) defines a Hamiltonian flow Ψt on the phase space ∗ T X  (x, ξ) (X = Rd ); however after φ is selected we have a flow Φt on the configuration space X  x: dx = ∇ξ a0 (x, ∇φ(x, t)). dt

(1.2.10)

Remark 1.2.1. It is known that Ψt is a symplectomorphism and so preserves the volume formon the phase space; however the similar statement is not (j) true for Φt and j ∂xj a0 (x, ∇φ) is a divergence of this flow: (1.2.11)



(j)

∂xj a0 (x, ∇φ) =

j

d log | det dΦt |. dt

To eliminate the second term in the right hand expression of (1.2.3) we need to impose condition to f :   ∂t + (1.2.12) (∂ξj a0 )(x, ∇φ) + iK f = 0 j

64

CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

which is equivalent to equation along trajectories of Φt (1.2.13)

 d + iK f = 0 dt

and its called transport equation. Together with (1.2.11) it implies that (1.2.14)

  d − 2 Im ias |f |2 · | det dΦt | = 0; dt

in particular for real as we get a kind of energy conservation (1.2.15)

 d  2 |f | · | det dΦt | = 0. dt

To eliminate all other term in (1.2.3) we pick up (1.2.16)

f ∼



fn h n

n≥0

where f0 we already defined and all other terms are defined recursively from transport equations (1.2.17)

d  + iK fn = −i Lr fn+1−r . dt r ≥2

Remark 1.2.2. This construction is obviously extended to the matrix operators provided a0 is scalar. Furthermore, this construction is extended even to more general matrix operators provided (1.2.18)

  dim ker φt + a0 (x, ∇φ) = const.

So, we found asymptotic solutions to a equation (1.2.1) with initial condition (1.2.19)

u|t=0 = e ih

−1 φ

0

f0 (x).

However this solution is only local because of the blow-up.

1.2. FOURIER INTEGRAL OPERATORS

65

Blow-Up The trouble is that solution to eikonal equation (1.2.6) is only local. The geometric reason is very transparent: while Ψt is a symplectomorphism, (1.2.20)

Φt : x → πx Ψt (x, ∇φ0 (x))

is a smooth map but may cease to be a diffeomorphism for |t| ≥ t0 ; here πx : (x, ξ) → x is a projection map. At the moment when Φt ceases to be a diffeomorphism the energy density |f |2 surges due to (1.2.14) and this is the reason why propagators fail to be bounded operators from Lp (Rd ) to Lp (Rd ) for p = 2. One can describe it in a different way: (1.2.21)

Λ0 = {(x, ∇φ0 (x)), x ∈ X } ⊂ T ∗ X

is a Lagrangian submanifold , i.e. submanifold of the maximal dimension (thus d) on which symplectic form (1.2.22) σ= (dξj ∧ dxj ) j

identically vanishes. Then since Ψt is a symplectomorphism Λt = Ψt Λ0 is a Lagrangian manifold as well. If πx : Λt → Rd is a local diffeomorphism one can (locally) describe Λt by (1.2.21) with φ0 replaced by φt . Blow-up happens when rank(dπx |Λt ) < d. Very geometrical discussion of blow-up (caustics, focussing) the reader can find in V. I. Arnold [3]. Extensive analytic theory could be found in V. M. Babich [1]. Maslov Canonical Operator To tackle blow-ups V. Maslov [1] (see also M. Fedoryuk, V. Maslov [1]) suggested Maslov canonical operator method. Idea of this method is to pass to the different representation near the blow-up. While we do not use this method in our book, we sketch it here. So, let us go to p-representations of (1.2.2):  −1 (1.2.23) F(u)(ξ) = (2πh)−d/2 e ih (−x,ξ +φ(x)) f (x) dx

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

where we ignore t-dependence. To calculate oscillatory integral (1.2.23) we need to apply stationary phase formula (see Appendix 1.A.3):   (1.2.24) ∇x −x, ξ + φ(x) = 0 ⇐⇒ ξ = ∇x φ(x) and under the nondegeneracy assumption (1.2.25) we get (1.2.26)

det Hess φ = 0 F(u)(ξ) ∼ e ih

−1 φ(ξ) ˜



f˜(ξ) + hf˜1 (ξ) + ...



with x(ξ) defined by (1.2.24) (locally) uniquely under assumption (1.2.25); if there are several solutions to (1.2.24) we arrive to the sum of several expressions of (1.2.26) type; further   ˜ = −ξ, x(ξ) + φ x(ξ) , (1.2.27) φ(ξ) and  πi f˜0 (ξ) = e 4 sgn Hess φ | det Hess φ|−1/2 f0 x=x(ξ) . (1.2.28) ˜ Transition from (x, φ(x)) to (ξ, φ(ξ)) described above is called Legendre transformation. Remark 1.2.3. The meaning of the second factor in (1.2.28) is also obvious in virtue of | det Hess φ| = | det(ξ  (x))| where ξ  is a Jacobi matrix of x → ξ = ∇x φ. Due to this factor when we pass from x to ξ coordinates amplitude behaves as a half-density rather than a function: |f˜|2 dξ = |f |2 dx in the corresponding points where dξ and dx are Euclidean measures on Rdx and Rdξ respectively. Finally, in the first factor of (1.2.28) sgn b is a signature of a real symmetric matrix b, which is the number of its positive eigenvalues minus number of its negative ones. This factor manifests itself prominently when we return back to x after blow-up. Similar formulae hold for partial h-Fourier transform and for (partial) inverse h-Fourier transform. Furthermore, locally on every Lagrange manifold Λ one can introduce coordinates (xJ , ξJ  ) where xJ = (xj1 , ... , xjr ), {j1 , ... , jr } = J ⊂ {1, ... , d}, J  = {1, ... , d} \ J and therefore we can always find an appropriate representation. To every such representation we

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67

attribute locally a phase function and also an amplitude; moreover, modulo factor i p (with p ∈ Z/4Z) this amplitude can be considered as half-density on Λ. To understand the role of the first factor in (1.2.28) let us consider case d = 1. Then this factor is ±i if ± dξ > 0. F.e. if Λ is as on picture below. dx ξ dξ dx

>0

Λ

dξ dx

B dξ dx

0} to dx dξ { dx < 0} increases by −2 πi and thus factor e − 2 = −i is acquired. Therefore after going counter-clockwise around Λ factor −1 is acquired.

More generally: consider multidimensional case. Let Λ be a Lagrangian sub∗  manifold in T X and z, z ∈ Λ such that rank dπx (z) = rank dπx (z  ) = d. Consider path (z  , z) from z to z  . One can prove that this path could be deformed to the path along which rank dπx = d in all but a discrete set of points z1 , ... , zr and rank dπx (zj ) = d − 1. Figure 1.1: Maslov index

Near point zj one can select k such that rank dπxkˆ ,ξk (zj ) = d; here xkˆ means “all coordinates but k-th”. At this points Maslov index of the path k k grows (decreases) by 2 if we pass from { dξ < 0} to { dξ > 0} (in the dxk dxk opposite direction, respectively). This procedure introduces Maslov index of the path (z  , z) and allows to introduce Maslov index of the closed path  which does not depend on the choice of initial point. The most important conclusion we carry from this construction, that solution for any time is expected to be (a sum of terms)

(1.2.29)

(2πh)

−m

 e ih

−1 ϕ(x,θ)

f (x, θ, h) dθ

with θ = ξJ  , ϕ(x, θ) = xJ  , θ + φ(xJ , θ), m = #J  .

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

Propagator Another approach is to consider initial phase x − y , ξ and initial function (2πh)−d and construct local solution in the form K(x, y , t) = (2πh)−d e ih

(1.2.30)

−1 (φ(x,ξ,t)−y ,ξ )

b(x, ξ, t, h) −1

and this would be a Schwartz kernel of the propagator e −ih tA . Then for any t = t1 + t2 + ... + tm we can construct a Schwartz kernel of −1 −1 −1 propagator e −ih tA = e −ih t1 A · · · e −ih tm A  −1 −dm e ih ϕ(x,θ,y ,t) b(x, θ, y , t, h) dθ (1.2.31) K(x, y , t) = (2πh) where we dumped all intermediate variables into θ = (ξ, x(1) , ξ(1) , ... , x(m−1) , ξ(m−1) ) and ϕ = φ(x, ξ, t1 ) − x(1) , ξ + φ(x(1) , ξ(1) ) − x(2) , ξ(1)  + ... − y , ξ(m−1) . −1 Further, applying it to initial function e ih ψ f0 we get a solution in the form  −1 −dm e ih ϕ(x,θ,t) f (x, θ, t, h) dθ (1.2.32) u(x, t) = (2πh) where here we dumped y into θ as well and included ψ(y ) into ϕ. Expressions (1.2.32) and (1.2.31) will be used to define Lagrangian distributions and Schwartz kernels of Fourier integral operators. Remark 1.2.4. There is a disparity between Lagrangian distributions and Lagrangian distributions which are Schwartz kernels of operators in L2 (X ): the former with m-dimensional parameter theta have factor h−m/2 while the latter have factor h−(m+d)/2 where d = dim(X ). Later we will consider Lagrangian distributions which are Schwartz kernels of operators L2 (X ) to L2 (Y ) and those will have factor h−(m+dX )/2 where dX = dim(X ) and dY = dim(Y ).

1.2.2

Lagrangian Distributions and Manifolds

Definition We start with the theory of Lagrangian distributions which are given by (1.2.32) without t and with φ satisfying non-degeneracy condition below.

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69

Definition 1.2.5. A smooth real-valued function φ(x, θ) defined for (x, θ) ∈ G  Rd × Rr is called non-degenerate if (1.2.33)

(x, θ) ∈ G¯, φθ (x, θ) = 0 =⇒ rank(φθx , φθθ ) = r

where φθx , φθθ are d × r and r × r matrices of the second derivatives, so (φθx , φθθ ) is (d + r ) × r matrix. The quantified version of (1.2.33) is (1.2.34) If |φθ (x, θ)| ≤ , then the matrix (φθx , φθθ ) has a minor (determinant) of dimension r , the absolute value of which is not less than  with a small enough constant  > 0. Let H be an auxiliary Hilbert space as before. Definition 1.2.6. An H-valued Lagrangian distribution is a distribution of the form  r −1 (1.2.35) u(x) = (2πh)− 2 a(x, θ)e ih φ(x,θ) dθ with the smooth, real-valued and nondegenerate phase function φ ∈ CN+K (G¯) and smooth amplitude a ∈ C0N+K (G , H). The quantitative versions of these conditions are: (1.2.36)

diam G ≤ c,

(1.2.37)

|Dxα Dθν φ| ≤ L ∀α, ν : |α| ≤ N + K , |ν| ≤ N + K

(1.2.38)

||Dxα Dθν a||

≤M

dist(supp(a), ∂G ) ≥ , ∀α, θ : |α| ≤ N + K , |ν| ≤ N + K

∀(x, θ) ∈ G , ∀(x, θ).

Definition 1.2.7. More generally, let us consider amplitudes of the form (1.1.30) where aj satisfy the previous conditions with N replaced by (N − j). The class of these amplitudes we denote by S˙ h,N (G , H) and the least M for which these estimates hold we denote by |||a|||. It is rather obvious that the same Lagrangian distribution could be defined (modulo O(h∞ )) by different phase functions and amplitudes and the first question to answer is “How one can characterize phase functions defining the same Lagrangian distributions”?

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Definition 1.2.8. Lagrangian manifold generated by a phase function φ is (1.2.39) where (1.2.40)

Λφ = {(x, φx (x, θ)) : (x, θ) ∈ Σφ } ⊂ T ∗ Rd Σφ = {(x, θ) ∈ G , φθ (x, θ) = 0}.

Then Λ = Λφ and Σφ are respectively CN+K −2 and CN+K −1 manifolds of dimension d. Furthermore (1.2.41) Λφ is a Lagrangian manifold which  means that dim(Λ) = d and the restriction of the symplectic form σ = j dξj ∧ dxj to Λ equals 0, (1.2.42) The map ι : Σφ  (x, θ) → (x, φx (x, θ)) ∈ Λφ is a local diffeomorphism. One can prove easily that (1.2.43)

r − rank φθθ (x, θ) = d − rank πΛ,x (ι(x, θ)),

where πΛ,x : Λ  (x, ξ) → x ∈ Rd and πΛ,x is its differential. Main Properties For a justification of all our future calculations we need Proposition 1.2.9. Let conditions (1.2.34) and (1.2.36)–(1.2.38) be fulfilled with K = K0 (d, r ), N = 0. Then the L2 (Rd , H)-norm of u does not exceed C |||a||| with C = C (d, r , c, ). Proof. The proof of this proposition follows from the arguments of Chapter 21.1 of L. H¨ormander [1]. Namely, consider  −1  2 −r (1.2.44) u = (2πh) a(x, θ), a(x, θ )e ih (φ(x,θ)−φ(x,θ )) dxdθdθ which is an oscillatory integral with the phase function ϕ(z) = φ(x, θ) − φ(x, θ ), z = (θ, θ , x) (and with the “deficient” power of (2πh)). One can see easily that the (quantified) nondegeneracy condition for φ implies (quantified) condition ϕz = 0 =⇒ rank ϕzz ≥ 2r . Then sthe tationary phase method reduces this integral to the finite sum of (oscillatory) integrals with d variables and 0-th power of (2πh).

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71

Remark 1.2.10. Since (hD)α u is a distribution of the same form again we obtain that its L2 (Rd , H)-norm also does not exceed C |||a||| with C = C (d, r , N, c, ) provided |α| ≤ N in the conditions. Then the stationary phase method immediately implies Proposition 1.2.11. Let ι be a one-to-one map, and let πΛ,x be a diffeomorphism8) . Then for K = K (d, r ) a distribution u of the form (1.2.35) can be rewritten in the following form: (1.2.45) where (1.2.46)

u(x) = a˜(x)e ih

−1 φ(x) ˜

+ U(x)

˜ φ(x) = φ(ι−1 π−1 Λ,x (x)),

a||| ≤ C |||a||| where the norms are a˜ ∈ S˙ h,N (πΛ,x ιG ) with K = K (d, 0), |||˜ calculated in the corresponding classes,  πi 1  (1.2.47) a˜0 (x) = e 4 sgn(φθθ ) | det(φθθ )|− 2 a0 (x,θ)=ι−1 π−1 (x) , Λ,x

and the L2 (Rd , H)-norm of U does not exceed ChN |||a||| with C = C (d, r , N, c, ); moreover the same is true for (hD)α U provided |α| ≤ k and we add k to K and take C also depending on k 9) . Remark 1.2.12. One can see easily that Λ = {(x, φ˜x (x)) : x ∈ πΛ,x ιG } in this case. Definition 1.2.13. A point z ∈ Λ is called regular provided rank πΛ,x (z) = d. Therefore Proposition 1.2.11 states that if all points of a Lagrangian manifold are regular then one can take r = 0 in (1.2.35) (we assume that ι is a one-to-one map and that K = K (d, r )). Let us consider the case when rank πΛ,x (z) = k with 0 ≤ k ≤ d − 1. Then there exists a partition of variables x = (x  , x  ) ∈ Rk × Rd−k (preceded I.e., let πΛ,x be a one-to-one map with rank πΛ,x = d at every point of Λ; then | det πΛ,x | ≥  at every point of Λ. 9) Such functions we call (k, N)-negligible or simply negligible. 8)

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

by a change of enumeration of the coordinates if necessary) such that if ξ = (ξ  , ξ  ) is the corresponding partition of the dual variables in Tx∗ Rd then (1.2.48)

rank πΛ,(x  ,ξ ) (z) = d

where πΛ,(x  ,ξ ) : Λ  (x, ξ) → (x  , ξ  ) and the prime means differential (i.e. we assume that the absolute value of the determinant of this matrix is not less than ). Then one can prove easily that the r ∗ × r ∗ matrix    φθθ φθx  (1.2.49) Φ= φx  θ φx  x  with r ∗ = r + d − k is non-degenerate. Let us consider function  k−d −1   e ih x ,ξ u(x) dξ  v (x  , ξ  ) = (2πh) 2 which is a partial h−1 -Fourier transform according to Maslov’s terminology. Then v is a distribution of the form (1.2.35) with r , x, θ, a(x, θ) and φ(x, θ) replaced by r ∗ , y = (x  , ξ  ), θ∗ = (θ, x  ), a∗ (y , θ∗ ) = a(x, θ) and φ∗ (y , θ∗ ) = φ(x, θ) − x  , ξ   respectively. Surely the new amplitude is not compactly supported with respect to ξ  but it can easily be improved by multiplying by the cutting function ζ(ξ  ) with a fixed ζ ∈ C0∞ (Rd−k ) supported in the ball of appropriate radius R and equal to 1 in the ball of radius 12 R; one can prove easily by the stationary phase method that the error function V as well as ξk V are negligible. Applying Proposition 1.2.11 one can rewrite v in the form (1.2.45) with the phase function       (1.2.50) S(x  , ξ  ) = φ ι−1 π−1 Λ,(x  ,ξ  ) (x , ξ ) − x , ξ    with x  calculated at the point x = πΛ,x π−1 Λ,(x  ,ξ  ) (x , ξ ) instead of the phase ˜ Hence we obtain function φ.

Proposition 1.2.14. Let conditions (1.2.36)–(1.2.38) be satisfied with K = K (d, r ); let ι be a one-to-one map and for all z ∈ Λφ . Finally. let condition (1.2.48) be fulfilled. Then one can rewrite (modulo negligible functions) a distribution u of the form (1.2.35) in the same form with θ replaced by ξ  , with the phase function (1.2.51)

ψ(x, ξ  ) = S(x, ξ  ) + x  , ξ  

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73

and with a new amplitude b satisfying the same inequality for amplitude norms, with the same constants here and in the negligibility condition as before. One can see easily that (1.2.52)

Λφ = Λψ = {(x, ξ) : x  = −Sξ  (x  , ξ  ), ξ  = Sx  (x  , ξ  )};

so despite ever changing phase function, corresponding set Σ. and dimension of θ, Lagrangian manifold Λφ stays the same. Also any distribution of the form (1.2.35) can be rewritten modulo negligible functions as the sum of distributions of the same form with r ≤ d; recall that amplitudes can be chosen compactly supported. Here assumption that ι is one-to-one is no more needed. The arguments used in the proofs of Propositions 1.2.11 and 1.2.14 imply that the terms in the decomposition of the amplitude b(x  , ξ  ) are defined by the formula (Ln−j aj ) ◦ ι−1 π−1 (1.2.53) bn = Λ,(x  ,ξ  ) 0≤j≤n

where Lj are (2j)-th order linear differential operators depending on φ. In particular, (1.2.54)

b0 = e

πi 4

sgn Φ

| det Φ|− 2 a0 ◦ ι−1 π−1 Λ,(x  ,ξ  ) . 1

This implies that Corollary 1.2.15. Classes of distributions defined by different phase functions coincide modulo negligible functions provided these functions generate the same Lagrangian manifold and we take appropriate K . On the other hand, it is well known10) that if Λ is a Lagrangian manifold and if condition (1.2.48) is fulfilled at z ∈ Λ then Λ is generated in a neighborhood of z by the phase function (1.2.51) with S defined uniquely modulo an additive constant. Therefore we conclude that 10)

L. H¨ormander [1], Chapter XXV and Fedoryuk-Maslov [1].

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

Corollary 1.2.16. For a compact subset Λ0 of the Lagrangian CM manifold Λ, the spaces I h,N,k (Rd , Λ0 , H) of Lagrangian distributions are well defined. Here we assume that in the -neighborhood of each point z ∈ Λ0 the Lagrangian manifold coincides with {lj (x, ξ) = 0 ∀j = 1, ... , d} with all the derivatives up to order K + N of the functions lj not exceeding c and with the absolute value of some d-dimensional minor of the d × 2d matrix (ljxk , ljξk )j,k=1,...,d greater than or equal to ; then locally after an appropriate partition of coordinates in Rd × Rd Lagrangian manifold Λ is defined by (1.2.52) with appropriate function S. Let us consider the density δΣ on the manifold Σ = Σφ which is the push-forward of the Dirac measure δ in Rr under the map (x, θ) → φθ (x, θ); then |dz| δΣ =  D(λ,φ )  θ   D(x,θ)

where z = (z1 , ... , zd ) is an arbitrary local coordinate system on Λ continued as C1 functions in a neighborhood of Λ and |dz| is the Lebesgue measure on Λ. Then (1.2.54) implies that 1

1

a0 δΣ2 ◦ ι−1 = a˜0 δΣ˜2 ◦ ˜ι−1 provided the manifold Λ in a neighborhood of z is generated by two phase functions φ and φ˜ with coinciding r , sgn Φ and with the coinciding values on Λ 11) and a0 and a˜0 are the leading terms of the symbols a and a˜ and ˜ define the same Lagrangian distribution (modulo negligible (a, φ) and (˜ a, φ) functions). More generally, (1.2.54) implies that the half-density (1.2.55)

a0 = e

πi 4

sgn Φ

1

a0 δΣ2 ◦ ι−1

does not depend (modulo a factor i p with p ∈ Z/4Z) on the choice of φ generating Λ where here    φθθ φθx Φ= φxθ φxx is a (d + r ) × (d + r ) matrix (see Proposition 25.1.5 of L. H¨ormander [1]). Definition 1.2.17. H-valued half-density a0 defined on Λ modulo factor i p with p ∈ Z/4Z is called principal symbol of the Lagrangian distribution u. 11) The values of the functions at Λ are well defined since Λ and Σ are identified by means of ι; one can prove easily that two phase functions with coinciding r , sgn Φ and Λ coincide at Λ provided they coincide at z.

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75

Maslov Index Locally on Λ a branch of i p can be chosen uniquely; however it is not always possible globally (see below). To avoid this “modulo factor” one can use Maslov bundle M defined on Λ; so principal symbol is actually H⊗M-valued (see L. H¨ormander [1], Definition 21.6.5). However notion of Maslov index seems to be more transparent. Namely, assume that ι : Σ → Λ one-to-one map. Let us consider some path  without self-intersections; consider two regular points z1 and z2 . Then (1.2.56)

ι−1 z sgn φθθ ι−1 z21 = 2 indM (z1 , z2 )

where indM is the Maslov index and (z1 , z2 ) is any curve from z1 to z2 . In the general case one can define Maslov index of any path and then one can select branch i p globally if and only if (1.2.57)

indM  ≡ 0

mod 4Z

for every closed curve  on Λ (more precisely, for every class [] of homotopic closed curves since Maslov indices of homotopic closed curves coincide). This condition means exactly that Maslov bundle is trivial. Since dφ = ξ, dx on Λ 12) it follows that the global unique continuous definition of a0 is impossible unless  (1.2.58) ξ, dx = 0 

for every closed curve  on Λ (more precisely, for every class [] of homotopic closed curves since these integrals for homotopic closed curves coincide): for a Lagrangian manifold the exterior form ξ, dx|Λ is closed13) . Definition 1.2.18. On the other hand, under quantization conditions (1.2.57) and (1.2.58) the symbol a0 can be chosen globally uniquely for u and conversely starting from a0 one can define u ∈ I h,N,k (Rd , Λ, H) with principal symbol a0 . The isomorphic map a0 → u is called the Maslov canonical operator . 12) 13)

In fact, dφ = φx , dx + φθ , dθ = ξ, dx. In fact, dξ, dx = −σ.

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Remark 1.2.19. The same arguments imply that if we want to consider not all the values of h ∈ (0, h0 ] but only a certain sequence tending to +0 then we can replace conditions (1.2.57) and (1.2.58) by one quantization condition  1 −1 indM [] + (2πh) (1.2.59) ξ, dx ≡ 0 mod Z 4 [] for every class [] of homotopic closed curves and for every h from this sequence. Condition (1.2.59) is called Bohr-Sommerfeld quantization condition. This condition plays a crucial role in calculation of eigenvalues of onedimensional operators. Other Properties In conclusion we give some other properties of Lagrangian distributions: Theorem 1.2.20. Let u be a distribution of the form (1.2.35) and φ, a satisfy conditions (1.2.36)–(1.2.38) with M = K + N, a ∈ S˙ h,N (G , H) with K = K (d, r , k). Moreover, let us assume that for every n ≤ s the amplitude an has a (2s − 2n + 1)-th order zero at Σφ . Then (i) There exists an amplitude a ∈ S˙ h,N (G , H) with an = 0 for n ≤ s such that |||a ||| ≤ C |||a||| and a defines u modulo negligible functions14) . (ii) The terms in the decomposition of a are defined by the formula (1.2.60) an = Lj an−s−j 0≤j≤n−s−1

where Lj are 2j-th order linear differential operators.  Proof. Obviously an ∼ |α|≥(2s−2n+1) bnα (∇θ φ)α with bnα being |α|-th order differential operators applied to an . Integration |α| times by parts concludes the proof. Theorem 1.2.21. Let u bes a distribution of the form (1.2.35) and φ satisfy condition (1.2.37) with M = K + N. Let a ∈ S˙ h,N (G , H) with K = K (d, r , k). Let P ∈ Sh,N (Rd × Rd , H , H). Then 14) More precisely, the L2 (Rd , H )-norm of (hD)α (u − v ) does not exceed Cα |||a||| · hN+1 for |α| ≤ k.

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77

(i) There exists a symbol b ∈ S˙ h,N (G , H ) such that |||b||| ≤ C |||a||| · |||P||| (we recall that all norms are calculated in the corresponding classes) and b defines a Lagrangian distribution v with the same phase function coinciding modulo negligible functions with P(x, hD, h)u 15) . Moreover, b0 (x, θ) = P0 (x, φx (x, θ))a0 (x, θ)

(1.2.61) and therefore (1.2.61)

b 0 (x, ξ) = P 0 (x, ξ)a0 (x, ξ)

on Λ.

(ii) Finally, if P 0 (x, ξ) = 0 on Λ then v = hw where w is also a Lagrangian distribution with the principal symbol c 0 = −i{P 0 , a0 } + Ka0

(1.2.62)

on Λ

where the Poisson bracket does not depend on a smooth continuation of a0 from Λ and K is a symbol depending on P 0 and Λ. Proof. (i) Statement (i) follows immediately from Theorem 1.1.14. (ii) Since all objects engaged in (1.2.62) are invariant with respect to action of unitary Fourier integral operator u → F ∗ u, P → F ∗ PF (see next subsection) then one can assume without any loss of the generality that u = b(x) and Λ = {ξ = 0}. In this case formula (1.2.62) becomes trivial. Remark 1.2.22. Obviously all these statementss remain true if we replace H-valued distributions and symbols (amplitudes) by L(H, H0 )-valued ones where H0 is also an auxiliary Hilbert space. Complex Phase The similar construction works provided φ is complex-valued function satisfying condition Im φ ≥ 0. Then one should redefine (1.2.63)

Σφ = {(x, θ) : φθ (x, θ) = 0, Im φ(x, θ) = 0}

15) More precisely, the L2 (Rd , H )-norm of (hD)α (P(x, hD, h)u − v ) does not exceed C |||a||| · |||P|||hN+1 for |α| ≤ k.

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(instead of (1.2.40)), while preserving (1.2.39) for Λ. However Λ is not a Lagrangian manifold anymore, but a “real part” of a positive Lagrangian ideal (Section 25.2 of L. H¨ormander [1]). In this case all non-degeneracy conditions are supposed to be fulfilled at the real points (i.e. of Σ and Λ). We only mention that such distributions allow to generalize construction of the oscillatory solutions for operators with scalar principal symbols with Im a0 (x, ξ) ≤ 0 and with initial data of the same type as before. Then Hamiltonian trajectories while going into complex domain where Im φ > 0 would never become real again and they do not contribute in Λt . Coherent States Out of all such Lagrangian distributions a special role is played by those with φ(x, θ) = φ(x) with (1.2.64)

Im φ(x) ≥ |x − y |2

(i.e. φ = 0 ⇐⇒ x = y and moreover Im φ (y ) > 0). In this case (1.2.65)

Λ = {(y , ∇φ(y ))}

consists of a single point. The simplest example of such distribution is the one with quadratic phase functions (1.2.66)

1 φ(x) = φ¯ + x − y , η + Q(x − y ), (x − y ) 2

where Q is symmetric matrix and Im Q is positive. Lagrangian distribution with the general phase function φ(x) satisfying (1.2.64) could be reduced to one with the quadratic phase function and with amplitudes (1.2.67) fh (x) ∼ cαn (x − y )α h−n α,n≤|α|/3

where here n ∈ Z may be negative. Such functions are called Gaussian beams (due to the obvious reason) or coherent states (not sure why). However, we add an extra factor h−d/4 since (1.2.68)

e ih

−1 φ

f  ≤ Chd/4

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79

provided φ satisfies (1.2.64). The remarkable property of such distributions is that such form is preserved in the propagation with the scalar real-valued a0 for arbitrarily ¯ evolve long time (despite all the caustics). In this case y (t), η(t) and φ(t) according to the Hamiltonian equation (1.2.8) (with (y , η) instead of (x, ξ)) ¯ = η(t), dy (t) and Q(t) evolves according to and d φ(t) (1.2.69)

d Q(t) = −Q(t)A(t)Q(t) − B(t)Q(t) − Q(t)B T (t) − C (t), dt

with 0  (y (t), η(t)), (1.2.70) A(t) = aηη

B(t) = ay0 η (y (t), η(t)), 0  C (t) = ayy (y (t)η(t)),

where B T (t) = ay0 η (y (t), η(t)) means transposed matrix. The proof of (1.2.69)–(1.2.70) is based on the linearization of the Hamiltonian system at (y (t), η(t)) (1.2.71)

d δx = A(t)δx + B(t)δξ, dt

which preserves both forms (1.2.72) δξj ∧ δxj

and

d δξ = −B T (t)δx − C (t)δξ, dt

Im



j

δξj · δxj† ;

j

and therefore on the tangent plane to Λt the first one is 0 and the second one is positive definite and therefore this plane is given by δξ = Q(t)δx with symmetric Q(t), Im Q(t) > 0. Coherent states surely look more like particles than waves. Another (closely related) property is that T u is a coherent state for any Fourier integral operator T (see next subsection).

1.2.3

Fourier Integral Operators

Let us now proceed to the theory of h-Fourier integral operators. Generally speaking (see Section 25.2 of L. H¨ormander [1]) Fourier integral operators are operators whose Schwartz kernels are Lagrangian distributions in Rd × Rm corresponding to Lagrangian manifold ΛT with respect to symplectic form

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σ = σX + σY . However we consider a slightly more narrow but far the most important (and the only one we need in this book) case of Λ being a canonical graph. In this case automatically m = d. Definition 1.2.23. A Lagrangian manifold Λ in X ×Y = Rd ×Rd equipped with a symplectic form (1.2.73) σX − σY = dξj ∧ dxj − dηj ∧ dyj j

j

is a canonical graph, if projectors πX : Λ → T ∗ X and πY : Λ → T ∗ Y ∗ ∗ are local diffeomorphisms and therefore ΨπX π−1 Y : T Y → T X is a local diffeomorphism (and then Λ = graph(Ψ) is its graph). ∗ On the other hand, let X and Y be domains in phase spaces T  X and T ∗ Y  equipped with symplectic forms σX = j dξj ∧ dxj and σY = j dηj ∧ dyj respectively; here X = Rd  x and Y = Rd  y . Let Ψ : Y → X be a diffeomorphism such that

(1.2.74)

|Dyα,η Ψ(y , η)| ≤ c

∀α : |α| ≤ M

∀(y , η) ∈ Y

and the same inequalities remain true for the inverse diffeomorphism Ψ−1 and (x, ξ) ∈ X . Remark 1.2.24. Then the following statements are equivalent: (i) Ψ is a symplectomorphism, i.e., Ψ preserves the symplectic structure (Ψ∗ σY = σX ) or (what is the same) Ψ preserves Poisson brackets ({f ◦ Ψ, g ◦ Ψ} = {f , g } ◦ Ψ for all functions f , g on ΩX ; one need check only f , g ∈ {x1 , ... , xd , ξ1 , ... , ξd }). (ii) Λ = graph(Ψ) is a Lagrangian manifold with respect to the symplectic form σ = σX −σY or (what is the same) ΛT = {(x, ξ, y , η) : (x, ξ, y , −η) ∈ Λ} is a Lagrangian manifold with respect to the symplectic form σ T = σX + σY . Definition 1.2.25. Let Ψ : Y → X be a symplectomorphism, Λ its graph, ¯ \ Λ) ≥  > 0. A Fourier inteΛ0 a compact subset of Λ with dist(Λ0 , Λ gral operator is an operator A with Schwartz kernel KA (x, y ) such that  (2πh)d/2 KA (x, y ) ∈ I h,N,k Rd × Rd , ΛT0 , L(H , H) .

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81

On the other hand, consider φ(x, y , θ) is a phase function satisfying (1.2.36)–(1.2.37) with (x, y ) instead of x and generating Lagrangian manifold ΛT then  −1 − r +d 2 a(x, y , θ)e ih φ(x,y ,θ) dθ (1.2.75) KA (x, y ) = (2πh) with a ∈ S˙ h,N (G0 , L(H , H)) where G0 is an admissible domain in Rd ×Rd ×Rr . Then locally  (1.2.76) Λ = (x, φx (x, y , θ), y , −φy (x, y , θ)) : φθ (x, y , θ) = 0 . Example 1.2.26. (i) If the differential of the map Λ  (x, ξ, y , η) → (x, η) is non-degenerate then we can take φ(x, y , θ) = S(x, η) − y , η with θ = η and then  d −1 a(x, y , η)e ih (S(x,η)−y ,η ) dη (1.2.77) KA (x, y ) = (2πh)− 2 and in this case Λ = {(x, Sx (x, η), Sη (x, η), η)}. (ii) In particular, for S(x, η) = y , η we obtain an h-pseudodifferential operator with compactly supported symbol. (iii) The map Ψ : (y , η) = (y  , y  , η  , η  ) → (y  , η  , η  , −y  ) with y  ∈ Rk , y  ∈ Rd−k corresponds to the phase function φ(x, y , η  ) = x  − y  , η   − x  , y   and for a symbol equal to 1 we obtain the operator which is the partial h-Fourier transform:  −1    (k−d)/2 e −ih x ,y u(x  , y  ) dy  (1.2.78) (F u)(x) = (2πh) (surely 1 is not a compactly supported function but we can save this by multiplying F  on the left or right by an h-pseudodifferential operator with compactly supported symbol (see Theorem 1.2.29(ii) below)). With this choice of the scalar factor F  is an unitary operator. (iv) Suppose U and V are domains in X and Y respectively and ψ : V → U is a diffeomorphism; let us introduce the operator A : u → P(fug ◦ ψ −1 ) with f , g ∈ C0M (V) and an h-pseudodifferential operator P with compactly supported symbol. Then A corresponds to the symplectomorphism Ψ : (y , η) → (ψ(y ),t ψ −1 (y )η) where the notation of Section 1.1 is used.

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(v) The special case of a symplectomorphism described in (i) is a map (y , η) → (y , η − ψ  ) with a real-valued scalar function ψ; the corresponding −1 −1 Fourier integral operator is of the form Pe ih ψ (or e ih ψ P) where P is an h-pseudodifferential operator with compactly supported symbol. Remark 1.2.27. Egorov’s theorem (see L. H¨ormander [1] states that every symplectomorphism locally is a composition of those described in Example 1.2.26(i), (iii) and (iv); moreover, microlocally (see the next Subsection) every Fourier integral operator is a product of those described in (i), (iii) and (iv). Theorem 1.2.28. Let φ(x, y , θ) be a phase function which satisfies conditions (1.2.36)–(1.2.37) and suppose Λ is the Lagrangian manifold given by (1.2.76). Assume that Λ is a canonical graph, i.e. (1.2.79) At every point (x, ξ, y , η) ∈ Λ the differentials of the maps πX : Λ  (x, ξ, y , η) → (x, ξ) and πY : Λ  (x, ξ, y , η) → (y , η) have rank 2d 16) . Further, let a ∈ S˙ h,N (G , L(H , H)) with N = 0, K = K (d, r ) and A is an operator with Schwartz kernel defined by (1.2.75).   Then A ≤ C |||a||| where . means the L L2 (Rd , H ), L2 (Rd , H) -norm and C = C (d, r , c, ) 17) . Proof. By the arguments of Section 25.2 of L. H¨ormander [1]. First, due to Theorem 1.2.29(i),(ii) below A∗ A is an h-Fourier integral operator with the identical canonical map, i.e. pseudodifferential operator. Applying Theorem 1.1.19 we arrive to the statement. Applying this Theorem one can prove easily that all the statements of Proposition 1.2.14, Corollaries 1.2.15 and 1.2.16 and Theorems 1.2.20 and 1.2.21 (with P staying either to the left or to the right of A) remain true for Fourier integral operators; one should only replace (1.2.35) by (1.2.74) and  the L2 (Rd , H)-norm by the L L2 (Rd , H ), L2 (Rd , H) -norm and in (1.2.61)

   φθθ φθ x Equivalently, at every point (x, y , θ) ∈ Σφ the matrix is non-degenerate. φxθ φxx     φθθ φθx | ≥  > 0 ∀(x, y , θ) ∈ Σφ . In quantitative terms: | det φxθ φxx 17) Moreover, the same is true for the operator (hD)α A(hD)β provided |α| + |β| ≤ k and K is replaced by K + k; in the conditions and C depends on k now. 16)

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83

one should replace P0 (x, φx ) by P0 (y , −φy ) in the case when P is to the right of A; formulas (1.2.61) and (1.2.62) should be changed in the same manner. Moreover, we obtain the following statements which one can prove in the same manner Theorem 1.2.29. (i) Let A be a Fourier integral operator with the phase function φ(x, y , θ) and symbol a(x, y , θ). Then the adjoint operator A∗ is also a Fourier integral operator with the phase function −φ(y , x, θ) and symbol a† (y , x, θ) and (1.2.80)

ΨA∗ = Ψ−1 A

where here and below ΨA is the symplectomorphism corresponding to A. (ii) Let A and B be Fourier integral operators with the phase functions φA (x, y , θ ) and φB (y , z, θ ) and symbols a(x, y , θ ) and b(y , z, θ ) respectively. Then AB is also a Fourier integral operator with phase function φA (x, z, θ ) + φB (z, y , θ ) and symbol a(x, z, θ )b(z, y , θ ), where now θ = (θ , z, θ ) ∈ Rr , r = r  + d + r  . Moreover, if φA and φB satisfy (1.2.36), (1.2.37) and (1.2.79) then the new phase function also satisfies those conditions (maybe with other constants c and ) and (1.2.81)

ΨAB = ΨA ◦ ΨB .

Further, if a ∈ S˙ h,N (G  , H , H ) and b ∈ S˙ h,N (G  , H , H) respectively and we remove from with degree of h greater than   the product ab all terms N then the L L2 (Rd , H ), L2 (Rd , H) -norm of the error does not exceed C |||a||| · |||b|||hN+1 ; moreover, footnote 17) remains true. Furthermore, (1.2.82)

c 0 (x, ξ, z, ζ) = a0 (x, ξ, y , η)b0 (y , η, z, ζ)

for (x, ξ) = ΨA (y , η), (y , η) = ΨB (z, ζ). (iii) Let A be a Fourier integral operator with phase function φ(x, y , θ ) which satisfies (1.2.36), (1.2.37) and (1.2.79) and with symbol a(x, y , θ ) and let u be a Lagrangian distribution with phase function ψ(y , θ ) which satisfies (1.2.36), (1.2.37) and (1.2.33) and with amplitude b(y , θ ).

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

Then Au is a Lagrangian distribution with phase function ψ  (x, θ) = φ(x, y , θ ) + ψ(y , θ ) and amplitude b(x, θ) = a(x, y , θ )b(y , θ ), where θ = (θ , y , θ ) ∈ Rr , r = r  + d + r  . Moreover, ψ  (x, θ) satisfies (1.2.36), (1.2.37) and (1.2.33) and (1.2.83)

ΛAu = ΨA (Λu )

where here Λu is the Lagrangian manifold corresponding to the Lagrangian distribution u. Further, if a ∈ S˙ h,N (G  , H , H) and b ∈ S˙ h,N (G  , H) and we remove from the product ab all terms with degree of h greater than N then the L2 (Rd , H )norm of the error does not exceed C |||a||| · |||b|||hN+1 ; moreover, the natural modification of footnote 17) remains true. Furthermore (1.2.82)

d 0 (x, ξ) = a0 (x, ξ, y , η)b0 (y , η)

for (x, ξ) = ΨA (y , η) where c 0 and d 0 are the principal symbols of AB and Au respectively. Theorem 1.2.30. Let A be a Fourier integral operator with phase function φ satisfying (1.2.36), (1.2.37) and (1.2.79) with with K = K (d, r ) and with symbol a ∈ S˙ h,N (G , H , H) with K = K (d, r ). Moreover, let p ∈ Sh,N (Rd × Rd , H , H ) and q ∈ Sh,N (Rd × Rd , H, H ). Then there exist symbols b ∈ S˙ h,N (G , H , H) and b  ∈ S˙ h,N (G , H , H )  with and  2|||b|||d≤ C |||P|||2· |||a|||  |||b ||| ≤ C |||P||| · |||a||| such that the d LL (R , H ), L (R , H) -norm of the operator p(x, hD, h)A − B and the L L2 (Rd , H ), L2 (Rd , H ) -norm of the operator Aq(x, hD, h) − B  do not exceed C |||P||| · |||a|||hN+1 and, moreover, footnote 17) remains true. Here B and B  are Fourier integral operators with phase function φ and symbols b and b  respectively. Moreover, (1.2.84)

b0 (x, y , θ) = p0 (x, φx (x, θ)) · a0 (x, y , θ),

(1.2.84) and hence (1.2.85)

b0 (x, y , θ) = a0 (x, y , θ) · q0 (x, −φy (x, θ)) b 0 (x, ξ, y , η) = p0 (x, ξ) · a0 (x, ξ, y , η),

(1.2.85)

b0 (x, ξ, y , η) = a0 (x, ξ, y , η) · q0 (y , η)

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85

where the principal symbol of a Fourier integral operator equals the principal symbol of its Schwartz kernel at the T -transposed point. In what follows we assume that the domains and ranges of all symbols and operators agree. Corollary 1.2.31. (i) Let A and B be Fourier integral operators with symplectomorphisms Ψ and Ψ−1 respectively and let p(x, hD, h) be an hpseudodifferential operator. Then modulo negligible operators q(x, hD, h) ≡ Bp(x, hD, h)A is also an h-pseudodifferential operator and Ln−j (pj ) ◦ Ψ (1.2.86) qn = 0≤j≤n

where Lj are (2j)-th degree linear differential operators of the form Lj (q) =



α  cjkα (Dx,ξ q)cjkα

k,|α|≤2j  with operator-valued coefficients cjkα and cjkα (for scalar A and B we obtain scalar differential operators); in particular,

(1.2.87)

q0 (x, ξ) = b 0 ((x, ξ), Ψ(x, ξ)) · p0 (Ψ(x, ξ)) · a0 (Ψ(x, ξ), (x, ξ)).

(ii) Assume that the operator A is elliptic for (x, ξ, y , η) with (x, ξ) = ΨA (y , η) and (y , η) ∈ Y   Y, i.e., a0 (x, ξ, y , η) is invertible and the norm of the inverse symbol does not exceed c (as well as |||a|||). Then there exists a Fourier integral operator B corresponding to the  symplectomorphism Ψ−1 A such that BA ≡ I in Y (i.e., the equality is true modulo the sum of an h-pseudodifferential operator with symbol vanishing in this neighborhood and a negligible operator) and AB ≡ I in X  = ΨA (Y  ). In particular, if A is a scalar operator and P = p(x, hD, h) is an hpseudodifferential operator then in Y for the principal symbol of Q ≡ BPA the following equality holds: (1.2.88)

q0 = p0 ◦ Ψ.

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

Let us note that for a given symplectomorphism Ψ and domain Y  an elliptic symbol does not necessarily exist because of obstacles of a topological nature. However, such symbols certainly exist (even in the scalar case) provided quantization conditions (1.2.57) and (1.2.58) are fulfilled; in particular, such symbols exist provided Y  is simply connected. In this case A∗ A is an h-pseudodifferential operator with a Hermitian symbol and with the positive definite principal symbol in the vicinity of Y  . Then we arrive to Theorem 1.2.32. Let Ψ : Y → X be a local symplectomorphism satisfying condition (1.2.74) with graph satisfying (1.2.57) and (1.2.58). Suppose Y   Y and let X  = Ψ(Y  ). Then there exists a scalar Fourier integral operator A corresponding to Ψ such that A∗ A ≡ I and AA∗ ≡ I in neighborhoods of Y  and X  respectively. One can easily rewrite all these assertions in the uniform form which we used before. Using a partition of unity in Rd × Rd and Corollary 1.2.31 one can prove Theorem 1.2.33. Let A and B be scalar Fourier integral operators with symplectomorphisms Ψ and Ψ−1 respectively and let p be a symbol supported in G  Rd × Rd belonging to Sh,N (Rd , H , H) after rescaling x → x/ρ, ξ → ξ/ρ, ρ ∈ (0, 1]. Let  = hρ−2 ∈ (0, 1]. Then there exists symbol supported in G  = Ψ(G ) and belonging to Sh,N (G  , H , H) after the same rescaling such that |||q|||ρ ≤ C |||p|||ρ (where |||.|||ρ denotes the norm of the rescaled symbol) (1.2.89)

q(x, hD, h) − Bp(x, hD, h)A ≤ C ρ−4d N+1 |||p|||

with C = C (d, c, N, G , A, B) and K = K (d) in the conditions to Ψ, A, B. Moreover, footnote 17) remains true. Remark 1.2.34. (i) One can easily replace in (1.2.89) factor ρ−4d by the factor (ρ−2d mes supp(P) + 1). I do not know whether it is possible to replace the factor ρ−4d by 1 here. However we have no need of these improvements. (ii) Since when transforming pseudodifferential operators by Fourier integral operators we mess x and ξ, in this and similar theorems one needs to replace in the definition of Sh,.,.,N range |α| ≤ N + K , |β| ≤ N + K by |α| + |β| ≤ 2N + K .

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87

However we are not concerned about N and we will skip N in notations at all (or assume N = ∞). The following statement is well known (see, e.g., L. H¨ormander [1]): Theorem 1.2.35. Let A ∈ Ψh,N (Rd , H, H) and suppose its principal symbol a0 is real-valued and scalar. Let P ∈ Ψh,N (Rd , H, H) be an operator with −1 −1 compactly supported symbol. Then e ith A P and Pe ith A are Fourier integral operators (for any t ∈ R) with symplectomorphism Ψ = Φt where Φt = exp(tHa0 ) is the Hamiltonian flow generated by the Hamiltonian field −∇# a0 ; recall that ∇# a0 = (∇ξ a0 , −∇x a0 ). In conclusion let us note that the notations for negligible functions and negligible operators do not agree completely: if hd/2 KA is an (N, k)-negligible function then A is an (N − d2 , k)-negligible operator; conversely, if A is an (N, k)-negligible operator then hd/2 KA is an (N, k − " d2 # − 1)-negligible function. However this fact should not lead to confusion. Final Remarks −1

Remark 1.2.36. (i) So far we constructed propagator U(t) = e −ih tA of operator A which does not depend on t. Construction of propagator U(t, t  ) instead of U(t − t  ) of t-dependent A is done by a similar way. (ii) As in Subsection 1.2.2 we considered Lagrangian distributions with the complex phase, we can construct h-Fourier integral operators with the complex phase. They appear as parametrix for operators with scalar principal symbol a with Im a ≤ 0. (iii) Since any function could be decomposed into coherent states due to  1 −1 −1 2 e ih x,ξ − 2 h |x−z| dξdz (1.2.90) δ(x) = (2πh)−3d/2 one can construct propagator constructing solutions with the initial data 1 −1 −1 2 e ih x−y ,ξ − 2 h |x−y −z| (instead of the usual procedure of plane waves decomposition. (iv) One can construct the theory for “analytic” phase functions and symbols using Fourier-Bros-Iagolnitzer transform (see f.e. A. Martinez [1]). (v) Non semiclassical theory is well known.

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1.2.4

Metaplectic Operators

Let us consider operators with quadratic symbols:  1  αij xi xj + αij (xi hDj + hDj ai ) + αij h2 Di Dj 2 i,j  1  αaij xi xj + 2αij xi ξj + αij ξi ξj a = σ(Fz, z) = 2 i,j

(1.2.91)1 A = σ(FZ , Z ) = (1.2.92)1

where σ(z, z  ) = ξ, x   − x, ξ   for z = (x, ξ) and z  = (x  , ξ  ) is the skew-symmetric bilinear symplectic form, Z = (x, hD) is a vector-valued operator from H 1 (Rd ) := {u : u, xj u, Dj u ∈ L2 (Rd , C2d ) ∀j = 1, ... , d} −t α −α to L2 (Rd , C2d ), F = is the skew-Hessian of a, α = (αij ), α α     α = (αij ), α = (αij ) are d × d matrices and (α), (α ) are assumed to be symmetric and all the coefficients here and below are assumed to be real. Moreover, let us consider operators with linear symbols: A = σ(, Z ) = (1.2.91) 1 (αj xj + αj hDj ), , 2

j

(1.2.92) 1 2

a = σ(, z) =



(αj xj + αj ξj ),

j

where  = (−α, α ), α = (α1 , ... , αd ), α = (α1 , ... , αd ) and the subscript means the genuine order of the operator in the framework of the “correct” calculus. One can see easily that (1.2.93)

σ(Fz, z  ) = −σ(z, Fz  ),

(1.2.94)

σ(∇# f , ∇# g ) = {f , g }

∀z, z 

and (1.2.95) ∇# a = −Fz, ∇# a = − for quadratic and linear symbols respectively. Let A1 and A 1 be the spaces of operators with real-valued quadratic and 2 real-valued linear symbols respectively and let A0 = R. Then

1.2. FOURIER INTEGRAL OPERATORS

89

˜ = A1 ⊕ A 1 ⊕ A0 are Lie algebras with Lie operation (1.2.96) A = A1 and A 2 ˜ is a graded Lie algebra) A, B → ih−1 [A, B] (moreover, A and (1.2.97)

dim A = d(2d + 1),

˜ = (d + 1)(2d + 1). dim A

On the other hand, there are the natural isomorphisms Aj % Bj , etc., where B1 is the space of all real 2d × 2d matrices satisfying (1.2.93), B 1 = Rd , B0 = R equipped by induced Lie operations F , F  → −[F , F  ] and 2 ˜ respectively. (F ; ; c), (F  ;  ; c  ) → (−[F , F  ]; −Fl  + F  l; σ(l, l  )) on B and B The following statements are either well known or easy (see, e.g., Leray [1]): ˜ and D(A) = {u ∈ L2 (Rd ) : Au ∈ Proposition 1.2.37. (i) Let A ∈ A 2 d L (R )}. Then A is a self-adjoint operator. ˜ of all operators of the form e ih−1 A with A ∈ A, A ∈ A ˜ (ii) The sets G and G ˜ = (d + 1)(2d + 1). are Lie groups and dim G = d(2d + 1), dim G (iii) The set G of all matrices of the form exp(−F ) with F ∈ B coincides with the Lie group of all symplectic matrices (i.e., matrices S such that σ(Sz, Sz  ) = σ(z, ζ  )); further dim G = d(2d + 1). Moreover, the set G˜ of all transformations of the form exp(−F˜)|R2d with ˜ ˜ coincides with the Lie group of all symplectic affine transformations F ∈B 2d of R (i.e., transformations of the form z → Sz + L with S ∈ G, L ∈ R2d ). For F˜ = (F ; ; c) we have  1 S = exp(−F ), L = exp(−tF ) dt; 0

further, dim G˜ = d(2d + 3). ˜ %B ˜ generate natural maps θ : G → G (iv) The isomorphisms A % B and A ˜ ˜ ˜ ˜ and θ : G → G where θ is two-sheeted covering and for any g ∈ G, −1 1 ˜ θ g % S ⊂ C. ˜ and let P = p w (x, D) be a pseudodifferential (v) Suppose G ∈ G or G ∈ G operator. Then G ∗ PG = q w (x, D), q(x, ξ) = q(S(x, ξ)) ˜ ) ∈ G˜ respectively. where S = θ(G ) ∈ G or S = θ(G (1.2.98)

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One can find the complete description of spectra of such operators in M. Zaretskaya [1]. ˜ and let P be an hProposition 1.2.38. (i) Let G ∈ G (or G ∈ G) pseudodifferential operator with compactly supported symbol. Then PG and GP are Fourier integral operators with symplectomorphism S = θ(G ) (or ˜ ) respectively). They can be defined by a quadratic phase function S = θ(G (with linear part in the second case). Moreover, GZ = (SZ )G .

(1.2.99)

(ii) On the other hand, (1.2.99) with linear (or affine) transformation S ˜ ) respectively). ˜ S = θ(G yields that G ∈ G, S = θ(G ) (or G ∈ G, Definition 1.2.39. (i) Elements of G are metaplectic operators. ˜ are generalized metaplectic operators. (ii) Elements of G Example 1.2.40. (i) Partial h-Fourier transforms (with an appropriate numerical factor) are metaplectic operators. Further, operators of the form (Gu)(x) = | det β|1/2 u(βx) with nondegenerate matrix β are metaplectic operators. 1

Finally, operators of the form (Gu)(x) = e 2 ih matrix W are metaplectic operators.

−1 Wx,x

u(x) with symmetric

Conversely, all metaplectic operators are products of operators of these three types. −1

(ii) Operators of the form (Gu)(x) = e ih (bx +φ) u(x −a) are (all) generalized metaplectic operators with linear phase functions. (iii) All generalized metaplectic operators are products of operators of form (i) and (ii).

1.2.5

Germs

Lemma 1.1.17 yields the following

1.2. FOURIER INTEGRAL OPERATORS

91

Proposition 1.2.41. Let X = Rd and pj (x, ξ) with j = 1, ... , s − 1 be scalar linear functions on T ∗ X . Then (1.2.100)

(p1 · · · ps )w (x, hD) =

1 w pj1 (x, hD) · · · pjws (x, hD) s! |J|=s

where the summation is taken over all the permutations J. Definition 1.2.42. The right-hand expression of (1.2.100) is a symmetric product of (operators) p1w (x, hD), ... , psw (x, hD). Moreover (1.1.37)3 implies Proposition 1.2.43. Let pj (x, ξ, h), j = 1, ... , s > 1 be scalar symbols and ¯ ∈ T ∗ Rd . Let q be the Weyl symbol of the suppose pj0 (x, ξ) vanish at (¯ x , ξ) symmetric product of p1w (x, hD, h), ... , psw (x, hD, h). Then α ¯ =0 qn (¯ x , ξ) Dx,ξ

(1.2.101)

∀n ≥ 1, α : |α| + 2n ≤ s.

Remark 1.2.44. These equalities remain true for other types of quantization only provided |α| + 2n < s. Let pj (x, ξ, h) be the symbols described in Proposition 1.2.43 and let p˜j (x, ξ, h) be Weyl symbols of Bpjw (x, hD, h)A where A and B are Fourier integral operators corresponding to symplectomorphisms Ψ and Ψ−1 respec¯ and Y  (¯ ¯ tively such that AB ≡ I and BA ≡ I in X  (¯ x , ξ) y , η¯) = Ψ−1 (¯ x , ξ) respectively. Then symbols p˜j0 = pj0 ◦ Ψ vanish at (¯ y , η¯). On the other hand,    w  pj (x, hD, h) A Bpjw (x, hD, h)A ≡ B 0≤j≤s

0≤j≤s

and the same is also true for a symmetric product. Applying Corollary 1.2.31(i) and Pproposition 1.2.43 to both sides of the latter equality we obtain that the degrees of the operators Lj in (1.2.86) do not exceed (2j − 1) for j ≥ 1. Moreover, the general case of Corollary 1.2.31(i) can be easily reduced to the case considered (when A and B are scalar and mutually inverse in the indicated sense) by replacing A and B by A = AR1 and B  = R2 B respectively with h-pseudodifferential operators R1 and R2 . So we arrive to

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92

Proposition 1.2.45. For Weyl quantization Lj in (1.2.86) (and hence in (1.2.74) also) are (2j − 1)-th degree operators for j ≥ 1. Remark 1.2.46. One can observe easily that this assertion fails for other types of the quantization (it fails even for j = 1). Definition 1.2.47. (i) Let P = p(x, hD, h) be an operator with symbol p such that (1.2.102)

¯ =0 ∂xα ∂ξβ pn (¯ x , ξ)

∀α, β, n : l|α| + r |β| + n(l + r ) < m

¯ ∈ T ∗ Rd 18) . Then the operator x , ξ) in Rd where l > 0, r > 0, (¯ (1.2.103) q(z, hDz , h) :=



α,β,n:l|α|+r |β|+n(l+r )=m

1  α β  ¯ x , ξ)αβ (z, hDz )hn ∂ ∂ pn (¯ α!β! x ξ

¯ here αβ (z, ζ) = z α ζ β is called an (l, r )-germ of operator P at the point (¯ x , ξ); and we use the same type of quantization for p and z α ζ β . (ii) This procedure is called microlocalization. One can show easily that (1.2.104) The microlocalization does not depend on the type of quantization. Remark 1.2.48. (i) One can also introduce anisotropic microlocalizations (with vector l and r ); moreover it is possible to consider the case in which some components of l and r vanish, but we have no need of it. (ii) One can see easily that the product of operators corresponds to the product of their microlocalizations; (l, r ) are supposed to be the same and then degrees (m) sum. Corollary 1.2.31, Theorem 1.1.35 and Proposition 1.2.43 imply Proposition 1.2.49. Let (1.2.102) be fulfilled. Then 18)

Without any loss of generality one can assume that l + r = 1.

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93

(i) Let A and B be scalar Fourier integral operators corresponding to symplectomorphisms Ψ and Ψ−1 respectively such that BA ≡ I on Y and AB ≡ I ¯ on X  (¯ x , ξ). ¯ Then for l = r the operator P˜ = BPA satisfies (1.2.102) at (¯ y , η¯) = Ψ(¯ x , ξ) ˜ and the germ of operator P at point (¯ y , η¯) is linked to the the germ of operator ¯ by the equality P at point (¯ x , ξ) q˜(z, hDz , h) = (q ◦ Ψ(¯y ,¯η) )(z, hDz , h) = G ∗ q(z, hDz , h)G

(1.2.105)

where Ψ(¯y ,¯η) is the differential of Ψ at (¯ y , η¯) (so Ψ(¯y ,¯η) is a linear symplectomorphism in Rd × Rd ) and G is the corresponding metaplectic operator. (ii) The above statement remains true provided l < r and Au = u ◦ κ, Bu = u ◦ κ −1 where κ is a diffeomorphism19) .

1.3 1.3.1

Wave Front Sets and Related Topics Principal nNotions

Philosophy In this book we consider various objects depending on a parameter (or parameters) m ∈ M of an arbitrary nature (i.e., we consider families of the individual objects); one of these parameters is the semiclassical parameter h ∈ (0, 1] 20) and the exact nature of other parameters is as a rule unessential21) . Objects not depending on the parameters (i.e., individual objects) are called fixed while numbers depending on parameters are also called parameters 22) . If B is a Banach space of individual objects then B is the Banach space of these objects uniformly bounded (in which case uB = supm∈M um B In this case (Gu)(x) = | det β| 2 u(βx) with β = κ  (¯ y ). Surely, h can run over a certain subset of (0, 1] with an accumulation point 0. 21) For example, for h-differential operators one can treat coefficients as parameters belonging to certain functional spaces. On the other hand, we can consider coefficients also depending on h via other parameters. 22) There is no room for any confusion: fixed numbers are called constants and some of these constants are called exponents. 19) 20)

1

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

for u = (um )m ); if λ = (λm ) is a (numerical) parameter then λB is the space of families of the form λu with u ∈ B and v λB = λ−1 uB . In what follows we speak of objects but mean families of these objects 23) . In this book as admissible we consider tempered functions, i.e., functions u ∈ h−L L2 (Rd , H) where H, H , etc., are auxiliary Hilbert spaces. Sometimes we replace H in this definition by L(H, H ) 24) . Similarly, consider tempered operators, i.e., operators  as admissible we   A ∈ h−L L L2 (Rd , H), L2 (Rd , H ) .  d  s 2 d s 2  R Functions u ∈ h L (R , H) or u ∈ h L , L(H, H ) and operators    A ∈ hs L L2 (Rd , H), L2 (Rd , H ) are called negligible; in these cases we write u ≡ 0 and A ≡ 0 respectively. Exponents l and s enter into these definitions (so in fact we speak about l-admissible or s-negligible functions, etc.). Smooth (Standard) Theory As an admissible box we consider a parallelepiped  (1.3.1) B = (x, ξ) : |xj − x¯j | ≤ γj , |ξj − ξ¯j | ≤ ρj ∀j = 1, ... , d ¯ ρ = (ρ1 , ... , ρd ), γ = (γ1 , ... , γd ) are parameters such that where (¯ x , ξ), (1.3.2)

∞ ≥ γj > 0,

∞ ≥ ρj > 0,

ρj γj ≥ h1−δ

∀j = 1, ... , d

where here and in what follows δ > 0 is always an arbitrarily small positive exponent; so an admissible box is not necessarily fixed. Definition 1.3.1. (i) Let u be an admissible function and B an admissible box. Then u is negligible in B (u ≡ 0 in B) if there exists a symbol ϕ ∈ Sh,ρ,γ,N (Rd × Rd , C) with K = K (d, l, s, δ) in the definition of the class such that ϕ = 1 in B and ϕ(x, hD, h)u ≡ 0 (this agrees with the definition given in the introduction)25) . (ii) Similarly, let A be an admissible operator and B = B × B  an admissible box. Then A is negligible in B (A ≡ 0 in B) if there exist symbols ϕ ∈ 23) This casualness does not lead to any confusion because we use the word fixed for individual objects. 24) In what follows we use plain letters instead of bold. 25) Here and below extra indices ρ, γ mean that ϕ would belong to the standard class Sh,N after rescaling xj → xj /γj , ξj → ξj /ρj .

1.3. WAVE FRONT SETS AND RELATED TOPICS 





95



Sh,ρ ,γ  ,N (Rd ×Rd , C) and ϕ ∈ Sh,ρ ,γ  ,N (Rd ×Rd , C) with K = K (d, l, s, δ) in the definition of the class such that ϕ = 1 in B  , ϕ = I in B  and Op(ϕ )A Op(ϕ ) ≡ 0. (iii) On the other hand, we call an admissible function u essentially supported in B if ϕ(x, hD, h)u ≡ u for some φ with a symbol, supported in B. Also we call an operator A essentially supported in B if Op(φ )A Op(φ ) ≡ A for some φ , φ with symbols, supported in B  and B  respectively. Remark 1.3.2. (i) Surely these definitions depend on exponents l and s; the results of Section 1.1 imply that these definitions do not depend on the choice of K large enough; (ii) Though all admissible (negligible) functions are Schwartz kernels of admissible (negligible) operators the converse assertion fails to be true. Definition 1.3.3. (i) Let u be an admissible function. Let us introduce ¯ ∈ the set WFs (u) ⊂ T ∗ Rd : (¯ x , ξ) / WFs (u) if and only if u ≡ 0 in some fixed ¯ ¯ is also fixed); box centered at (¯ x , ξ) (so, the point (¯ x , ξ) (ii) Similarly, let A be an admissible operator. Let us introduce   ¯ y¯, η¯) ∈ x , ξ, / WFs (A) iff A ≡ 0 in some fixed box WFs (A) ⊂ T ∗ (Rd × Rd ) : (¯ ¯ y¯, η¯). centered in (¯ x , ξ, (iii) In both cases WFs (.) are called wave front sets. Remark 1.3.4. (i) In the previous book I called these sets oscillation front sets and denote them as OFs to avoid confusion with the standard wave front sets. Here on the contrary I would like to emphasize similarity and I use the standard term wave front set and notation WFs . (ii) Still, there is a very essential difference: negligible functions are not smooth anymore. This leads to certain problems, usually related to the restrictions of functions. Therefore we can consider as a main space  (1.3.3) H m = u : um = (1 + h|D|)m/2 u < ∞ where m ∈ R. This leads to the modification of all our definitions and to sets WFs,m (u).

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

(iii) Respectively for operators we consider L(H m , H k ) of (uniformly with respect to h) bounded operators from H k to H m and to wave front sets WFs,m,k (A). (iv) The notion of wave (oscillation) front set is rather rough and it would be desirable to give a more refined and sophisticated definition also based on Definition 1.3.1 and leading to no difficulty in the advanced development of the theory. Maybe non-standard analysis will be useful? However we simply use Definition 1.3.1 instead of Definition 1.3.3 if the latter is too rough for our needs. Analytic Theory In the framework of the theory of operators with analytic symbols we replace the standard microlocal uncertainty principle ρj γj ≥ h1−δ by the logarithmic uncertainty principle ρj γj ≥ Cs h| log h| and thus we replace (1.3.2) by (1.3.4)

∞ ≥ γj > 0,

∞ ≥ ρj > 0,

ρj γj ≥ Ch| log h|

∀j = 1, ... , d

Remark 1.3.5. This leads to the question: can we go further? The answer is No. Microlocal Analysis starts from logarithmic uncertainty principle; see Appendix 1.A.4.

1.3.2

Properties

The results of Sections 1.1 and 1.2 imply immediately Theorem 1.3.6. (i) Let p ∈ Sh,ρ,γ,N (Rd × Rd , H , H) with K = K (d, s, δ). Then P = p(x, hD, h) is negligible in every box B = B  × B  with (1.3.5)

 B  = (x, ξ) : |xj − x¯j | ≤ γj , |ξj − ξ¯j | ≤ ρj 

B = {(y , η) : |yj − y¯j | ≤

γj ,

|ηj − η¯j | ≤

∀j = 1, ... , d},

ρj

provided there exist γj , ρj such that (1.3.6) (1.3.7)

γj  γj ,

ρj  ρj ,

min(ρj , ρj ) · min(γj , γj ) ≥ h1−δ

∀j

∀j = 1, ... , d}

1.3. WAVE FRONT SETS AND RELATED TOPICS

97

and (1.3.8)

1  1 (|xj − yj | + |xj − zj |) + (|ξj − ηj | + |ξj − ζj |) ≥  γj ρj 1≤j≤d ∀(x, ξ) ∈ B  ,

∀(y , η) ∈ B  ,

∀(z, ζ) ∈ supp(p);

recall that  is an arbitrary positive constant. (ii) Further, if p ∈ Sh,ρ,γ,an (Rd × Rd , H , H) 26) one can replace (1.3.7) by (1.3.9)

min(ρj , ρj ) · min(γj , γj ) ≥ Ch| log h|

∀j.

Theorem 1.3.7. (i) Let A be a Fourier integral operator corresponding to the symplectomorphism Ψ and let all the conditions on phase functions and symbols be fulfilled with M = K = K (d, r , s, δ) uniformly with respect to all the parameters. Then A is negligible in every box B  × B  of the form (1.3.5) such that (1.3.10)

γj  γj  γ,

ρj  ρj  ρ,

γ = ρ ≥ h(1−δ)/2 ,

    (1.3.11) dist (x, ξ), Ψ(y , η) + dist (x, ξ), (z, φz (z, z  , θ) ≥ ρ ∀(x, ξ) ∈ B  ,

∀(y , η) ∈ B  ,

∀(z, z  , θ) ∈ supp(a) ∩ {|φθ (z, z  , θ)| ≤ ρ}. (ii) Let Au = (fu) ◦ κ −1 where κ is a CK diffeomorphism from Y ⊂ Rd to X ⊂ Rd and f ∈ C0K (Y ) uniformly with respect to all the parameters, K = K (d, s, δ). Then A is negligible in every box B  × B  of the form (1.3.5) such that (1.3.12)

(1.3.13)

26)

γj  γj  γ, ρj  ρj  ρ, γ ≤ ρ, min(γ, 1) · min(ρ, 1) ≥ h1−δ ,  1 1 |x − κ(y )| + |y − z| + |t κ  ξ − η| ≥  γ ρ ∀(x, ξ) ∈ B  , ∀(y , η) ∈ B  ,

We define these classes in the manner similar to

25)

.

∀z ∈ supp(f ).

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

(iii) Let u be a Lagrangian distribution with Lagrangian manifold Λ and let all the conditions on Λ, phase functions and amplitudes be fulfilled with K = K (d, r , s, δ) uniformly with respect to all the parameters. Then u is negligible in every box B of the form (1.3.5) provided (1.3.14)

γj  γ,

ρj  ρ

∀j,

γ = ρ ≥ h(1−δ)/2 ,

  (1.3.15) dist (x, ξ), (z, φz (z, θ) ≥ ρ ∀(x, ξ) ∈ Π ∀(z, θ) ∈ supp(a) ∩ {|φθ (z, θ)| ≤ ρ}. (iv) Moreover, let Λ = {x  = ξ  = 0} be a model Lagrangian manifold where x = (x  ; x  ) = (x1 , ... , xd  ; xd  +1 , ... , xd ), etc., with 0 ≤ d  ≤ d. Then u is negligible in every box of the form (1.3.5) such that (1.3.16) (1.3.17)

min(ρj , 1) · min(γj , 1) ≥ h1−δ ∀j, 1 1 |xj | + |ξj | ≥ . γj ρj 1≤j≤d  d  +1≤j≤d

Using statement (iv) one can easily find boxes  in which operators A: (Au)(x  ) = (f (hD)u)(x  , 0) and B: (Bu)(x  ) = f (x)u(x)|x=(x  ,x  ) dx  with compactly supported smooth function f are negligible (see also Theorem 1.3.13. It is also easy to prove that wave front sets introduced here have the following properties similar to properties of the standard wave front sets: Theorem 1.3.8. WFs (.) is a closed set; conversely, if M ⊂ T ∗ Rd is a closed set and s > 0 then there exists u ∈ L2 (Rd ) such that WFs (u) = M. Theorem 1.3.9. Let u, v etc be admissible with l = 0 functions and A, B etc be admissible with l = 0 operators. Further, let Q ∈ Ψh,N be an operator with symbol supported in {|x| + |ξ| ≤ C0 }. Then for s > 0 (1.3.18)

WFs (AQu) ⊂ WFs (A) ◦ WFs (u) =  (x, ξ : ∃(y , η) ∈ WFs (u) : (x, ξ, y , η) ∈ WFs (A) ,

(1.3.19) WFs (AQB) ⊂ WFs (A) ◦ WFs (B) =  (x, ξ, z, ζ) : ∃(y , η) : (x, ξ, y , η) ∈ WFs (A), (y , η, z, ζ) ∈ WFs (B) ; (1.3.20)

WFs (u ⊗ v ) = WFs (u) × WFs (v ).

1.3. WAVE FRONT SETS AND RELATED TOPICS

99

Theorem 1.3.10. Let P = p(x, hD, h) with p ∈ Sh,ρ,γ,N (Rd × Rd , H , H) with K = K (d, s, δ) such that (1.3.21) Then (1.3.22)

min(1, ρj ) · min(1, γj ) ≥ h1−δ

∀j.

WFs (P) = {(x, ξ, x, ξ) : (x, ξ) ∈ supp(p)}

and for every admissible function u (1.3.23)

WFs (Pu) ⊂ WFs (u) ⊂ WFs (Pu) ∪ Char(P)

provided K also depends on l and Char(P) is the set of all the points (x, ξ) at which the symbol p(x, ξ, 0) is not invertible with the norm of the inverse symbol not exceeding C . Theorem 1.3.11. Let A be an admissible operator with Schwartz kernel KA . Then the following inclusions hold: (1.3.24)

WFs (A) ⊂ WFs (KA )T

and for admissible KA (1.3.25)

d

WFs (KA )T ⊂ WFs+ 4 (A)

with d = d  + d  provided KA ∈ h−l L2 . Theorem 1.3.12. (i) Let u be a Lagrangian distribution with Lagrangian manifold Λ ⊂ T ∗ Rd . Then (1.3.26)

WFs (u) ⊂ Λ

provided K = K (d, r , s); (ii) Let A be a Fourier integral operator with Lagrangian manifold Λ ⊂ T ∗ R2d . Then (1.3.27)

WFs (u) ⊂ ΛT .

Theorem 1.3.13. (i) Let a ∈ Sh,N (Rd × Rd , H , H) with K = K (d, l, s) and supp(a) ⊂ {|ξ  | ≤ c}. Let   v (x  ) = a(x, hD, h)u (x  , 0). (1.3.28) Then  d  WFs (v ) ⊂ (x  , ξ  ) : ∃ξ  : (x  , 0, ξ  , ξ  ) ∈ WFs+ 2 (u) ; (1.3.29)

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

(ii) Further, let a ∈ Sh,N (Rd × Rd , H , H) with K = K (d, l, s) and supp(a) ⊂ {|x  | ≤ c}. Let  (1.3.30) (a(x, hD, h)u)(x  , x  )dx  . v (x  ) = Rd 

Then (1.3.31)

 WFs (v ) ⊂ (x  , ξ  ) : ∃x  : (x  , x  , ξ  , 0) ∈ WFs (u) .

Remark 1.3.14. (i) Let u ∈ L2 (X , H) with X ⊂ Rd . Then Definition 1.3.1(i) remains reasonable provided Bx = {|xj − x¯j | ≤ (1 + )γj

∀j = 1, ... , d} ⊂ X

with some small positive constant ; in this case one should replace the cutting operator ϕ(x, hD, h) by ϕ(x, hD, h)ψ(x) where ψ = 1 in Π, ψ is supported in Πx and belongs to the same symbol class. Therefore Definition 1.3.3(i) remains reasonable provided dist(¯ x , X ) ≥ .  2   (ii) Let A ∈ L L (Y , H ), L2 (X , H) with X ⊂ Rd and Y ⊂ Rd then   Definition 1.3.1(ii) remains reasonable provided Bx ⊂ X and By ⊂ Y ; one    should only take cutting operators of the form Op(ϕ )ψ and ψ Op(ϕ ). So Definition 1.3.3(ii) remains reasonable provided dist(¯ x , X ) ≥  and dist(¯ y , Y ) ≥ . (iii) Let v ∈ L2 (X , H) be a fixed function. Let u = uh = φ(h|D|)v be a family of functions where φ ∈ C0∞ (R+ ), φ =  0. Then (WF(u))cone = WF(v ) where Mcone is the cone hull (with respect to ξ) of the set M and WFcl (v ) is a standard wave front set.

1.3.3

Propagator

Theorem 1.3.12(ii) and the construction of propagator as a Fourier integral operator imply immediately Theorem 1.3.15. Let A ∈ Ψh,N with a scalar principal symbol a0 , which is real. Then  −1 (1.3.32) WFs (e −ith A ) = (x, ξ; y , η) : (x, ξ) = Ψt (y , η) where Ψt is a forward Hamiltonian flow described by (1.2.8). In particular, (1.3.33)

WFs (e −ith

−1 A

f ) = Ψt ◦ WFs (f ).

1.4. ELLIPTIC BOUNDARY VALUE PROBLEMS

1.3.4

101

Alternative Theories

We are sending the reader to any Microlocal Analysis textbook for the theory of wave front sets associated with the classical pseudodifferential operators. In this case h = 1 and A ∈ Ψ1cl . Further, one can con construct the theory of global operators (whose symbols satisfy (1.1.213) or (1.1.214) with m = 1: order of operator must be 1. In this case we assume that modulo 0-th order symbol a is positively quasihomogeneous as |x| + |ξ| ≥ C . Then Hamiltonian flow generated by quasihomogeneous principal symbol acts on R2d /R where R means a group (x, ξ) → (λx, λν ξ) with any λ > 0; see V. Ivrii, [12].

1.4

Elliptic Boundary Value Problems and Associated Operators

1.4.1

Pseudodifferential Operators with Transmission Property

The purpose of this Subsection is to introduce pseudodifferential operators in the half-space and then in the domains with the smooth boundaries. Causality Consider classical double symbol a(x, y , ξ) such that (1.4.1) ||∂xα ∂yβ ∂ξγ a(x, y , ξ)|| ≤ M(1 + |ξ|)m ∀α, β, γ : |α| + |β| ≤ N + K , |γ| ≤ N + K and a vector ν ∈ Rd \ 0. Assume that (1.4.2) a(x, y , ξ + τ ν) has an analytic continuation to τ ∈ C+ 27) such that (1.4.3) ||∂xα ∂yβ ∂ξγ a(x, y , ξ + τ ν)|| ≤ M(1 + |ξ| + |τ |)m ∀α, β, γ : |α| + |β| ≤ N + K , |γ| ≤ N + K . 27)

Where we here and in what follows use notations C± := {τ ∈ C : ± τ > 0}.

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We arrive immediately to Proposition 1.4.1. Let symbol a satisfy (1.4.1)–(1.4.2) and let KA (x, y ) be 3

2

1

a Schwartz kernel of operator A = a(x, hD, x). Then  (1.4.4) supp(KA (x, y )) ⊂ (x, y ) : x − y , ν ≤ 0 . Proof. Without any loss of the generality one can assume that ν = (1, 0, ... , 0). Then the proof follows from deforming the contour of integration with respect to ξ1 in expression (1.1.18) for KA (x, y ). We refer to property (1.4.4) as acausality. One can prove easily that Proposition 1.4.2. (i) If a double symbol a satisfies (1.4.1)–(1.4.2) then the same is true for all three ordinary symbols (pq-, qp- and Weyl) of operator A. (ii) If ordinary symbols a, b satisfy (1.4.1)–(1.4.2) then the same is true for a • b for any method of quantization; (iii) If an ordinary symbol a satisfies (1.4.1)–(1.4.2) then the same is true for a† and vector −ν instead of ν. Transmission Property While causality is a useful and deserving study property, it is too restrictive for our needs. Instead we introduce transmission property essentially following L. Boutet de Monvel [1]: Definition 1.4.3. (i) Symbol a ∈ Sm σ,ς with 1 ≥ σ > ς ≥ 0 has a transmission property with respect to half-space R˙ + × Rd−1 = {x + 1 ≥ 0} if one can decompose a and all its derivatives in {|ξ1 | ≥ C |ξ  | + C }: ak (x  , ξ  )(ξ1 + i0)m−k (1.4.5) a∼ k≥1 d d−1 with ak ∈ S−k , H , H) where asymptotic decomposition (1.4.5) σ,ς (R × R means that   ak (x  , ξ  )(ξ1 + i0)m−k x1 =0 || ≤ (1.4.6) ||∂xα ∂ξγ a − 1≤k≤r −1

Mr αγ (1 + |ξ  |)r +ς|α| |ξ1 |m−r −σ|γ|

1.4. ELLIPTIC BOUNDARY VALUE PROBLEMS

103

for all r , α, γ. d  Class of such symbols is denoted by Sm σ,ς,trans (X × R , H , H) where here + d−1 := {x : x1 > 0}. X = R˙ × R

(ii) Operator A ∈ Ψm σ,ς with 1 ≥ σ > ς ≥ 0 has a transmission property with respect to a half-space X = R˙ + × Rd−1 if A ≡ Op(a) modulo infinitely smoothing operator where a ∈ Sm σ,ς satisfies (i).  Class of such operators is denoted by Ψm σ,ς,trans (X , H , H). m (iii) Finally, Sm cl,trans and Ψcl,trans denote classes of classical symbols and operators with transmission property.

Remark 1.4.4. (i) Surely one can introduce similar classes with r , |α|, ν bounded by N + K . (ii) Obviously transmission property is preserved under affine changes of variables preserving half-space R+ × Rd−1 . Therefore one can introduce transmission property with respect to any half-space X = {x : x, ν > c}. One can prove easily that Proposition 1.4.5. For X = {x : x, ν > c} the following properties hold: m2    1 (i) If A1 ∈ Ψm σ,ς,trans (X , H , H ) and A2 ∈ Ψσ,ς,trans (X , H , H) then m1 +m2  A1 A2 ∈ Ψσ,ς,trans (X , H , H).  (ii) If A ∈ Ψm σ,ς,trans (X , H , H) is elliptic and m ∈ Z then it has a parametrix −m   A ∈ Ψσ,ς,trans (X , H , H).

(iii) Therefore if A ∈ Ψm σ,ς,trans (X , H, H) and m ≤ 0 then f (A) − f (0) ∈ Ψm σ,ς,trans (X , H, H) as well for all “admissible” functions f (in the sense of the functional calculus of Section 1.1.   (iv) If A ∈ Ψm σ,ς,trans (X , H , H ) and κ is a diffeomorphism preserving X then −1 Aκ = κ∗ Aκ∗ also belongs to this class; m  ∗   (v) If A ∈ Ψm σ,ς,trans (X , H , H) with then A ∈ Ψσ,ς,trans (X , H, H ) with an  opposite half-space X = {x, ν ≤ c}.

(vi) Classes Sm and Ψm for half-space X  coincide with those for X if and only if m ∈ Z.

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Remark 1.4.6. Statement (ii) and a pseudo-locality allow us to extend the notion of transmission property to any bounded open domain X ⊂ Rd with a smooth boundary in such way that all statements of Proposition 1.4.5 still hold. We need to assume that either X is a half-space or ∂X is bounded or to assume that ∂X is “uniformly C∞ ” (the notion we do not want to exploit). Pseudodifferential Operators in a Domain However we need to explain how actually one applies such operators to functions defined in such domain (or in an open half-space). The main idea is to introduce action of operator A by (1.4.7) AX = ðX AX where X is an operator of extension from X to Rd by 0 and ðX is an operator of restriction from Rd to X . This formula would definitely brake composition and related operations. We can take different extension operators X but since we develop mainly L2 -theory we will take X = ð∗X which is continuation by 0. (1.4.8) For s ≥ 0 let us introduce H s (X ) = {u : ∃v ∈ H s (Rd ) : ðX v = u} with uHs (X ) = inf v :ðX v =u v Hs (Rd ) . As u = ðX v with v ∈ H s (Rd ) we get X u = v + = v − v − with v + = v θX (x) and v − = v θX (x) where θY (x) denotes characteristic function of Y and therefore AX u = ðX v + = ðX Av − ðX Av − . Let us consider first the case of half-space.  Proposition 1.4.7. Let X = {x : x1 > 0} and A ∈ Ψm σ,ς,trans (X , H , H). j + D1 ðX A is a bounded operator from  Then for any j ∈ Z operator  L2 R− , H s (Rd−1 , H) to L2 R+ , H s−m+j (Rd−1 , H ) .

  Obviously D1j A would be a bounded operator from L2 R, H s (H) to Proof.   L2 R− , H s−m+j (Rd−1 , H ) if we cut off it in zone {|ξ1 | ≥ C |ξ  |} replacing a(x, ξ) by a(x, ξ)ϕ(ξ1 /|ξ  |) with ϕ ∈ C0∞ (R). Therefore we need to consider only a(x, ξ) = ak (x, ξ  )(ξ1 + i0)m−k . On the other hand, operator A with  has a causality property  such symbol and therefore ðX A restricted to L2 R˙ − , H s (H) is simply 0.

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105

Observing that H j (R+ , H s−j (Rd−1 , H )) ⊂ H s (X , H )

(1.4.9)

as s ≥ 0

j≥0

we conclude that the following statement holds for half-space X = {x : x1 > 0} and therefore for any bounded domain with a smooth boundary.  Proposition 1.4.8. Let A ∈ Ψm σ,ς,trans (X , H , H). Then for any s ≥ max(m, 0) s AX is a bounded operator from H (X , H)) to H s−m (X , H)).

Now we want to consider product a AX BX . According to (1.4.7) (1.4.10)

AX BX = ðX AX ðX BX = (AB)X − ðX A(I − X ðX )BX .

One can apply proposition 1.4.7 to such operator. We investigate it later.

1.4.2

Related Operators

Boundary (Poisson) Operators Let us start from equation (1.4.11)

A(x, D)u = f

in X

where A is a differential operator of order m. To apply a pseudodifferential parametrix to it we need to rewrite it first as an equation in Rd . Consider continuation X u to Rd by 0. Then (j) (1.4.12) AX u = X f + δ∂X (x)gj (x) j≤m−1

with gj = Aj (x, D)u where Aj (x, D) are differential operators of order (m − j − 1), j = 0, ... , m − 1. More precisely, if X coincides locally with a half-space {x : x1 > 0} (j) 1 then δ∂X (x) = δ(j) (x1 ) and Aj (x, ξ) = (j+1)! Dξj+1 P(x, ξ) provided we use 1 qp-quantization. Then δ(j) (x1 )gj (x) = δ(j−k) (x1 )ð0 ∂xk1 gj (−1)k = j≤m−1

k≤j≤m−1

k≤j≤m−1

  δ(j−k) (x1 )ð0 ∂xk1 Aj (x, D)uj (−1)k

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where ð0 is an operator of restriction to ∂X . Assume that A is elliptic; let A be its parametrix. Then A ∈ Ψ−m cl,trans . Applying ðX A to (1.4.12) we arrive to   (j) (1.4.13) u ≡ ðX A X f + ðX A δ∂X (x)gj (x) j≤m−1

where the first term is exactly AX f . Consider other terms. To do this we need to analyze ðX A δ(j) (x1 )gj (x  ) where gj = ð0 Lj (x, D)u and Lj are differential operators of order (m − 1 − j). Departing from the previous considerations and notations we need to study operators of the type P : g → A(δ(j) ⊗ g ) where g = g (x  ) and A is operator of order (m − j). This operator has a Schwartz kernel      −d KP (x, y ) = (2π) bj (x, y  , ξ)e ix −y ,ξ +ix1 ξ1 dξ (1.4.14) where bj = (−∂y1 + iξ1 )j a(x, y , ξ); one can rewrite it as      1−d k(ξ  x1 , x  , y  , ξ  )e ix −y ,ξ dξ  KP (x, y ) = (2π) (1.4.15) with (1.4.16)







k(t, x , y , ξ ) = (2π)

−1





bj (x, y  , ξ)e itξ1 /ξ dξ1 .

Remark 1.4.9. While this construction works for any symbol a, for symbols with transmission property resulting function k(t, x  , y  , ξ  ) is “nice”; one can prove easily that for a ∈ Ψm−1 σ,ς,trans with m ∈ Z (1.4.17) ||Dxα Dyβ Dξγ Dtj k(t, x  , y  , ξ  )|| ≤ Cαβγjr (1 + t)−s (1 + |ξ  |)m+ς(|α|+|β|)−σ|γ|

∀α, β, γ, j, r .

Definition 1.4.10. (i) Operator with the Schwartz kernel KP (x, y  ) given modulo C∞ (X¯ × ∂X ) by (1.4.15) with k(t, x  , y  , ξ  ) satisfying (1.4.17) is called Poisson operator of order m. m Let P m σ,ς denote the class of such operators.  Further, by P cl we denote class of operators with functions k ∼ ks where ks are positively homogeneous of degrees (m − s) with respect to ξ  . m (ii) One can define P m σ,ς and P cl for any half-space and for any domain X with the smooth compact boundary in the natural way.

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107

Trace and Singular Green Operators Let us introduce other classes of operators: Definition 1.4.11. (i) Operator with the Schwartz kernel KT (x  , y ) defined modulo C∞ (∂X × X¯ ) by (1.4.15) with x1 replaced by y1 and with k(t, x  , y  , ξ  ) satisfying (1.4.17) with m replaced by (m + 1) is called trace operator of order m. m Let T m σ,ς denote the class of such operators.  Further, by T cl we denote class of operators with functions k ∼ ks where ks are positively homogeneous of degrees (m + 1 − s) with respect to ξ  . m (ii) One can define T m σ,ς and T cl for any half-space and for any domain X with the smooth compact boundary.

Sometimes Poisson operators are called coboundary operators and trace operators are called boundary operators. Definition 1.4.12. (i) Operator with the Schwartz kernel KG (x, y ) given modulo C∞ (X¯ × X¯ ) by     1−d (1.4.18) KG (x, y ) = (2π) k(ξ  x1 , x  , ξ  y1 , y  , ξ  )e ix −y ,ξ dξ  with k(t, x  , t  , y  , ξ  ) satisfying j     (1.4.19) ||Dxα Dyβ Dξγ Dt,t  k(t, x , t , y , ξ )|| ≤

Cαβγjr (1 + t + t  )−s (1 + |ξ  |)m+1+ς(|α|+|β|)−σ|γ|

∀α, β, γ, j, r .

is called singular Green operator of order m. m Let G m σ,ς denote the class of such operators.  Further, by G cl we denote class of operators with functions k ∼ ks where ks are positively homogeneous of degrees (m + 1 − s) with respect to ξ  . m (ii) One can define G m σ,ς and G cl for any half-space and for any bounded domain X with the smooth boundary.

One can prove (see L. Boutet de Monvel [1]) the following statements:

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Proposition 1.4.13. Let P ∈ P cl (X ), T ∈ T cl (X ), G , G  ∈ G cl (X ). Then KP , KT and KG are smooth in ∂X × X¯ , X¯ × ∂X and X¯ × X¯ respectively with the exception of the diagonal of ∂X × ∂X (i.e. the set {(x, x), x ∈ ∂X }).  Proposition 1.4.14. (i) Let P ∈ P m cl (X , H , H). Then P is a bounded operator from H s−1/2 (∂X , H) to H s−m (X , H ).

Further, as X = {x : x1 > 0} P is a bounded operator from H s−1/2 (∂X , H) to

  H j R+ , H s−m−j (Rd−1 , H ) .

(1.4.20) j≥0

 s (ii) Let T ∈ T m cl (X , H , H). Then T is a bounded operator from H (X , H) s−m−1/2  to H (∂X , H ).  s (iii) Let G ∈ G m cl (X , H , H). Then G is a bounded operator from H (X , H) s−m  to H (X , H ).

Further, as X = {x : x1 > 0} P is a bounded operator from H s (X , H) to (1.4.20). m+1  ∗ Proposition 1.4.15. (i) P ∈ P m (X , H, H ). cl (X , H , H) if and only if P ∈ T cl m  ∗  (ii) G ∈ G m cl (X , H H) if and only if G ∈ G cl (X , H, H ).

Proposition 1.4.16. Let P ∈ P cl (X ), T ∈ T cl (X ), G , G  ∈ G cl (X ) and let A, A ∈ Ψcl,trans (X ) with ord(A), ord(A ) ∈ Z, B ∈ Ψcl (∂X ) 28) . (i) Then as ordA ≤ 0 (1.4.21) (1.4.22) (1.4.23) (1.4.24) and (1.4.25)

AX P, GP, PB TAX , TG , BT AX G , GAX , GG  , PT TP

∈P cl (X ), ∈T cl (X ), ∈G cl (X ), ∈Ψcl (∂X )

AX AX − (AA )X ∈G cl (X )

and in each statement the rule “the order of the product equals to the sum of the orders” holds; 28) Here and below we deliberately skip H, H and H in our notations as otherwise it would be too tedious.

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109

(ii) On the other hand, as m = ordA ≥ 1 (1.4.26) Bj ð∂X D1j + T  , TAX = 0≤j≤m −1

(1.4.27)

GAX − (GA )X =



Tj ð∂X D1j + G  ,

0≤j≤m −1

(1.4.28)

AX AX − (AA )X =



Tj ð∂X D1j + G 

0≤j≤m −1 



−1−j −1−j with operators Bj ∈ Ψm+m (∂X ) and Tj , Tj ∈ T m+m (X ) and T  , G  cl cl belonging to the corresponding classes.

Proposition 1.4.17. Let conditions of proposition 1.4.16 be fulfilled. Assume that locally X = {x1 > 0}. (i) Let j : v → (δ(j) ⊗ v ). Then ðX Aj , G j ∈ P cl (X ) and T j ∈ Ψcl (∂X ) and the rule “the order of the product equals to the sum of the orders” holds if we assign an order (j + 1) to operator j . (ii) Let ð∂X be an operator of restriction to ∂X . Then ð∂X G ∈ T cl (X ) and ð∂X AX ∈ T cl (X ) provided ord(A) < 0. On the other hand, for m = ord(A) ≥ 0 m Qj ð∂X D1j + T , Qj ∈ Ψm−j (1.4.29) ð∂X AX = cl (∂X ), T ∈ T cl (X ). 0≤j≤m

The rule “the order of the product equals to the sum of the orders” holds in both cases if we assign an order 0 to ð∂X . (iii) x1 G , Gx1 ∈ G cl (X ), x1 P ∈ P cl (X ) and Tx1 ∈ T cl (X ) and the rule “the order of the product equals to the sum of the orders” holds if we assign an order −1 to an operator of multiplication by x1 . Theorem 1.4.18. Let κ : X → X  be a diffeomorphism of U  0 to V  0 such that κ(X ) = X  with X = U ∩ {x1 > 0}, X = V ∩ {x1 > 0}. Then m m m m  induced map A → κ−1 ∗ Aκ∗ transforms P cl (X ), T cl (X ) and G cl (X ) to P cl (X ), m m T cl (X  ) and G cl (X  ) respectively. Remark 1.4.19. Similar statements (may be with some modifications) hold m m m for Ψm σ,ς , P σ,ς , T σ,ς and G σ,ς .

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Boutet de Monvel Algebra In L. Boutet de Monvel [1] the notion of trace and singular Green operator are extended to include operators as in the right hand expressions of (1.4.26) and (1.4.27) respectively. Further, operators of the type A + G are called Green operators in contrast to singular Green operatorswith A = 0. AX + G P Furthermore, the matrix operators of the form A = from T B H ⊕ Hb to H ⊕ Hb are introduced where Hb and Hb are also axillary Hilbert spaces.Recall that B ∈ Ψcl (∂X , H , Hb ); T is a trace operator of extended type. This leads to Boutet de Monvel algebra B(X ) of such operators. Moreover, the notion of ellipticity is introduced and parametrices for elliptic Boutet de Monvel operators are constructed; these parametrices are also Boutet de Monvel operators; the usual rule ord(A−1 ) = −ord(A) holds. There are easy and natural generalizations to manifolds and to sections of the Hilbert bundles over such manifolds. We refer to L. Boutet de Monvel [1] for the further reading; the only things we am interested are Green operators of order 0 (see next Subsubsection) and applications to elliptic boundary problems (see Subsection 1.4.3). One can also prove similar results for operators of (σ, ς)-type. Ellipticity and Parametrices Consider A = AX + G operators with A ∈ Ψ0cl,trans (X , H , H) and G ∈ G 0cl (X , H , H). We want to construct parametrix of the same type. Obviously ellipticity of A is necessary (but not sufficient): (1.4.30) For each (x, ξ) ∈ S ∗ X symbol a0 (x, ξ) is invertible and satisfies ||a0 (x, ξ)−1 || ≤ c. On the other hand if condition (1.4.30) is fulfilled what we need is to construct parametrix locally in the vicinity of every point x¯ ∈ ∂X and therefore we can assume that X = {x : x1 > 0}. Consider first the “constant” and homogeneous case when a = a0 (ξ) and in definition (1.4.18) of KG one can take k(x1 |ξ  |, y1 |ξ  |, ξ  ) and this function is positively homogeneous of order 1 with respect to the third argument: |ξ  |κ(x1 |ξ  |, y1 |ξ  |, ξ  /|ξ  |).

1.4. ELLIPTIC BOUNDARY VALUE PROBLEMS

111

In this case we can make a Fourier transform with respect to x  → ξ  and the analysis of AX + G is reduced to the analysis of 1-dimensional operator a0 (D1 ; ξ  )R+ + g0 (ξ  ) where g0 (ξ  ) is one dimensional integral operator on R+ with Schwartz kernel κ(x1 , y1 ; ξ  ); we need to consider only |ξ  | = 1 because the general case is reduced by an obvious change of variables x1 and y1 . Obviously we need to assume that a0 (D1 ; ξ  )R+ + g0 (ξ  ) : L2 (R+ , H) → 2 L (R+ , H ) is invertible for all ξ  . Definition 1.4.20. Green operator AX + G with A ∈ Ψ0cl,trans (X , H , H) and G ∈ G 0cl (X , H , H) is elliptic if (i) a is elliptic in the sense of (1.4.30) and (ii) For each (x  , ξ  ) ∈ S ∗ ∂X operator a0 (D1 ; x  , ξ  )R+ + g0 (x  , ξ  ) : L2 (R+ , H) → L2 (R+ , H ) −1  is invertible and || a0 (D1 ; x  , ξ  ) + g0 (x  , ξ  ) || ≤ c where to construct g0 (x  , ξ  ) we locally make X = {x : x1 > 0}, take principal symbol of k (i.e. remove all the terms with an order of homogeneity less than 1) and freeze y  = x . Definition 1.4.21. One-dimensional integral operator g0 is principal symbol of G . One can see easily that under these assumptions an inverse operator is of the same type as the original one  −1 a0 (D1 ; x  , ξ  )R+ + g0 (x  , ξ  ) = a0−1 (D1 ; x  , ξ  )R+ + g0 (x  , ξ  ) and smoothly depends on (x  , ξ  ). This leads to an obvious construction of the singular Green operator G  ∈ G 0cl (X ) with the principal symbol g0 . One can see easily that operator (AX + G  ) where A is the parametrix of A satisfies (1.4.31)

(AX + G )(AX + G  ) ≡ (AX + G  )(AX + G ) ≡ I

−1  modulo G −1 cl (X ). Therefore one can adjust G (by G cl (X ) operator) so that ¯ (1.4.31) holds modulo operators with smooth in X × X¯ symbol. Therefore we have proved

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112

Theorem 1.4.22. Operator (AX + G ) with A ∈ Ψ0cl,trans (X , H , H) and G ∈ G 0cl (X , H , H) has a parametrix (AX + G  ) with A ∈ Ψ0cl,trans (X , H, H ) and G  ∈ G 0cl (X , H, H ) if and only if (AX + G ) is elliptic in the sense of Definition 1.4.20. Resolvent and Functional Calculus Now one can consider resolvent R(ζ) of operator of this type. Similarly to arguments for classical pseudodifferential and h-pseudodifferential operators we have: Theorem 1.4.23. Consider operator (AX + G ) with A ∈ Ψ0cl (X , H, H) and G ∈ G 0cl (X ). Let (1.4.32)

Σ(a) =



  Spec a0 (x, ξ)

(x,ξ)∈S ∗ X

and (1.4.33)

Σb (a0 , g0 ) =



  Spec a0 (D1 ; x  , ξ  )R+ + g0 (x  , ξ  ) .

(x,ξ)∈S ∗ ∂X

Then for any vicinity Ω of Σ(a0 , g0 ) = Σ(a) ∪ Σb (a0 , g0 ) one can adjust (AX + G ) by an operator with infinitely smooth Schwartz kernel so that the spectrum of the new (AX + G ) would be contained in Ω. Theorem 1.4.24. Consider operator (AX + G ) with A ∈ Ψ0cl (X , H, H) and G ∈ G 0cl (X ). Let f (z) be analytic function in the vicinity Ω of Spec(AX + G ). Then (i) f (AX +G ) also is operator of this type. Further, pseudodifferential part of it could be constructed as a (formal) f (A) and thus principal pseudodifferential symbol is f (a0 ). symbol (in the sense of defini(ii) Furthermore, its principal singular Green  tion 1.4.21) is a singular Green part of f a0 (D1 ; x  , ξ  )R+ + g0 (x  , ξ  ) ; (iii) In particular, as A = 0 pseudodifferential part of f (AX + G ) is f (0) and f (G ) − f (0) ∈ G 0cl (X ). Further, the resolvent based construction leads to

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113

Theorem 1.4.25. Consider operator (AX + G ) with A ∈ Ψ0cl (X , H, H) and G ∈ G 0cl (X , H.H). (i) Assume that it is Hermitian, positive and disjoint from 0 in operator sense operator. Then (AX + G )l is operator of the same type for any l ∈ R (and even l ∈ C); further, its principal pseudodifferential symbol is a0l and its principal sin l gular Green symbol is a singular Green part of a0 (D1 ; x  , ξ  )R+ + g0 (x  , ξ  ) . (ii) Assume that it is Hermitian and disjoint from 0 in the operator sense operator. Consider orthogonal (in L2 (X , H)) projectors Π± on positive and negative subspaces of A. Then Π± are operators of the same type; their principal pseudodifferential symbols are π ± (x, ξ) (orthogonal projectors to positive and negative subspaces of a0 (x, ξ)) and their singular Green principal symbols are  singular Green parts of the corresponding projectors for a0 (D1 ; x  , ξ  )R+ + g0 (x  , ξ  ) . (iii) In particular, Π± ∈ G 0cl (X , H, H) (i.e. pseudodifferential part is 0) if and only if ±a0 (x, ξ) < 0 for all (x, ξ). Remark 1.4.26. Arguments leading to the proof of Theorem 1.1.46 allows us to construct f (AX + G ) as (AX + G ) is Hermitian and f is smooth in the vicinity of Spec(AX + G ).

1.4.3

Applications to Elliptic Boundary Value Problems

Parametrix Construction Let us consider m = ordA > 0. In this case (AX + G ) is not necessarily elliptic (i.e. (AX + G ) : H s (X , H) → H s−m (X , H ) fails to be invertible) and one needs to consider in addition to (AX + G )u = f boundary value problem Tu = g where T : H s (X , H) → H s−μ−1/2 (X , Hb ) is a trace operator of extended type.

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Remark 1.4.27. (i) We are interested only in the case when A is a differential operator, G = 0 and T = B with (1.4.34) B= Bj ð∂X D1j with Bj ∈ Ψμ−j cl (∂X ). 0≤j≤m−1

(ii) There could be more than one boundary conditions of different orders but one can always make them of the same order (say μ = (m − 1)) by multiplying by elliptic pseudodifferential operators on the boundary and then join then by taking Hb = Hb1 ⊕ ... ⊕ Hbk . Assume that A is elliptic in the sense that ||a0 (x, ξ)−1 || ≤ c|ξ|−m . Let us consider in (x  , ξ  ) ∈ T ∗ ∂X ; without any loss of the generality one can assume that locally X = {x1 > 0}. Consider space Ker a0 (D1 ; x  , ξ  )R+ of exponentially decaying solutions of a0 (D1 ; x  , ξ  ). Alternatively one can consider Y(x  , ξ  ) = ð(m−1) Ker a0 (D1 ; x  , ξ  )R+ where ð(m−1) v = (v , D1 v , ... , D1m−1 v ). Definition 1.4.28. We call (A, B) elliptic boundary value problem if (i) A is elliptic in the sense that ||a0 (x, ξ)−1 || ≤ c|ξ|−m ; (ii) Map (bj (x  , ξ), ... bm−1 (x  , ξ  ) : Y(x  , ξ  ) → Hb is invertible with uniformly bounded inverse for all (x  , ξ  ) where bj are principal symbols of Bj are positively homogeneous with respect to ξ  of degrees μ − j. Assume that (A, B) is elliptic. Let (1.4.35)

β(x  , ξ) : Hb → Y(x  , ξ  ) ⊂ Hm = H ⊕ ... H    m copies

be an inverse map and βj (x  , ξ  ) (j = 0, ... , m − 1) be its components; then βj (x  , ξ  ) are positively homogeneous of degrees (j − μ) with respect to ξ  . Consider also maps (1.4.36) κj (x  , ξ  ) : H → Ker a0 (D1 ; x  , ξ  ) : Dtk κj (t, x  , ξ  )|t=0 = δjk j, k = 0, ... , m − 1 which has a Schwartz kernel kj (x1 |ξ  |, x  , ξ  ) with kj (t, x  , ξ  ) positively homogeneous of degree −j with respect to ξ  and satisfying (1.4.17) as |ξ  | = 1.

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115

Then operator (1.4.37)

κ(x  , ξ  ) =



κj (x  , ξ  )βj (x  , ξ  ) : Hb → Ker a0 (D1 ; x  , ξ  )

j

 has a Schwartz kernel k(|ξ  |x1 , x  , ξ  ) = j κj (|ξ  |x1 , x  , ξ  )βj (x  , ξ  ) with k(t, x  , ξ  ) positively homogeneous of degree −j with respect to ξ  and satisfying (1.4.17) as |ξ  | = 1.    Consider P ∈ P −μ cl (X , Hb , H) with the principal symbol k(|ξ |x1 , x , ξ ). Then (1.4.38)

AX P ≡ 0

mod P m−μ−1 (X , H , Hb ), cl

(1.4.39)

BP ≡ I

mod Ψ−1 cl (∂X , Hb , Hb ).

  Meanwhile let A ∈ Ψ−m cl (X , H, H ) be a parametrix of A. Then AX AX ≡ I modulo infinitely smoothing operator because A is a differential operator. On the other hand, BAX ∈ T μ−m (X , Hb , H ) and therefore G = −PBAX ∈ cl −m    G cl (X , H , H ) and R = AX + G satisfies

(1.4.40)

AX R ≡ I

  mod Ψ−∞ cl (X , H , H ),

(1.4.41)

BR ≡ 0

mod T μ−m−1 (X , Hb , H ). cl

Therefore (X , H, H ) and P by element (1.4.42) One can adjust R by element of G −m−1 cl −μ−1 of P cl (X , H, Hb ) so that (1.4.38)–(1.4.41) hold modulo operators with infinitely smooth Schwartz kernels. In other words, (1.4.43)

     I 0 A  R P ≡ 0 I B

and R = (R, B) is a left parametrix of boundary value problem (A, B). On the other hand, if Au ≡ f then due to (1.4.12) and consequent analysis (j) AX u ≡ X f + δ∂X (x)gj j≤m−1

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

modulo X C∞ (X¯ , H , H) where gj = ð∂X ,j u. Applying ðX A we arrive to (j) (1.4.44) u ≡ AX f + ðX A δ∂X (x)gj ≡ AX f + Pj gj j≤m−1

j

modulo C∞ (X¯ , H) where Pj ∈ P −j cl (X , H, H). Further, (1.4.45)

gk = ð∂X ,k u ≡ Tk f +



Qkj gj

j

where Tk ∈ T −m+k (X , H, H ) and Qkj ∈ Ψk−j cl cl (∂X , H, H). Furthermore (1.4.46) Bu ≡ B k gk . k

One can prove easily that ellipticity as in Definition 1.4.28 implies that if f ≡ 0, Bu ≡ 0 then (1.4.45)–(1.4.46) imply that gk ≡ 0. Therefore R = (R, B) is a right parametrix of boundary value problem as well: RA + PB ≡ I and therefore (1.4.47) R is a parametrix of boundary value problem. Meanwhile (1.4.48) R = AX + G is a parametrix of A under boundary condition Bu ≡ 0. Dirichlet-to-Neumann and Similar Operators On the other hand, as Au ≡ 0 we get with Lj ∈ Ψ−μ+j (∂X , H, Hb )  Example 1.4.29. Consider Laplace-Beltrami operator ΔX = j,k Dj g jk Dk with symmetric real positive definite matrix (g jk ) (considered as a Rie jk 1/2 g nj Dk u|∂X = iΔ∂X u|∂X mannian metrics). Then ΔX u ≡ 0 implies that where nj are elements on the unit normal to ∂X (with respect to this metrics) directed into X , Δ∂X is defined in the same way according to induced metrics on ∂X . The properly defined Laplace-Beltrami operator (which is self-adjoint in L2 (X ) with Riemannian density) differs from this one by lower order terms. (1.4.49)

ð∂X ,j u ≡ ð∂X ,j PBu ≡ Lj Bu

1.4. ELLIPTIC BOUNDARY VALUE PROBLEMS

117

Adjoint Operator; Self-Adjoint Operator Consider operator AB which is operator A with the domain  (1.4.50) D(AB ) = u ∈ H m (X , H) : Bu = 0 . We can introduce adjoint operator A∗B in the standard functional analysis way and we would say that AB is self-adjoint if A∗B = AB (as H = H). However we will give a slightly more restrictive definition. As far as we know, nobody explored the gap. So, for elliptic differential operator A let us consider (1.4.51) (Au, v )X − (u, A∗ v )X = (Bð(m−1),∂X u, ð(m−1),∂X v )∂X = (Bjk ðj,∂X u, ðk,∂X v )∂X j+k≤m−1

where A∗ is a formally adjoint differential operator, we pass from the left-hand expression to the middle one by integration by parts and Bjk are differential operators of order (m − 1 − j − k) on ∂X . Recall that ð(m−1),∂X u = (ð0,∂X u, ... , ðm−1,∂X ) is a function with values in Hm . Consider the middle expression of (1.4.51). It is a bilinear form of ð(m−1),∂X u and ð(m−1),∂X v . Let us consider boundary value problem Bu = 0 for A; Recall that actually Bj ðj,∂X u (1.4.52) Bu = Bð(m−1),∂X u = 0≤j≤m−1

with Bj ∈ Ψμ−j cl (∂X , Hb , H); there should be no confusion that we excluded ð(m−1),∂X from B. Further, let us consider for A∗ the boundary conditions B  v = 0 of the same type (1.4.52). Definition 1.4.30. Let us consider boundary conditions Bu = 0 for operator A and B  u = 0 for A∗ ; both conditions are of (1.4.52) type. Then B  = 0 is an adjoined boundary value problem for A∗ if (1.4.53)

BU = 0, B  V = 0 =⇒ (BU, V )∂X = 0,

(1.4.54)

(BU, V )∂X = 0

∀U : BU = 0 =⇒ B  V = 0

(1.4.55)

(BU, V )∂X = 0

∀V : B  V = 0 =⇒ BU = 0

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

and moreover (1.4.56) B = βB + (β  B  )∗ with β ∈ Ψm−1−μ (∂X , H m , Hb ), β  ∈ Ψm−1−μ (∂X , Hm , Hb ). cl cl One can prove easily that principal symbols of both β and β  should satisfy Ker β0 (x  , ξ  ) = 0,

(1.4.57)

Ker β0 (x  , ξ  ) = 0.

Definition 1.4.31. Let A = A∗ , H = H. Then B is skew-self-adjoint. We call Bu = 0 self-adjoint boundary value problem for A if B  = B satisfies definition 1.4.30.  Example 1.4.32. Let A = j,k (Dj − Vj )g jk (Dk = Vk ) + V with symmetric real positive definite matrix (g jk (x)) (considered as a Riemannian metrics) and real-valued Vj (x), V (x). Then A∗ = A. On the other hand, (1.4.58) (Au, v )X − (u, Av )X = i(Lu, v )∂X + i(u, Lv )∂X with Lu =



νj g jk (Dk − Ak )

j,k

where ν = (ν1 , ... , νd ) is an inner unit normal to ∂X . Therefore (i) Dirichlet boundary condition u|∂X = 0 is self-adjoint; (ii) Neumann boundary condition Lu|∂X = 0 is self-adjoint; (iii) Adjoint to boundary condition (L − iM)u|∂X = 0 with M ∈ Ψ1cl (∂X ) is (L − iM ∗ )v |∂X = 0; in particular it is self-adjoint if and only if M = M ∗ . One can see easily that this problem is elliptic if and only if 1/2  jk g˜ ξj ξk + M1 (x  , ξ  )| ≥  (x  , ξ  ) ∈ S ∗ ∂X (1.4.59) | j,k

where g˜

jk

is an induced metrics on ∂X .

Further, one can prove easily that (under condition (1.4.59)) AB is semi-bounded from below if and only if 1/2  jk g˜ ξj ξk + M1 (x  , ξ  ) ≥  (x  , ξ  ) ∈ S ∗ ∂X . (1.4.60) j,k

As d = 2 one can satisfy (1.4.59) and fail (1.4.60) even with differential operator M.

1.4. ELLIPTIC BOUNDARY VALUE PROBLEMS

119

Projectors Standard resolvent construction (like in the proof of Theorem 1.1.59) yields two following theorems (as H = H ): Theorem 1.4.33. Let A be an elliptic differential operator and B an elliptic boundary value problem for it. Let X be compact. Then for arbitrarily small  > 0 Spec(AB ) with the exception of finite subset is contained in -vicinity of Σ(a) ∪ Σ(ab ) where as before 

Σ(a) =

Spec a0 (x, ξ)

(x,ξ)∈T ∗ X

and (1.4.61)

Σ(ab ) =



Spec a0 (D1 , x  , ξ  )b0 (D1 ,x  ,x  )

(x  ,ξ  )∈T ∗ ∂X

where a0 and b0 are principal symbols of A and B and a0 (D1 , x  , ξ  )b0 (D1 ,x  ,ξ ) is boundary-value problem on R+ ; Theorem 1.4.34. Let A be an elliptic formally self-adjoint differential operator and B an elliptic boundary value problem for it. Let AB be selfadjoint. Let Π± be orthogonal projectors onto positive and negative invariant subspaces of AB . Then (i) Π± ≡ π ± (x, D)X mod G 0cl (X , H, H) where π ± (x, D) ∈ Ψ0cl,trans (X , H, H) and their principal symbols π0± (x, ξ) are projectors onto positive and negative subspaces of a0 (x, ξ). (ii) π ± = 0 if and only if ±a0 (x, ξ) is negative definite at every (x, ξ) ∈ T ∗ X \ 0. This condition is equivalent to Σ(a) ⊂ R∓ . In this case Π± ∈ G 0cl (X , H, H) and its Schwartz kernel is  1    1−d e ix −y ,ξ κ± (x1 |ξ  |, (x  + y  ), ξ  ) dξ  (1.4.62) KΠ± (x, y ) = (2π) 2 where modulo lower order symbol κ± (x1 |ξ  |, x  , ξ  ) is the Schwartz kernel of the spectral projector π± (x  , ξ) of a0 (D1 , x  , ξ  )b0 (D1 ,x  ,ξ ) .

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

(iii) Π± ≡ 0 modulo infinitely smoothing operator is and only if ±a0 (x, ξ) is negative definite at every (x, ξ) ∈ T ∗ X \ 0 and a0 (D1 , x  , xi  )b0 (D1 ,x  ,ξ ) is negative definite at every (x  , ξ  ) ∈ T ∗ ∂X \ 0. This condition is equivalent to Σ(a) ∪ Σ(ab ) ⊂ R∓ . Note however that |A| does not have a transmission property unless m is even and |A|l does not have a transmission property unless ml is even. Example 1.4.35. Consider operator AB from Example 1.4.32(iii) with violated condition (1.4.62). Without any loss of the generality one can assume that g 1k = δ1k as x ∈ ∂X 29) . Then modulo Schwartz kernel of operator G −1 cl one can define KΠ± (x, y ) by (1.4.62) with     (1.4.63) κ− (x1 |ξ  |, x  , ξ  ) ≡ 2λ(x  , ξ)e −λ(x ,ξ )x1 θ −M1 (x  , ξ  ) − λ(x  , ξ) with λ(x  , ξ) =

 j,k

g˜ jk ξj ξk

1/2

.

Semiclassical Theory Remark 1.4.36. Obviously all things stay the same for differential and pseudodifferential operators with D replaced by hD (and multiplied by h−s if it was operator belonging to Ψscl ). Then in the right-hand expressions of (1.4.15) of KP (x, y  ) and KT (x  , y ) and (1.4.15) of KG (x, y ) one should (i) Replace x1 and y1 by h−1 x1 and h−1 y1 respectively, 





(ii) Replace e ix −y ,ξ by e ih

−1 x  −y  ,ξ 

and dξ  by h1−d dξ  and

(iii) Multiply them by h−s as KP (x, y  ) is defined and by h−s−1 as KT (x  , y ) and GT (x, y ) are defined. AB from example 1.4.32(iii). Then in Example 1.4.37. Consider operator  the semiclassical theory a0 = j,k g jk (ξj − Vj )(ξk − Vk ) + V and positivity of a0 condition means that also V ≥ . Ellipticity condition (1.4.59) and positivity condition (1.4.62) are replaced by (1.4.64)

|λ(x  , ξ  ) + M1 (x  , ξ  )| ≥ 

∀(x  , ξ  ) ∈ T ∗ ∂X

29) One can always achieve it by taking x1 = dist(x, ∂X ) in the vicinity of ∂X where dist means the Riemannian distance.

1.A. APPENDICES and (1.4.65)

121

λ(x  , ξ  ) + M1 (x  , ξ  ) ≥ 

∀(x  , ξ  ) ∈ T ∗ ∂X

respectively and (1.4.63) is replaced by (1.4.66) κ− (h−1 x1 |ξ  |, x  , ξ  ) ≡ 2h−1 λ(x  , ξ)e −h

−1 λ(x  ,ξ  )x

1

  θ −M1 (x  , ξ  ) − λ(x  , ξ)

with (1.4.67)

λ(x  , ξ  ) =



g˜ jk (ξj − Vj )(ξk − Vk ) + V

1/2

.

j,k

1.A 1.A.1

Appendices Operator e itQ(D)

Let q(D) = QD, D be a nondegenerate real quadratic form in Rd with real symmetric matrix Q; then (1.A.1)

k(x, y ) = Ke iq(D) (x, y ) = (2π)d/2 | det Q  |1/2 e

πid − 4i q  (x−y ) 4

is the Schwartz kernel of e iq(D) where q  (z) = Q −1 z, z. We claim first that (1.A.2) e iq(D) f L∞ ≤ C D α f L∞ . |α|≤(d+2)/2



Indeed, an estimate for || B(x,1) k(x, y )f (y ) dy || is obvious; to estimate a similar integral over Rd \B(x, 1) we note that ∇y k(x, y ) = 2i Q −1 (y −x)k(x, y ); then k(x, y ) = ω−1 (x − y )∇y k(x, y ) with positive homogeneous of degree s functions ωs ; then after integration by parts  k(x, y )f (y ) dy = |x−y |≥1   ω−2 (x − y )k(x, y )f (y ) dy + ω−1 (x − y )k(x, y )∇f (y ) dy and we continue repeating this procedure until we get ω−d−1 in every term.

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

Estimate (1.A.2) after rescaling x → |t|−1/2 x, D → |t|1/2 D implies immediately (1.A.3) e tiq(D) f L∞ ≤ C |t||α|/2 D α f L∞ . |α|≤(d+2)/2

Using Taylor decomposition (with respect to t at t = 0) and applying (1.A.3) to the remainder estimate we arrive to  1 n  (1.A.4)  e tiq(D) − itQ(D) f  ≤ n! n≤N−1 1 C |t|N |t||α|/2 D α Q(D)N f L∞ . N! |α|≤(d+2)/2

Finally, we get immediately (1.A.5)

|β|=K

 1 n  itq(D) f  ≤ D β e tiq(D) − n! n≤N−1 C |t|N

1.A.2

1 N!



|t||α|/2 D α+β q(D)N f L∞ .

|β|=K ,|α|≤(d+2)/2

Almost Analytic Compactly Supported Symbols

Proposition 1.A.1. For any n there exists a function φn , such that supp(φn ) ⊂ [0, 1],  ∞ φn ≥ 0 , φn dx = 1,

(1.A.6) (1.A.7)

−∞ k

|∂ φn | ≤ c(cn) k

(1.A.8)

∀k ≤ n

with c which does not depend on n. Proof. Let us fix φ = φ1 supported in [0, 12 ] and satisfying (1.A.7). Then (1.A.9)

φn (t) = nφ(nt) ∗ nφ(nt) ∗ · · · ∗ nφ(nt) ∗φ(t)    n times

is a proper function.

1.A. APPENDICES

1.A.3

123

Stationary Phase Method

Let us consider I(x) = e ih

(1.A.10)

−1 φ(θ)

f (θ)dθ

with smooth and real-valued φ, smooth and compactly supported f and θ ∈ RN . Then under assumption (1.A.11)

|∇φ| ≥ 0

on supp(f )

I = O(hs ) for any s. Proof is trivial: decomposing f = f1 + ... + fN with |∂θk φ| ≥ 1 on supp(fk ) and integrating by parts. Therefore in the general case we need to consider f supported in the vicinity of the set C = {θ : ∇φ(θ) = 0}

(1.A.12) of stationary points of φ. Assume now that (1.A.13)

|∇φ| ≤ 0 =⇒ | det Hess φ| ≥ 0

on supp(f )

which is a quantitative version of rank Hess φ = N on C ∩ supp(f ). Then C ∩ supp(f ) = {θ1 , ... , θr } is a finite set. So we can consider f supported in ¯ Without any loss of the vicinity a single non-degenerate stationary point θ. the generality we can assume that θ¯ = 0. Changing variables we can achieve φ(θ) = 12 ς1 θ12 + ... + 12 ςN θN2 where ςj are eigenvalues of Hess φ(0); then f (0) is preserved. Note that (1.A.14) If f = O(|∇φ|l ) then one can rewrite I = hl I  where I  is given by the same formula (1.A.10) with f  = Ll f with depending on φ linear differential operator Ll of order l. One can prove it using integration by parts. Decomposing f into Taylor series at θ = 0 we arrive to need to consider only case f = 1; then applying formula  πi sgn Q i (1.A.15) e 2 hQθ,θ dθ = (2πh)N/2 | det Q|−1/2 e 4 we arrive to the stationary phase principle

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CHAPTER 1. INTRODUCTION TO MICROLOCAL ANALYSIS

Proposition 1.A.2. Consider (1.A.10) with smooth and real-valued φ, smooth and compactly supported f and θ ∈ RN . Then under assumption (1.A.13)   −1 πi (1.A.16) I  e ih φ(θk ) × | det Hess φ(θk )|−1/2 e 4 sgn Hess φ(θk ) fk,n hn zk ∈Ck

n≥0

with fk,0 = f (θk ) and fk,n = (Ln f )(θk ). Proof. Proof of (1.A.10) is trivial: first, as Q is diagonal it is a product of one-dimensional integrals; further, changing z = |λj |1/2 h−1/2 θ we reduce it to  ∞ ±iz 2 /2 e dz. The contour of integration of the latter could be deformed to −∞ ∞ 2 ±iπ/4 e R; so we get e ±iπ/4 −∞ e −z /2 dz and the latter integral is well-known.

1.A.4

Function in the Box

Consider function u such that (1.A.17) u  1 such that is negligible in {|x| ≥ γ} and Fx→h−1 u is negligible in {|ξ| ≥ ρ}. Does such function exists? Obviously the same answerremains if instead of “Fx→h−1 ξ u is negligible in {|ξ| ≥ ρ}” we request “supp Fx→h−1 u ⊂ {|x| ≤ ρ}”. However then (see f.e. (3.3) of B. Paneah [2] and the references there): (1.A.18)

u{|x|≥γ} ≥ e −h

−1 ργ

u

with  = (d). Therefore −1

(1.A.19) u  1 and u{|x|≥γ} ≤ Chs imply e −h ργ ≤ Chs which is equivalent to h−1 ργ ≥ s| log h|; so we must assume ργ ≥ sh| log h|. On the other hand, (1.A.20) Consider a coherent state u = (2πα)−d/2 e −|x| /2α . Then Fx→h−1 ξ = 2 2 (2πβ)−d/2 e −|ξ| /2β with β = h/α. Picking α = γ(s| log h|)−1/2 we get that u satisfies (1.A.17) as long as β ≤ ρ(s| log h|)−1/2 which holds as ργ ≥ Cs| log h|. 2

2

1.A. APPENDICES

125

Comments We have sent the reader to the book of L. H¨ormander [1] for details, proofs and advanced results. One can find relatively easy exposition in the monographs of F. Treves [1] and M. Taylor [7]. One can find interesting approaches in the monograph of M. Shubin [3]. Unfortunately there seem to be no comprehensive and comprehensible book, covering relation between pseudodifferential operators and singular integral operators due to Calderon-Zygmund-Mihlin.

Chapter 2 Propagation of Singularities in the Interior of the Domain In this chapter we consider microlocal propagation of singularities (i.e., wave front sets) in the interior of a domain. In general in this book we consider two types of microlocal operator, namely, microelliptic and microhyperbolic. For microelliptic operators1) there is no propagation of singularities at all and we prove the different variants of this fact which we need by means of either a parametrix construction or the Ga ˚rding inequality; sometimes we combine both methods. In this chapter we treat microhyperbolic operators. There are different approaches. One based on Fourier integral operators we already considered. While delivering the most information about solutions it also imposes the most conditions to operator. We will apply different approaches instead. In Sections 2.1–2.3 we use the most general approach based on the energy estimates. First of all, in Section 2.1 we prove the main general Theorem 2.1.2 and certain of its corollaries which will be necessary for our study of spectral asymptotics; this Theorem is formulated in the terms of auxiliary real-valued functions φj (x, ξ) with j = 1, ... , J. Next, in Section 2.2 we reformulate this Theorem in more geometric terms, namely, in the terms of bicharacteristics and generalized bicharacteristics and consider a host of different examples. Finally, in Section 2.4 we generalize this Theorem to the case of the 1) I.e. operators elliptic in the vicinity of the point in question; so all operators are microelliptic everywhere except their characteristic varieties.

© Springer Nature Switzerland AG 2019 V. Ivrii, Microlocal Analysis, Sharp Spectral Asymptotics and Applications I, https://doi.org/10.1007/978-3-030-30557-4_2

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2.1. ENERGY ESTIMATES APPROACH

127

“rough” symbols. In Section 2.4 we consider Heisenberg approach for the propagation of singularities for scalar and related operators. It will allow us to consider a large time interval. We will use not only results but also the methods of this chapter to prove different type of propagation later; in particular in perturbed case of the periodic Hamiltonian flow the very long-time propagation results will be obtained by the same and other methods.

2.1 2.1.1

Energy Estimates Approach Main Theorem

Let us start with the key notion and key statement of this chapter: Definition 2.1.1. Let p ∈ Sh,1 (T ∗ Rd , H, H) be a Hermitian symbol. Then p is microhyperbolic at the point z = (x, ξ) ∈ T ∗ Rd in the direction  = (x , ξ ) ∈ Tz T ∗ Rd iff (2.1.1)

− (p)(z)w , w  ≥ 0 ||w ||2 − c1 ||p(z)w ||2

∀w ∈ H

where p = (x ∂x  + ξ , ∂ξ )p and 0 > 0 and c1 are appropriate constants. Using this definition we are going to prove the following main Theorem: Theorem 2.1.2. Suppose P ∈ Ψh (Rd , H, H) and assume that (2.1.2) p(z) = P0 (z) is a Hermitian symbol. Let Ω  T ∗ Rd and let φ1 , ... , φJ be real-valued CbN functions in T ∗ Rd such ¯ the symbol p is microhyperbolic in the directions that at every point z ∈ Ω # # ∇ φ1 , ... , ∇ φJ where ∇# φ = (∇ξ φ), ∇x −(∇x φ), ∇ξ  is the Hamiltonian field generated by φ. Let u be tempered, i.e., (2.1.3)

u ≤ h−l ,

and (2.1.4)

WFs+1 (Pu) ∩ Ω ∩ {φ1 ≤ 0} ∩ · · · ∩ {φJ ≤ 0} = ∅,

(2.1.5)

WFs (u) ∩ ∂Ω ∩ {φ1 ≤ 0} ∩ · · · ∩ {φJ ≤ 0} = ∅.

128 Then (2.1.6)

CHAPTER 2. PROPAGATION IN THE INTERIOR...

WFs (u) ∩ Ω ∩ {φ1 ≤ 0} ∩ · · · ∩ {φJ ≤ 0} = ∅.

This picture shows domains Ω and {φj ≤ 0}: so the dark shaded Ω area is where we assume that Pu is negligible and we deduct that {φ1 ≤ 0} {φ2 ≤ 0} u is negligible while we also assume that u is negligible in the small vicinity of the thick line Γ = ∂Ω ∩ {φ1 ≤ 0} ∩ {φ2 ≤ 0}. Γ It looks like Holmgren uniqueness Theorem but now Ω resides Figure 2.1: Domains to Theorem 2.1.2 in the phase space rather than configuration space and “u ≡ 0” replaces “u = 0” which does not have sense with the relation to phase space. Proof. Let q1 , q2 , q3 ∈ Sh (Rd , C) be arbitrary symbols such that q1 = 1 in Ω ∩ {φ1 ≤ } ∩ · · · ∩ {φJ ≤ }, q2 = 1 in (∂Ω) ∩ {φ1 ≤ } ∩ · · · ∩ {φJ ≤ } and supp(q3 ) ⊂ Ω 1  ∩ {φ1 ≤ 12 } ∩ · · · ∩ {φJ ≤ 12 }. 2 Let Qk = Opw (qk ) and N = N(s, l) in the definition of the symbol classes, etc. One can easily see that in order to prove the Theorem it is sufficient to prove that (2.1.3) and (2.1.7)1,2

Q1 Pu ≤ hs+1 ,

Q2 u ≤ hs

yield (2.1.8)

Q3 u ≤ Chs

where . as usual means the L2 (Rd , H)-norm and C depends on d, s, l, 0 , c1 , , c2 = |||P||| + |||qk ||| + φj  C∞ ; k

j

however, (2.1.9) C depends neither on Ω nor even its diameter.

2.1. ENERGY ESTIMATES APPROACH

129

We prove (2.1.8) under the additional assumption (2.1.10)

Q1 u ≤ hs−δ ;

with any exponent δ > 0 which does not depend s; we can take δ = 12 here; obviously, the general case is reduced to this one by “induction” on s in (2.1.8). Let χ ∈ C∞ (R) be a fixed real-valued function such that (2.1.11)

supp(χ) ⊂ (−∞, 0], χ > 0 on (−∞, 0), χ ≥ c3 χ and χ1 = χ 2 , χ2 = (−χ ) 2 ∈ C∞ (R) 1

1

where the prime means derivative and the constant c3 can be chosen later (such a function χ exists for every c3 ). We can replace φj by (φj − 14 ) and  by 34 ; hence it is sufficient to prove (2.1.8) only for Q3 = Opw (q3 ) with  χ(φk (z)) (2.1.12) q3 (z) = ζ(z) 1≤k≤J

where ζ ∈ C0N (Ω ), 0 ≤ ζ ≤ 1, ζ = 1 in Ω 2  ; there exists such a function 3 with C∞ -norm not exceeding C1 = C1 (d, N, ). Let us consider the main identity of the energy estimate method (2.1.13)

− Re i([P, Q]u, u) = Re i((Q + Q ∗ )Pu, u) + Re i((P ∗ − P)Qu, u).

Let us plug Q = Q32 ; then Q = Q ∗ and the absolute value of the first term in the right hand expression in (2.1.13) does not exceed Q3 Pu · Q3 u. Then (2.1.3) and (2.1.6) imply that Q3 Pu ≤ 2hs+1 : indeed, q1 = 1 in the 1 -neighborhood of supp(q3 ) with 1 = 1 (d, 0 , c1 , J, ) > 0). Therefore the absolute value of the first term in the right hand expression in (2.1.13) does not exceed 2hs+1 Q3 u2 . Meanwhile the second term in the right hand expression in (2.1.13) equals (2.1.14)

Re i((P ∗ − P)Q3 u, Q3 u) + Re i([P ∗ − P, Q3 ]Q3 u, u).

Then (2.1.2) yields that the absolute value of the first term here does not exceed C2 hQ3 u where C2 does not depend on c3 . Further, (2.1.2), (2.1.3)

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CHAPTER 2. PROPAGATION IN THE INTERIOR...

and (2.1.10) imply that the absolute value of the second term does not 3 exceed Chs+ 2 Q3 u2 where C depends on c3 . Thus (2.1.13) implies that (2.1.15)

| − Re ih−1 ([P, Q]u, u)| ≤ C2 Q3 u2 + Ch2s .

Now our goal is to estimate − Re ih−1 ([P, Q]u, u) from below. Let us note that the principal symbol of the operator −ih−1 [P, Q] is equal to

(2.1.16)



2{p, φj }rj2 − {p, ζ 2 }

1≤j≤J

χ2 (φk )

1≤k≤J

where rj = ζχ1 (φj )χ2 (φj )



χ(φk )

k =j

and χ1 , χ2 are defined by (2.1.11). Since q2 = 1 in the 1 -neighborhood of the support of the second term in (2.1.16) we conclude from (2.1.3), (2.1.7)1,2 , (2.1.10) and (2.1.15) that (2.1.17)

|



(Sj Rj u, Rj u)| ≤ C2 Q3 u2 + Ch2s

1≤j≤J

where Rj = Opw (rj ), Sj = Opw ({p, φj }). The microhyperbolicity of p in the directions ∇# φj yields that there exists a symbol q4 vanishing in the 1 -neighborhood of the supports of rj for all j such that |||q4 ||| ≤ C and the following inequalities are fulfilled: (2.1.18) ({p, φj } + cp † p + (q4 − 0 )I )w , w  ≥ 0 ∀w ∈ H ∀z ∈ T ∗ Rd

∀j = 1, ... , J.

Then passing from symbols to operators we obtain by means of the Ga ˚rding inequality (Theorem 1.1.29) that (2.1.19)

Re(Sj v , v ) + c1 Pv 2 + Re(Q4 v , v ) ≥ (0 − Ch)v 2 .

Let us plug v = Rj u here. Observe that (2.1.3), (2.1.6) and (2.1.10) yield that (2.1.20)

Pv  ≤ Rj Pu + [Rj , P]u ≤ Chs .

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131

Moreover, observe that |(Q4 v , v )| ≤ Ch2s because the supports of Q4 and Rj 1 are disjoint and that v  ≤ Chs− 2 in virtue of (2.1.10) and (2.1.3). Therefore (2.1.21)

Re(Sj Rj u, Rj u) ≥ 0 Rj u2 − Ch2s .

This inequality and (2.1.17) yield that Rj u ≤ C2 Q3 u2 + Ch2s . (2.1.22) 1≤j≤J −1 2 Moreover, (2.1.11) yields that q32 ≤ −1 3 rj with 3 = c3 . Therefore the Ga ˚rding inequality yields that

Q3 v 2 ≤ 3 Rj v 2 + Chv 2 . Let us plug v = Q5 u here where q5 = 1 in the 1 -neighborhood of supp(q5 ); then this estimate and (2.1.3) and (2.1.6) yield that Q3 u2 ≤ c3−1 Rj u2 + Ch2s . This estimate, (2.1.6) again, the possibility of choosing 3 arbitrarily small and the fact that C2 does not depend on c3 (and 3 ) together imply (2.1.8). Remark 2.1.3. (i) In fact, we could prove this Theorem for J = 1 from which the general case follows. Indeed, one can reduce J ≥ 2 to J = 1 considering φ = f (φ1 , ... , φJ ) where f = fε ∈ C∞ such that ∂tj f (t) > 0 ∀j ∀t = (t1 , ... , tJ ) and {t : f (t) ≤ −ε} ⊂ {t1 ≤ 0, ... , tJ ≤ 0} ⊂ {t : f (t) ≤ ε} with arbitrarily small constant ε > 0. (ii) We can replace “φj ≤ 0” by “φj < 0” in (2.1.4)–(2.1.6). We can use one type of inequalities for some indices j and another type fo other indices. (iii) The Theorem and Statement (ii) here remain true if we assume that p ¯ in the direction ∇# ψ with a real-valued function is microhyperbolic in Ω ¯ in the ψ with ψ C∞ ≤ c2 and that p is non-strictly microhyperbolic in Ω # directions ∇ φj for j = 1, ... , J; the non-strict microhyperbolicity means that for  = ∇# φj (2.1.23)

− (p)(z)w , w  ≥ −c1 ||p(z)w ||2

∀w ∈ H;

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132

we reduce this case to one considered above replacing φj by (φj + εψ) with small enough ε > 0. We need however to assume that sets {φj ≤ t} depend continuously on t at t = 0 i.e. {z, φj (z) < 0} = {z, φj (z) ≤ 0} and {z, φj (z) > 0} = {z, φj (z) ≥ 0}. In what follows we need Proposition 2.1.4. Statement “(2.1.7)1,2 & (2.1.3) =⇒ (2.1.8)” remains true if in the definition of c2 and hence in the definition of C we replace ¯ with an arbitrary Hermitian matrix P¯ ∈ L(H, H). |||P||| by |||P − P||| Proof. Let P = P¯ + P  , |||P  ||| ≤ c2 − 1. Without any loss of the generality ¯ j ⊂ Hj and Pw ¯ 1  ≤ c2 w1  , one can assume that H = H1 ⊗ H2 , PH ¯ Pw2  ≥ c2 w2  ∀wj ∈ Hj . Then  P=

P1 P12 P21 P2



and the principal symbol p2 of P2 is invertible and the norm of the inverse symbol does not exceed 1; then its parametrix R2 ∈ Ψh (Rd , H2 , H2 ) exists with |||R2 ||| ≤ C0 = C0 (d, N, c, c1 , c2 ). Thus the treatment of the propagation of singularities for P is reduced to that for P  = P1 − P12 R2 P21 ∈ Ψh (Rd , H1 , H1 ) with |||P  ||| ≤ C0 . The principal symbol p  = p1 − p12 p2−1 p21 of P  is Hermitian because † p12 = p21 . Moreover, the equality {p  , φ}w1 , w1  = {p, φ}w , w  holds for w = (w1 , w2 ) = (w1 , −p2−1 p21 w1 ) and hence condition (2.1.1) for p implies the same condition for p  with the constants c1 and 0 replaced by C0 c1 and C0−1 0 .

2.1. ENERGY ESTIMATES APPROACH

2.1.2

133

Cauchy Problem. I

Let us now consider the Cauchy problem for the operator P = hDt + A  with A ∈ C [0, T ], Ψh (Rd , H, H) where (2.1.24)

 1

(2.1.25) a(t, x, ξ) = A0 (t, x, ξ) is a Hermitian symbol for all (t, x, ξ).   We assume that Ω  R × T ∗ Rd and φj ∈ C1 [0, T ], CbN (T ∗ Rd ) are realvalued functions such that (2.1.26)

{p, φj }(t, z)w , w  ≥ 0 ||w ||2

∀w ∈ H.

Then we replace (2.1.13) by the identity (2.1.27) h Re(Qu, u)T − Re i([P, Q]u, u)[0,T ] = h Re(Qu, u)0 + Re i((Q + Q ∗ )u, u)[0,T ] + Re i((P − P ∗ )Qu, u)[0,T ] where (., .)[0,T ] and (., .)t are inner products in L2 ((0, T ) × Rd , H) and in L2 (Rd , H) for the indicated t respectively; for the norms similar notation will be used. Repeating the proof of Theorem 2.1.2 with the obvious modifications we obtain Theorem 2.1.5. Let conditions (2.1.24) and (2.1.25) be fulfilled and suppose Ω  R×T ∗ Rd . Let φj ∈ C1 ([0, T ], Cb∞ (T ∗ Rd )) be real-valued functions satisfying (2.1.26) in Ω where M denotes -vicinity of M. Let q1 , q2 , q3 ∈ C([0, T ], Sh (T ∗ Rd , C)) be symbols such that in Ω ∩ {φ1 ≤ } ∩ · · · ∩ {φJ ≤ }, in (∂Ω) ∩ {φ1 ≤ } ∩ · · · ∩ {φJ ≤ }

q1 = 1 q2 = 1 and

  supp(q3 ) ⊂ Ω 2 ∩ {φ1 ≤ } ∩ · · · ∩ {φJ ≤ }. 2 2 Let Qk = Opw (qk ) and N = N(s, l) in the definition of symbol classes, etc. Then (2.1.28)

sup ut ≤ 1,

Q1 u0 ≤ hs ,

t∈[0,T ]



T

(Q1 Put + hQ2 ut ) dt ≤ hs+1

(2.1.29) 0

CHAPTER 2. PROPAGATION IN THE INTERIOR...

134 imply that (2.1.30)

sup Q3 ut ≤ Chs t∈[0,T ]

with the same type constant C as in Theorem 2.1.2 2) . Here instead of Ω we consider strip Ω ∩ {t0 ≤ t ≤ T } t and initial condition are on the Ω bottom lid {t = t0 } while the {φ ≤ 0} top lid {t = T } is free. So, we Γ (x, ξ, τ ) assume that Pu ≡ 0 at Ω ∩ {φ ≤ 0} ∩ {t = t0 } Ω ∩ {φ ≤ 0} ∩ {t0 ≤ t ≤ T } and conclude that u ≡ 0 there, Figure 2.2: Domains to Theorem 2.1.5 assuming also that u ≡ 0 at Γ = ∂Ω ∩ {φ ≤ 0} ∩ {t0 ≤ t ≤ T } and also that u|t=0 ≡ 0 at Ω ∩ {φ ≤ 0} ∩ {t0 ≤ t ≤ T }. Remark 2.1.6. (i) Theorem 2.1.5 remains true if we in the conditions and in the conclusion replace supt∈[0,T ] .t by √1T .[0,T ] . (ii) Moreover, Proposition 2.1.4 remains true3) . Let us note that if (2.1.31)

||∇z a|| ≤ c

∀(t, z) ∈ [0, T ] × T ∗ Rd

then (2.1.26) holds for a function φ = t + ψ(z) provided ∇z ψ ≤ c −1 (1 − ). Corollary 2.1.7. Let conditions (2.1.24), (2.1.25) and (2.1.31) be fulfilled. Then the statements of Theorem 2.1.5 and Remark 2.1.6 hold for Ω = [0, T ] × T ∗ Rd , qk = qk (z) (k = 1, 3), q2 = 0 provided dist(supp(1 − q1 ), supp(q3 )) ≥ CT +  with an arbitrarily small constant  > 0. 2) 3)

Surely with |||P||| replaced by |||A||| in the definition of c2 . ¯ replaced by |||A − A||| ¯ in the definition of c2 . Surely with |||P − P|||

2.1. ENERGY ESTIMATES APPROACH t {φ ≤ 0} (x, ξ, τ ) Figure 2.3: To the proof of Corollary 2.1.7

135 One can prove this Corollary using ψ =   ¯ 2 + ε2 1/2 −  |x − x¯|2 + |ξ − ξ| with an arbitrarily small constant  and the domain {t + ψ ≤ 0} is drawn here.

We will need the following 



Definition 2.1.8. Let x = (x  , x  ) ∈ Rd × Rd be a partition of variables, d = d  + d  . Let u be an admissible (in the previous sense) function. Let   us introduce WFs (u) ⊂ T ∗ Rd × Rd . Namely, (¯ x , ξ¯ ) ∈ / WFs (u) iff there   exists a symbol q(x, ξ  ) ∈ C1 (Rd , Sh (T ∗ Rd , C)) such that q(¯ x , ξ¯ ) = 0 and  s 2 d q(x, hD )u ≤ Ch where . means the L (R , H)-norm. We will call WFs (u) partial wave front set. Partial wave front sets are useful when studying boundary value problems, including those in the domains with edges or conical points. Using the natural partition (t, x) and in the definition replacing the 2 L (Rd , H)-norm by the L2 ([0, T ] × Rd , H)-norm we obtain Corollary 2.1.9. Under the assumptions of Corollary 2.1.7  WFs (u) ⊂ (t, x, ξ) : ∃(t  , y , η) ∈ WF(s+1) (Pu) and

with

± t  ∈ [0, ±t]

dist((x, ξ), (y , η)) ≤ c|t − t  |}∪

{(t, x, ξ) : ∃(y , η) ∈ WFs ((u|t=0 )

with

dist((x, ξ), (y , η)) ≤ c|t| ;

this assertion holds for intervals [0, T ], [−T , 0], [−T , T ]; in particular in the last case we obtain that     (2.1.32) π WFs (u)t=0 ⊂ WFs (u)t=0 ⊂ WFs (u t=0 ) ∪ WF(s+1) (Pu)t=0 where π : (t, x, τ , ξ) → (t, x, ξ) is a natural map. Corollary 2.1.9 states the finite speed of propagation of singularities. The second of its assertions also is useful: it justifies the application of Theorem 2.1.2 for the propagation of singularities of solutions to the Cauchy problem without loss in the order of “smallness”.

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136

2.1.3

Cauchy Problem. II

Let us now consider the Cauchy problem for the operator P = B −1 hDt + A

(2.1.33)

where A ∈ Ψh (Rd , H, H) satisfying (2.1.25) and (2.1.34) B is a self-adjoint operator in H such that Bw , w  ≥ ||w ||2

∀w ∈ D(B).

Let u ± = θ(±t)u where here and in what follows θ is the Heaviside function. Then (2.1.35)

Pu ± = (Pu)± ∓ ihδ(t)B −1 u0 ,

u0 = ð 0 u

where ð0 is the operator of restriction to {t = 0}. Then we obtain easily Proposition 2.1.10. Let conditions (2.1.33), (2.1.25) and (2.1.34) be fulfilled and N = N(s, l). Then (2.1.36)

WFs+1 (Pu ± ) ⊂ WFs+1 (Pu) ∩ {±t ≥ 0}∪

 1 (s+1) (Pu)t=0 ∪ WFs+ 2 (ð0 u)) π−1 0 (WF

where π0 : (0, x, τ , ξ) → (x, ξ) is a natural map. Remark 2.1.11. (i) Obviously function φ = t satisfies the microhyperbolicity condition in the domain Ω × {|τ | ≤ 2c  } if and only if (2.1.37)

||B − 2 w || + ||a(x, ξ)w || ≥ 0 ||w || 1

∀w ∈ H ∀(x, ξ) ∈ Ω .

More precisely, the microhyperbolicity condition (for φ = t) at some energy level τ ∈ [−2c  , 2c  ] 4) yields (2.1.37) while (2.1.37) yields the microhyperbolicity condition at every level τ ∈ [−2c  , 2c  ]; without any loss of generality we assume that c  = c here. (ii) One can check easily that replacing in (2.1.37) B − 2 by B −δ with an exponent δ > 0 we obtain exactly the same condition (2.1.37). 1

4)

I.e., the standard microhyperbolicity condition for the operator A − B −1 τ I .

2.1. ENERGY ESTIMATES APPROACH

137

Observe that if conditions (2.1.37) and (2.1.31) are fulfilled then microhyperbolicity condition (2.1.1) is fulfilled for φ = t + ψ(x, ξ) with any real-valued function ψ such that |∇x,ξ ψ| ≤ 2 with 2 = 2 (0 , c) > 0. This observation, the uniform version of Proposition 2.1.10, Theorem 2.1.2 and the fact that operators P and ϕ(hDt ) commute imply Proposition 2.1.12. Let conditions (2.1.31), (2.1.33), (2.1.34), (2.1.25) and (2.1.37) be fulfilled. Let T ∈ [, c1 ], T  = (1 + )T and Qj = Op(qj ) with qj ∈ Sh (T ∗ R, C) such that dist(Ω , supp(1 − q1 ), supp(q2 )) ≥ C0 T

(2.1.38)

with C0 = C0 (c, 0 ). Let χ ∈ C0N ([−1, 1]) and ϕ ∈ C0N ([−c, c]) be fixed functions, χT (t) = χ( Tt ). Then (i) The inequalities (2.1.39)

u[−T  ,T  ] ≤ h−l ,

(2.1.40)

B − 2 Q1 u0 ≤ hs+ 2 1

Q1 Pu[−T  ,T  ] ≤ hs+1 , 1

imply that (2.1.41)

ϕ(hDt )χT (t)Q2 u ≤ Chs

¯ +  |||qk |||, A¯ where C depends on d, s, l, c, 0 , c1 , c2 , and , c2 = |||A − A||| k is an arbitrarily chosen (constant) Hermitian operator in H and N = N(s, l) in the conditions. (ii) In particular, for u ≤ 1 we obtain that (2.1.42)

 WFs (u) ∩ |t| ≤ T , |τ | ≤ c, (x, ξ) ∈ Ω  ⊂ (t, x, ξ) : ∃(t  , y , η) ∈ WFs+1 (Pu) with ± t  ∈ [0, ±t]

and dist((x, ξ), (y , η)) ≤ C0 |t − t  | ∪  1 1  (t, x, ξ) : ∃(y , η) ∈ WFs+ 2 (B − 2 u t=0 ) ∪ ∂Ω with dist((x, ξ), (y , η)) ≤ C0 |t| .

So we state here the finite speed of propagation of singularities for energy levels τ ∈ [−c, c].

CHAPTER 2. PROPAGATION IN THE INTERIOR...

138

2.1.4

Properties of Fundamental Solutions

Now let us treat the (approximate) fundamental solution to the Cauchy problem for operator (2.1.33). Let us consider L(H, H)-valued distribution U(t, x, y ) such that an operator U : L2 (Rd , H) → L2 (Rd+1 ,H) with Schwartz kernel U(t, x, y ) satisfies the following estimates in the L L2 (Rd+1 , H), L2 (Rd , H) U ≤ M,

(2.1.43)

χ¯T0 (t) Op(ζ)PU ≤ Mhs+1   and the estimate in the L L2 (Rd , H), L2 (Rd , H) -norm (2.1.44)

(2.1.45)

1

 Op(ζ)(ð0 U − I ) ≤ Mhs+ 2

where ζ = 1 in Ω  T ∗ Rd , here and below χ¯ (and χ as well) is a fixed function (2.1.46)

χ¯ ∈ C0∞ (R),

1 1 χ¯ = 1 on (− , ), 2 2

χ¯T (t) = χ(t/T ¯ )

 ≤ T0 ≤ c1 , |||ζ||| ≤ c1 and the constant M will be discussed later. −1 In particular, the Schwartz kernel u(x, y , t) of e ih BA cut outside of {|t| ≤ T , |ξ| ≤ C0 } satisfies these assumptions. Then Proposition 2.1.12 yields Proposition 2.1.13. Let conditions (2.1.31)–(2.1.34), (2.1.25), (2.1.37) and (2.1.43)–(2.1.45) be fulfilled with M ≥ 1, N = N(s, l). Let ϕ ∈ C0N ([−c, c]) be a fixed function and Qk = Op(qk ) where qk ∈ Sh (T ∗ Rd , H, H) satisfy condition   dist supp(q1 ), supp(q2 ) ∪ Ω ≥ C0 T (2.1.47) with C0 = C0 (c) and T = T0 /2. Then (2.1.48)

ϕ(hDt )χ¯T /2 (t)Q1 UQ2  ≤ CMhs .

This Proposition and Proposition 1.1.11 yield Corollary 2.1.14. Let conditions (2.1.31)–(2.1.34), (2.1.25), (2.1.37) and (2.1.43)–(2.1.45) be fulfilled with M ≥ 1, N = N(s, l). Let qk ∈ Sh (T ∗ Rd , H, H) and diam(supp(qk )) ≤ c, k = 1, 2. Then

2.1. ENERGY ESTIMATES APPROACH

139

(i) If (2.1.43) holds then for T ∈ [, T0 ] estimate (2.1.49) ||Ft→h−1 τ χT (t)Q1x U tQ2y || ≤ CMh−d

∀τ : |τ | ≤ c

∀x, y ∈ Rd

holds where here and in what follows tQ is a dual operator to Qdefined by t Qu = (Q ∗ u † )† , “ † ” is a natural antilinear map from H to the dual space H† and inversely, and operators with respect to y are written to the right of the function5) . (ii) Further, under assumption (2.1.47) the following estimate holds: (2.1.50) Ft→h−1 τ χT (t)Q1x U tQ2y  ≤ CMhs 3

2

1

∀τ : |τ | ≤ c 1

2

∀x, y ∈ Rd 3

Remark 2.1.15. (i) If Q = q(x, hD, x) then tQ = q(x, −hD, x); (ii) C is the same type of constant as before. Let us assume that a is microhyperbolic in Ω with respect to ∇# φ. Let us apply Theorem 2.1.2 with φ1 = ±t and φ2 = ±(φ − 1 t) with a small enough constant 1 to u ± and equation (2.1.35) replacing P by ±P. We then obtain immediately Proposition 2.1.16. Let conditions (2.1.31)–(2.1.34), (2.1.25), (2.1.37) and (2.1.43)–(2.1.45) be fulfilled with M ≥ 1, N = N(s, l). Let φ ∈ CbN (T ∗ Rd ) be a real-valued function such that φ C∞ ≤ c1 and a is microhyperbolic in Ω with respect to ∇# φ.6) Let qk ∈ Sh (T ∗ Rd , H, H) satisfy the conditions (2.1.51)

± φ(x, ξ) ≤ ±φ(y , η) + 1 T −  ∀(x, ξ) ∈ supp(q1 ) ∀(y , η) ∈ supp(q2 )

and (2.1.52) 5)

  dist supp(q1 ), Ω ≥ C0 T + 

This is a convenient notation because it agrees with the matrix theory notation even if it contradicts to the standard operator theory notation. 6) Then p is also microhyperbolic in Ω = {(t, x, τ , ξ) : (x, ξ) ∈ Ω , |τ | ≤ 21 } with respect to ∇# φ where 1 = 1 (0 , c, c1 ).

CHAPTER 2. PROPAGATION IN THE INTERIOR...

140

where T ∈ [, T0 ] is arbitrary and T0 = T0 (c) > 0 is a small enough constant,  > 0 is an arbitrarily small constant. Moreover, let χ± and ϕ ∈ C0N ((−21 , 21 )) be fixed functions where here and below (2.1.53)

1 χ+ ∈ C0∞ ([ , 1]), 2

χ− (t) = χ(−t),

χ = χ + + χ− .

Then s ϕ(hDt )χ± T (t)Q1 UQ2  ≤ CMh .

(2.1.54)

t

t {φ < 0}

{φ < 0}

{φ < 0} O

x1

(a) gray shaded zone is free of WFs

O

(b) now dark-shaded zone is also free of WFs

Figure 2.4: Illustration to the proof of Proposition 2.1.16 and Corollary 2.1.17; (x  , ξ, τ ) are not shown. All zones are open. Representing q1 and q2 as sums of symbols with small enough supports and applying Proposition 1.1.11 we obtain immediately Corollary 2.1.17. Let conditions (2.1.31)–(2.1.34), (2.1.25), (2.1.37) and (2.1.43)–(2.1.45) be fulfilled with M ≥ 1, N = N(s, l). Let φ ∈ CbN (T ∗ Rd ) be a real-valued function such that φ C∞ ≤ c1 and a is microhyperbolic in Ω with respect to ∇# φ. Let diam(supp(qk )) ≤ c, k = 1, 2. Then (i) The estimate (2.1.55)

t s Ft→h−1 τ χ± T (t)Γ(Q1x U Q2y ) ≤ CMh

∀τ : |τ | ≤ ε0

holds where here and in what follows  (2.1.56)

(Γx u)(x) = u(x, x)

x1

and

(Γu) =

(Γx u)(x)dx.

2.1. ENERGY ESTIMATES APPROACH

141

(ii) Moreover, if φ = φ(x) and supp(qk ) ⊂ {|ξ| ≤ c1 } ∀k = 1, 2 then the following estimate holds: t s (2.1.57) Ft→h−1 τ χ± T (t)Γx (Q1x U Q2y ) ≤ CMh

∀τ : |τ | ≤ ε0

∀x ∈ Rd .

Remark 2.1.18. Since condition to sign ± disappeared when we passed from Proposition 2.1.16 to Corollary 2.1.17, estimates (2.1.55) and (2.1.57) hold for both signs ± and thus for function χ instead of χ± . Actually this Corollary easily implies a much stronger version of itself: Theorem 2.1.19. Let conditions (2.1.31)–(2.1.34), (2.1.25), (2.1.37) and (2.1.43)–(2.1.45) be fulfilled with M ≥ 1, N = N(s, l). Let φ ∈ CbN (T ∗ Rd ) be a real-valued function such that φ C∞ ≤ c1 and assume a is microhyperbolic in Ω with respect to ∇# φ. Let qk ∈ Sh (T ∗ Rd , H, H) satisfy (2.1.52), and let diam(supp(qk )) ≤ c, k = 1, 2. Let T0 = T0 (c, 0 ) > 0 be a small enough constant. Then (i) For T ∈ [h, T0 ] the following estimate holds: (2.1.58)

||Ft→h−1 τ χT (t)Γ(Q1x U tQ2y )|| ≤ CMh1−d

 h s T

∀τ : |τ | ≤ ε0 .

(ii) Moreover, if φ = φ(x) and supp(qk ) ⊂ {|ξ| ≤ c1 } ∀k = 1, 2 then the following estimate following holds: (2.1.59) ||Ft→h−1 τ χT (t)Γx (Q1x U tQ2y )|| ≤ CMh1−d

 h s T ∀τ : |τ | ≤ ε0

∀x ∈ Rd

Proof. Let us observe that we need to prove estimates (2.1.58) and (2.1.59) only provided diam(supp(qk )) ≤ 2 , k = 1, 2 with a small enough constant 2 = 2 (d, 0 , c, c1 ) > 0 (otherwise one can make a partition of unity). Then without any loss of generality one can assume that φ = x1 : in fact, we can reduce the case φ = φ(x) to this one by a change of the coordinate system (and (2.1.59) remains true); meanwhile we can reduce the most general case to this one by replacing U by U  = Tx U tTy where T is a scalar Fourier integral operator such that T ∗ T ≡ I in a neighborhood of supp(q1 ),

CHAPTER 2. PROPAGATION IN THE INTERIOR...

142

T T ∗ ≡ I in a neighborhood of ΨT (supp(q1 )) and φ = x1 ◦ ΨT where ΨT is the corresponding symplectomorphism. For a parameter  r ∈ [h, 1] let us consider a real-valued r -admissible partition of unity k ψk2 in the vicinity Ω of supp(q1 ) 7) . Obviously, for c3 = c3 (d, c) such functions exist. Let us replace Q1 and Q2 by ψk Q1 and ψk Q2 respectively and make the rescaling (dilatation) t  = t/r , x  = x/r , h = h/r (in which case h Dx  = hDx and h Dt  = hDt ). Now we are in precisely the framework of Corollary 2.1.17 and therefore we obtain the estimate (2.1.57) in the new variables. Then in the old variables we obtain the estimate  h s r ∀x ∈ Rd ∀k = 1, ... , K .

(2.1.60) Ft→h−1 τ χ±rT0 (t)Γx (ψk2 (x)Q1x U tQ2y ) ≤ CMr 1−d ∀τ : |τ | ≤ ε0

Remark 2.1.20. (i) The factor r −d in the right hand expression appears because (2.1.61) The Schwartz kernel is a function with respect to x and a density with respect to y and therefore the restriction of the Schwartz kernel to the diagonal is also a density. (ii) The factor r in the right hand expression appears because Ft→h−1 τ = rFt  →h−1 τ .

(2.1.62)

These two simple observations are very important because the rescaling technique will be used very often. Finally, let us sum with respect to k and pick r = T /T0 . We get estimate (2.1.59). Moreover, we need Remark 2.1.21. In addition to conditions of Theorem 2.1.19 let us assume that (2.1.63)

B12j ϕ(hDt )χT (t)U(t)B12j  ≤ M

As usual this means that ψk (x) (k = 1, ... , K ) are real-valued functions such that 2  α −|α| ψ ∀α : |α| ≤ K ; K = c3 (1+r −d ) k k (x) = 1 in Ω , diam(supp(ψk )) ≤ r , |Dx ψk | ≤ c3 r and the multiplicity of the covering by supp(ψk ) is less than c3 . 

7)

2.1. ENERGY ESTIMATES APPROACH

143

where ϕ = 1 at (−c, c) and B1 is an operator satisfying (2.1.34). Then the standard interpolation arguments imply that every estimate obtained remains true if we multiply the left-hand expression by B1j on both sides and replace the s-degree of h or Th in the right hand expression by the corresponding s/2-degree.

2.1.5

Overdetermined Systems

Overdetermined Systems In certain cases we have an extra conditions to solutions of the evolution equation: f.e. for the standard Maxwell system ∂t E = ∇ × H, ∂t H = −∇ × E there are extra conditions ∇ · E = ∇ · H = 0. To cover this case we consider original system with such extra conditions. Remark 2.1.22. Let us note that if in Theorem 2.1.2 we know that Q1 P  u ≤ hs

(2.1.64)

then one can weaken microhyperbolicity condition (2.1.1): (2.1.1) −(p)(z)w , w  ≥ 0 ||w ||2 − c1 ||p(z)w ||2 − ||p  (z)w ||2

∀w ∈ H

where p  is the principal symbol of P  ∈ Ψh (T ∗ X , H , H). Indeed, replacement of (2.1.1) by (2.1.1) leads to the additional term C ||q1 p  w ||2 in the left-hand expression of (2.1.18) and then to the additional term C Q1 P  v 2 in the left-hand expression of (2.1.21). Other details of the proof do not change. Multi-Time Systems We will need to consider such systems to analyze systems of commuting operators. Namely let us consider the solution of the multi-time Cauchy problem (2.1.65) (2.1.66)

hDtj U ≡ −Aj U  U

in Ω (j = 1, ... , m), t1 =···=tm =0

≡I

where Aj = Aj (x, hD, h) commute modulo negligible operators: (2.1.67)

[Aj , Ak ] ≡ 0

∀j, k = 1, ... , m.

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Then Proposition 2.1.13 and Corollary 2.1.14 remain true with χ(t) ¯ and χ(t) understood as the products of the corresponding one-dimensional functions with t = (t1 , ... , tm ), τ = (τ1 , ... , τm ). Moreover, Proposition 2.1.16 remains true under microhyperbolicity condition (2.1.1) with P = Aβ = β1 A1 + ... + βm Am , Pj = Aj for some β ∈ Rm with |β| = 1 and χ supported in the vicinity of β. Repeating the proof of Theorem 2.1.19 with rescaling rule (2.1.62)

Ft→h−1 τ = r m Ft  →h−1 τ

instead of (2.1.62) we arrive to Theorem 2.1.23. Let conditions (2.1.31), (2.1.25), (2.1.43) and (2.1.65)– (2.1.65) be fulfilled for j = 1, ... , m with M ≥ 1, N = N(s, l). Suppose φ ∈ CbN (T ∗ Rd ) is a real-valued function such that φ C∞ ≤ c1 and the microhyperbolicity condition (2.1.1) is fulfilled in Ω with T = ∇# φ, p = aβ = β1 a1 + · · · + βm am and pj = aj . Let qk ∈ Sh (T ∗ Rd , H, H) satisfy (2.1.52) and let diam(supp(qk )) ≤ c, k = 1, 2. Let χ ∈ C∞ be a fixed function supported in the neighborhood of β and T0 = T0 (c, 0 ) > 0 be a small enough constant. Then (i) For T ∈ [h, T0 ] the estimate (2.1.58) Ft→h−1 τ χ±T (t)Γ(Q1x U tQ2y ) ≤ CMhm−d

 h s T

∀τ : |τ | ≤ ε0

holds for a sufficiently small constant 0 = 0 (c, 0). (ii) Moreover, if φ = φ(x) and supp(qk ) ⊂ {|ξ| ≤ c1 } ∀k = 1, 2 then the estimate  h s T ∀τ : |τ | ≤ ε0

(2.1.59) Ft→h−1 τ χ±T (t)Γx (Q1x U tQ2y ) ≤ CMhm−d

∀x ∈ Rd

holds. (iii) If an appropriate function φ = φβ exists for every β ∈ Sm−1 then   the corresponding estimate   (2.1.58) or (2.1.59) holds for every function 1 ∞ χ ∈ C0 B(0, 1) \ B(0, 2 ) .

2.2. GEOMETRIC INTERPRETATION

2.2 2.2.1

145

Geometric Interpretation Discussion of the Microhyperbolicity Condition

Scalar Case In this Section we consider a more geometrical version of Theorem 2.1.2, the main Theorem of the previous section. Then we will discuss various special cases of this main Theorem. We assume throughout this section that Ω  T ∗ Rd and   ¯ L(H, H) is Hermitian and microhyperbolic at (2.2.1) Symbol p ∈ Cb1 Ω, each point z = (x, ξ) ∈ Ω in the direction ∇# x1 . So x1 will play a role of time. In the simplest case of scalar principal symbol p ∈ C2 microhyperbolicity condition {p, φ} > 0 Ψt means exactly that ddt φ > 0 along Hamiltonian trajectories (as p = 0) and then Theorem 2.1.2 immediately implies that wave fronts propagate along such trajectories as Figure 2.5: Scalar principal shown on the picture to the left: one can type; shaded zone is {φ < 0} find φ strongly increasing along Hamiltonian trajectories such that {φ < 0} is an arbitrarily narrow “finger” there. Now we want to provide a similar geometric description in the case of the matrix8) symbol p. Cones of Directions of Microhyperbolicity Let us introduce the closed invariant subspace Hρ (p, z) ⊂ H of the matrix p(z) corresponding to the interval [−ρ, ρ] of its spectrum where ρ > 0 and z ∈ ω; recall that p(z) is Hermitian. Let us define for ρ > 0, ε > 0 the sets (2.2.2) Kρε (p, z) = { ∈ Tz Ω \ 0 : −(0 p)(z)w , w  ≥ ε||w ||2 ∀w ∈ Hρ (p, z)} 8) It is more convenient to use term “matrix” for an operator acting in the auxiliary spaces.

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CHAPTER 2. PROPAGATION IN THE INTERIOR...

where 0 = /||. Moreover, let us introduce the set   Kρε (p, z) = Kρ0 (p, z) (2.2.3) K (p, z) = ρ>0,ε>0

ρ>0

where the second equality is due to (2.2.1) (and we temporarily extended (2.2.2) to ε = 0). Then the following assertion is obvious: Lemma 2.2.1. (i) If || ≤ c satisfies (2.1.1) then  ∈ Kρε (p, z) with ε = ρ = ρ(0 , c1 ) > 0. On the other hand, if || ≥ ρ and  ∈ Kρε (p, z) then (2.1.1) holds with 0 = 0 (ρ, ε) > 0 and c1 = c1 (ρ, ε). (ii) Kρε (p, z) is a conical strictly convex respectively closed subset of Tz Ω \ 0 where here and below 0 means the 0-section of this bundle. (iii) For ρ < ρ and ε < ε the set Kρ ,ε (p, z) contains the ε -neighborhood of Kρε (p, z) ∩ {|| = 1} with ε = ε (0 , c1 , ρ, ρ , ε, ε ). (iv) Hence K (p, z) is a conical convex open subset of Tz Ω \ 0; (v)  ∈ ∂K (p, z) if and only if lim

inf

ρ→+0 w ∈Hρ (p,z), ||w ||=1

(p)(z)w , w  = 0.

Definition 2.2.2. Let M(z) be set-valued function; then (i) As zn → z we define for such functions limits 

lim M(zn ) :=

zn →z

ε>0 n

M(zn )

 ε

  where Mε is ε-vicinity of M; for conical set we set Mε = (M ∩{|ζ| = 1})ε cone instead where Mcone = {tζ, ζ ∈ M, t > 0} is a conical hull of M; (ii) We say that M(z) is upper semicontinuous, lower semicontinuous and continuous at z if for any sequence zn → z limzn →z M(zn ) ⊂ M(z), limzn →z M(zn ) ⊃ M(z) and limzn →z M(zn ) = M(z) respectively. Obviously, K (p, z) introduced by (2.2.2)–(2.2.2) is lower semicontinuous; moreover

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Lemma 2.2.3. (i) 9) {(z, ) ∈ T Ω :  ∈ Kρε (p, z)} is a respectively closed subset of T Ω \ 0 where 0 means the 0-section of this bundle. Furthermore, let the index set M be a Hausdorff topological space and assume that for all m ∈ M the symbol pm satisfies (2.2.1) and that the map M  m → pm ∈ Cb1 (T ∗ Rd ) is continuous; then the set {(m, z, ) ∈ M × T Ω :  ∈ Kρε (p, z)} is a respectively closed subset of M × (T Ω \ 0). (ii) Statement (iii) of Lemma 2.2.1 remains true if we replace Kρ ε (p, z) by {(z, ) ∈ T Ω :  ∈ Kρε (p, z)} and even by {(m, z, ) ∈ M × T Ω :  ∈ Kρε (p, z)} provided the index set is a metric space, and for all m ∈ M the symbol pm satisfies (2.2.1) and the map M  m → pm ∈ Cb1 (T ∗ Rd ) is uniformly continuous (surely in this case  depends on this map also). (iii) 10) {(m, z, ) ∈ M×T Ω :  ∈ K (p, z)} is an open subset of M×(T Ω\0) provided the additional conditions at the end of Statement (ii) are fulfilled. (iv) 11) (z, ) ∈ ∂{(z, ) ∈ T Ω :  ∈ K (p, z)} iff lim

inf

ρ→+0,z  →z w ∈Hρ (p,z  ), ||w ||=1

(p)(z  )w , w  = 0.

Definition 2.2.4. (i) We call K (p, z) microhyperbolicity cones. # (ii) Furthermore, let us introduce the propagation cones Kρε (p, z) and

(2.2.4)

K # (p, z) =

# Kρ0 (p, z)

# Kρε (p, z) = ρ>0,ε>0

ρ>0

where for any set G ⊂ T ∗ Rd the dual cone G # is defined by G # = {ζ ∈ T ∗ Rd \ 0, σ(ζ, ζ  ) ≤ 0 ∀ζ  ∈ G } and σ(., .) is a bilinear symplectic form12) . Since K (p, z) is lower semicontinuous (see definition 2.2.2) we conclude that (2.2.5) K # (p, z) is upper semicontinuous. 9) 10) 11) 12)

Cf. Lemma 2.2.1(ii). Cf. Lemma 2.2.1(iv). Cf. Lemma 2.2.1(v). Surely, G # is always conic convex closed set in T ∗ Rd .

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Examples Example 2.2.5. (i) If p is elliptic at z, i.e., if (2.2.6)

||p(z)w || ≥ ||w ||

∀w ∈ H

then K (p, z) = T ∗ Rd \ 0 and K # (p, z) = 0. (ii) Suppose that for z ∈ Ω the symbol p(z) is unitarily equivalent to   0 q(z)(ξ1 − λ(z  )) , H = H ⊕ H (2.2.7) 0 p  (z) where the symbols q and p  are bounded and Hermitian, λ is a real-valued function with z  = (x, ξ  ), ξ  = (ξ2 , ... , ξd ) in this example and p  is elliptic i.e. satisfies (2.2.6) on H , (2.2.8)

q(z)w , w  ≥ ||w ||2

∀w ∈ H .

Then in a neighborhood of Σ = {ξ1 = λ(z  )} all those symbols and functions (including the symbol of the operator effecting the unitary equivalence) can be chosen as smooth as p; moreover, for z ∈ Σ (2.2.9) and (2.2.10)

K (p, z) = {( , η1 ) ∈ T ∗ Rd : η1 > ( λ)(z  )} K # (p, z) = (∇# (ξ1 − λ))cone .

(iii) Suppose that for z ∈ Ω, p(z) is unitarily equivalent to the symbol ⎛ ⎞ p1 (z) ⊗ I1 · · · 0 0 ⎜ ⎟ .. .. .. ⎜ . . . 0 ⎟ (2.2.11) ⎜ ⎟ ⎝ 0 · · · pn (z) ⊗ In 0 ⎠ 0 ··· 0 p  (z) where H = H1 ⊕ ... ⊕ Hn ⊕ H , Hj = Hj ⊗ H0j , Hj are finite-dimensional spaces, pj (z) are matrices in Hj and Ij are the identity operators in H0j . Let us assume that all the symbols are Hermitian and smooth, the symbol effecting the unitary equivalence is smooth too and p  satisfies (2.2.6) at H . Then (2.2.12)

K (p, z) =

K (pj , z) 1≤j≤n

2.2. GEOMETRIC INTERPRETATION and (2.2.13)

K # (p, z) = convex hull

 

149

 K # (pj , z) = K # (pj , z)

1≤j≤n

1≤j≤n

 where Gj := {ζ = ζ1 ... + ζn : z1 ∈ G1 , ... , zn ∈ Gn } is an arithmetical sum of sets. Microhyperbolicity: Algebraic Theory In the framework of (2.2.1) and (2.2.11) let us consider the characteristic symbol (2.2.14)

g (z) =



det pj (z).

1≤j≤n

Proposition 2.2.6. Let g have a zero of order exactly r at z¯ ∈ Ω. (i) Then13) (Hxr1 g )(¯ z ) = 0

(2.2.15) and  ∈ K (p, z¯) if and only if (2.2.16)

(j Hxr1−j g )(¯ z ) · (Hxr1 g )−1 (¯ z) > 0

∀j = 1, ... , r .

Moreover,  ∈ K (p, z¯) if and only if the corresponding non-strict inequalities are fulfilled. (ii) Further, K (p, z¯) coincides with the connected component of the set {ζ : gz¯(ζ) = 0} where gz¯(ζ) is a germ of g at z¯: (2.2.17)

gz¯(ζ) =

1 (∂ α g )(¯ z )ζ α α! z

|α|=r

(iii) Furthermore, gz¯(ζ) is a hyperbolic polynomial of ζ with respect to ∇# x1 . 13)

Here and below sometimes we use notation Hf g := (∇# f )(g ) = {f , g }.

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150

Proof. Let us first consider the special case r = r1 + ... + rn where rj = dim Hj . Then (2.2.1) easily implies that p1 (¯ z ) = ... = pn (¯ z ) = 0 and (2.2.15) holds. In this case  ∈ K (p, z) iff −(p)(¯ z ) is a positive definite matrix, i.e., iff   (− − tHx1 )p (¯ z ) is a positive definite matrix for every t ≥ 0 and that is obviously equivalent to (2.2.16). Moreover, Statements (ii) and (iii) are also obvious in this case. On the other hand, if r < r1 + ... + rn then in a neighborhood of z¯ the finite-dimensional symbol ⎞ ⎛ p1 · · · 0 ⎟ ⎜ p  = ⎝ ... . . . ... ⎠ · · · pn    p 0 is unitarily equivalent to the matrix where p IV satisfies (2.2.6) 0 p IV on the corresponding space and the dimension of the first component of the decomposition is r . It is obvious that condition (2.2.15) or (2.2.16) for g = det p  is fulfilled if and only if the same condition is fulfilled for g  = det p  . Moreover, the germs of g and g  in z¯ coincide modulo of a constant factor. Finally, it is obvious that the microhyperbolicity conditions for p and p  are equivalent. 0

2.2.2

Main Theorem

Main Theorem To prove the main result of this section Theorem 2.2.10 we need first to reformulate Theorem 2.1.2 in the following way: Proposition 2.2.7. Let conditions (2.1.2) and (2.2.1) be fulfilled. Consider ¯ ∈ Ω, a proper convex closed cone G ⊂ {(x, ξ) ∈ T ∗ Rd , x1 > 0} ∪ 0 z¯ = (¯ x , ξ) and t¯ < x1 . Assume that for some ρ > 0, ε > 0 # (p, z) ⊂ G ∀z ∈ (¯ z − G ) ∩ {t¯ ≤ x1 ≤ x¯1 } (2.2.18) Kρε where z¯ − G := {¯ z − z, z ∈ G }. Let u be temperate and let (2.2.19)

WFs+1 (Pu) ∩ (¯ z − G ) ∩ {t ≤ x1 ≤ x¯1 } ∩ Ω = ∅,

(2.2.20)

z − G ) ∩ {t ≤ x1 ≤ x¯1 } ∩ ∂Ω = ∅, WFs (u) ∩ (¯

(2.2.21)

z − G ) ∩ {x1 = t} ∩ Ω = ∅. WFs (u) ∩ (¯

2.2. GEOMETRIC INTERPRETATION Then (2.2.22)

151

z − G ) ∩ {t ≤ x1 ≤ x¯1 } ∩ Ω = ∅. WFs (u) ∩ (¯

Proof. Due to our conditions G = {(x, ξ) : x1 ≥ ψ(x  , ξ)} with continuous positive homogeneous of degree 1 convex function ψ. One can smoothen ψ replacing it by ψη = ψ ∗ η −2d ω(z/η) with the smooth nonnegative ω ∈ C0∞ such that ω dz  = 1. Then {x1 ≥ ψη (x  , ξ) + β} ⊂ G ⊂ {x1 ≥ ψη (x  , ξ) − β} for arbitrarily small β > 0 and small enough η = η(β) > 0. Furthermore, due to (2.2.18) and properties of cones Kρε (p, z) we conclude that ∇# φ ∈ Kρ ε (p, z) for all z ∈ {φ ≤ β} ∩ {x1 ≥ t¯} where φ(x, ξ) = x1 − x¯1 + ψ(x  , ξ), ρ = ρ/2, ε = ε/2 and β > 0 is small enough. Then Theorem 2.1.2 yields the Proposition. # We are going to take G = Kρε (p, z¯) and apply Proposition 2.2.7 for t¯ = x¯1 − η with η = η(ρ, ε) so small that (2.2.18) holds with ρ, ε replaced by ρ/2, ε/2. Furthermore, we are going to repeat this procedure for any z ∈ (¯ z − G ) ∩ {x1 = t¯} and so on, getting the domain Gρε which is the union of small cut-off cones. Finally, we are going to consider its limit as ρ → +0, ε → +0. To codify this construction let us give the key

Definition 2.2.8. (i) A generalized bicharacteristics (generalized (ρ, ε)-bicharacteristics) of p is a Lipschitz curve {z = z(t)} such that for a.a. t, z(t) is differentiable and (2.2.23) (2.2.24)

dz ∈ K # (p, z), dt dz # ∈ Kρε (p, z) dt

respectively. # Since Kρε (p, z) ⊂ {|z  | ≤ Cx1 } then one can take x1 as a parameter t (let us recall that z = (x, ξ)). ± (ii) Let K± (p, z, Ω) (Kρε (p, z, Ω)) be the union of all the generalized bichar¯ and going out acteristics (generalized (ρ, ε)-bicharacteristics) of p lying in Ω ¯ in the direction of increasing ±t. from the point z ∈ Ω

Lemma 2.2.9. The introduced sets have the following properties:

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± (p, z, Ω) ⊂ Kρ± ε (p, z, Ω) for 0 < ρ < ρ , 0 < ε < ε ; (i) K± (p, z, Ω) ⊂ Kρε moreover, Kρ± ε (p, z¯, Ω) contains the η|x1 − x¯1 |-vicinity of every point z = ± (x, ξ) ∈ Kρε (p, z¯, Ω) intersected with Ω where η = η(0 , c1 , c2 , ρ, ρ , ε, ε ) > 0.

(ii) z  ∈ K± (p, z, Ω) if and only if z ∈ K∓ (p, z  , Ω). (iii) If z  ∈ K± (p, z  , Ω) and z  ∈ K± (p, z, Ω) then z  ∈ K± (p, z, Ω); State± ments (ii) and (iii) remain true for Kρε as well. ± (iv) K± (p, z, Ω) and Kρε (p, z, Ω) are closed sets.

(v) K± (p, z, Ω) =

± Kρε (p, z, Ω). ρ>0,ε>0

(vi) If zn → z, zn → z  and pn → p in Cb1 as n → +∞ and if zn ∈ K± (pn , zn , Ω) then z  ∈ K± (p, z, Ω) 14) . Proof. Statements (i)–(iii) are obvious; we prove (v); and (iv) and (vi) are proven in the same way. Let'us consider K+ (p, z, Ω). Then (i) implies that K+ (p, z, Ω) is contained in ρ>0,ε>0 Kρε (p, z,'Ω). + Let us prove the reverse inclusion. If z  ∈ ρ>0,ε>0 Kρε (p, z, Ω) then there exist sequences ρn → +0, εn → +0 and generalized (ρn , εn )-bicharacteristics z = zn (t) such that zn (0) = z, zn (tn ) = z  ; since one can take t = x1 + const then one can assume that tn = t0 . n (t) Since | dzdt | is uniformly bounded for all n and t ∈ [0, t0 ] then by the Ascoli-Arzela` Theorem one can assume that zn (t) → z(t) as n → +∞ uniformly with respect to t ∈ [0, t0 ] where z = z(t) is a Lipschitz curve; since z(t) is a function with bounded variation, by the Lebesgue Theorem (f.e. A. N. Kolmogorov, S. V. Fomin [1]) it is differentiable a.e. and we need only to prove that dz ∈ K # (p, z(t)) for a.a. t. dt It is obvious that for every ρ > 0, ε > 0 there exist  > 0 and n0 such # that zn − zn (t  ) ∈ ±Kρε (p, z(t  )) for all n ≥ n0 and t ∈ [t  − , t  + ] ∩ [0, t0 ] with ±(t − t  ) > 0. Then after the limit transition we obtain that z − z(t  ) ∈ # ±Kρε (p, z(t  )) for every ρ > 0, ε > 0 and indicated t, t  . This yields that if # z(t) is differentiable at t = t  then dz ∈ Kρε (p, z(t)) for t = t  and for all dt ∈ K # (p, z(t)) for t = t  . ρ > 0, ε > 0 and hence dz dt 14) However, even for pn = p it is possible that the set K± (p, z, Ω) is larger than the set of such limits.

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153

Now we can prove the central Theorem of this section: Theorem 2.2.10. Suppose Ω  T ∗ Rd and conditions (2.1.2) and (2.2.1) are fulfilled. Moreover, let z¯ ∈ Ω. Then if u is temperate and (2.2.25) and (2.2.26) then (2.2.27)

WFs+1 (Pu) ∩ K∓ (p, z¯, Ω) = ∅ WFs (u) ∩ K∓ (p, z¯, Ω) ∩ ∂Ω = ∅ WFs (u) ∩ K∓ (p, z¯, Ω) = ∅.

∓ Proof. Let us prove this statement with Kρε instead of K∓ . Then assertions (v) and (i) of Lemma 2.2.9 yield the statement in its original form. Let us − + consider Kρε ; replacing t, P by −t, −P one can reduce the Kρε case to this one. ¯ ∈Ω It is easy to see that if ρ > ρ > ρ , ε > ε > ε then for z¯ = (¯ x , ξ)     and t¯ = x¯1 − η with small enough η = η(0 , ρ, ρ , ρ , ε, ε , ε , c, c1 , c2 ) > 0 the following inclusions hold: # z − Kρε (p, z¯)) ∩ {x1 ≥ t¯} ⊃ Kρ− ε (p, z¯, Ω) ∩ {x1 ≥ t¯}. Kρ− ε (p, z¯, Ω) ⊃ (¯

Then the assertions of the Theorem for the domain Ωn = {z ∈ Ω, −c4 + ηn > x1 } instead of Ω are proven easily for an appropriate constant c4 by means of induction with respect to n: for n = 0 the assertion is trivial since Ω0 = ∅ for appropriate c4 , and the induction step is provided by Proposition 2.2.7. Therefore this assertion holds for Ω as well. Discussion Remark 2.2.11. (i) It is easy to see that in the assumptions of Theorem 2.1.2 the function φj increase along the generalized (ρ, ε)-bicharacteristics provided − ρ > 0, ε > 0 are small enough; hence if φ(¯ z ) < 0 and z ∈ Kρε (p, z¯, Ω) then φ(z) < 0. Therefore Theorem 2.1.2 and Theorem 2.2.10 are equivalent. (ii) One can generalize Theorem 2.2.10 to overdetermined systems. Assume that H = Hm and P  = (P1 , ... , Pm ) where Pj are self-adjoint and commute between themselves and with P. Then one should define Hρ (z) as an invariant  subspace of H corresponding to joint spectrum of p(z), p1 (z), ... , pm residing m+1 in [−ρ, ρ] .

CHAPTER 2. PROPAGATION IN THE INTERIOR...

154

Remark 2.2.12. (i) “Equations” of (2.2.23) type are called ordinary differential inclusions. They are very well known in the control theory. However our case is rather special: one can see easily that either in the generic point ∇# g = 0 and then in such points K # are 1-dimensional or at least g = g1r1 g2r2 · · · g k rk and again g redefined as g1 g2 · · · gk has the above property. So in our case in in generic points we have (up to parametrization15) ) ordinary differential equations. (ii) Let Σ(r ) (p) be a set of points in which g (z) has a zero exactly of order not exceeding r . One can see easily that K (p, z) and K # (p, z) are continuous functions of z ∈ Σ(r ) (p) \ Σ(r −1) (p). (iii) Let us define L(p, z) for z ∈ Σ(r ) (p) recurrently in the following way: (2.2.28)

z ∈ Σ(1) (p) =⇒ L(p) = K # (p, z),

(2.2.29)

z ∈ Σ(r ) (p) \ Σ(r −1) (p) =⇒    lim L(p, zn ) ∪ K σ (p, z) ∩ Tz Σ(r ) L(p, z) = {zn }⊂Σ(r −1) , zn →z

zn →z

where for any set M  z we define a conical subset in Tz Ω: (2.2.30) ζ ∈ / Tz M ⇐⇒ ∃ε > 0 such that dist(z + tζ, M) ≥ εt ∀t ∈ (0, ε). Due to (2.2.5) L(p, z) ⊂ K # (p, z). Obviously, L(p, z) is conic but not necessarily convex set. One can prove that actually generalized bicharacteristics satisfy a.e. dz ∈ L(p, z). dt

(2.2.31)

2.2.3

Examples of Propagation

We consider K− (p, z) in different cases. Constant Coefficients Case Example 2.2.13. Let p = p(ξ); then K (x, ξ) = K (ξ) = Rdx ⊕ Γ(ξ) with a conic set Γ(ξ) ⊂ Rdξ and K # (ξ) = Γ (ξ) ⊕ {0} where Rdx ⊃ Γ (ξ) is a dual cone to Γ(ξ). 15)

And we can take x1 as a parameter due to (2.2.1).

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155

  Finally, K± (x, ξ) = x ± Γ (ξ) × {ξ}. One can look at the phenomena of conical refraction (especially in Crystal Optics and Magnetic Hydrodynamics in R. Courant & D. Hilbert [2]. General theory is exposed in M F. Atiyah, R. Bott & L. Ga ˚rding [1] Scalar Principal Type (Again) Example 2.2.14. (i) If p satisfies (2.2.6) then K± (p, z, Ω) = z; (ii) If in Ω

  g (z)q(z) 0 p(z) = , 0 p  (z)

p  (z) and q(z) satisfy (2.2.6) and (2.2.8) respectively on the corresponding subspaces, g (z) is a real-valued scalar function with dg =  0 and if g (¯ z) = 0 then K± (p, z¯, Ω) = γ ± (p, z¯) is an interval of the semibicharacteristics of g ¯ i.e., of the curve z = z(t) with lying in Ω, (2.2.32)

dz = ∇# g (z), dt

z(0) = z¯

(±t ≥ 0).

In particular, if g = τ + a(x, ξ) where a is a scalar real-valued symbol and (x, ξ) is replaced by (t, x, τ , ξ) we obtain the system (2.2.33)

dz = ∇# a(z), dt

z = (x, ξ),

τ = −a(z) = const.

This case will be analyzed very carefully in Section 2.4. Until the end of subsection we are dealing in the framework of Example 2.2.5(iii). Operators with Completely Factorizing Characteristic Symbol: Non-Involutive Case We assume that characteristic symbol g (z) defined by (2.2.14) completely factorizes:  (2.2.34) gj (z), gj = 0 =⇒ Hgj =  0. g= 1≤j≤n

CHAPTER 2. PROPAGATION IN THE INTERIOR...

156 In this case

Σ := Char p =



Σj ,

Σj = {z : gj (z) = 0}.

1≤j≤n

 Then at Σj \ k =j Σk wave front sets propagate along bicharacteristics of gj and the question is what happens at Σj ∩ Σk with j = k. Obviously (2.2.35) g1 = ... = gn = 0 =⇒ K # (z) = {α1 ∇# g1 + α2 ∇# g2 (z) + ... + αn ∇# gn (z), α ∈ R+ n }. Let us consider the series of microlocal examples. So all statements below are true for guaranteed small time. Example 2.2.15. Let n = 2 and (2.2.36)

 0 gj = gk = 0 =⇒ {gj , gk } =

(j = k).

Then as z ∈ Σ1 ∩ Σ2 L(z) = {α1 ∇# g1 + α2 ∇# g2 , α ∈ R+ × 0 ∪ 0 × R+ } and K− (g , z) = γ1− (z) ∪ γ2− (z) where γj− (z) are semibicharacteristics of gj . The case of z ∈ Σ1 \ Σ2 is shown at pictures:

Σ2

Σ1

Σ2

Σ1

Σ2

Σ1

• z • Σ 1 ∩ Σ2

• Σ 1 ∩ Σ2

z • Σ 1 ∩ Σ2 • z

(a) γ1− (z) hits Σ1 ∩ Σ2

(b) z ∈ Σ1 ∩ Σ2

(c) γ1− (z) ∩ Σ1 ∩ Σ2 = ∅

Figure 2.6: Thick lines show K− (z) However Theorem 2.2.10 fails to notice that while branching at Σ1 ∩ Σ2 the singularity loses its power by 1/2: let z ∗ ∈ Σ1 ∩ Σ2 , zj± ∈ γj+ (z ∗ ) and different from z ∗ . Let γj (z, z  ) be a segment of the bicharacteristics with the ends z and z  .

2.2. GEOMETRIC INTERPRETATION

157

Proposition 2.2.16. In the framework of Example 2.2.15 / WFs (u), (2.2.37) γ1 (z1− , z1+ ) ∩ WFs+1 (Pu) = ∅, z1− ∈ γ2 (z2− , z ∗ ) ∩ WFs+1/2 (Pu) = ∅, z2− ∈ / WFs−1/2 (u) =⇒ z1+ ∈ / WFs (u). Proof. To prove this let us notice first that due to (2.2.36) (2.2.38) gj = gk = 0 = 0 =⇒ Hgj and ∇# gk are linearly independent for all j = k and then that multiplying p by elliptic symbols from the left and  right one can  P1 hB1 achieve p in the form (2.2.11) with n = 2 and pj = gj . Then P = hB2 P2 where Pj ∈ Ψh are operators with the principal parts gj and Bj ∈ Ψh . Let Rj be parametrices of Pj with the properly directed propagation; then one can run method of successive approximations andthus needs    to consider R1 0 0 hR1 B1 n and R = terms Q R where Q = . 0 R2 hR2 B2 0 Notice that Q 2 is diagonal with diagonal elements h2 R1 B1 R2 B2 and 2 h R2 B2 R1 B1 and QR has diagonal elements 0 and non-diagonal ones hR1 B1 R2 and hR2 B2 R1 . So one should consider two these later operators. Further, due to (2.2.36) one can assume without any loss of the generality that P1 = ξ1 and P2 = ξ1 − x1 and then one can take  x1 −1 −1 (2.2.39) (Rj u)(x) = h e ih (φj (x1 )−φj (y1 )) u(y1 , x  ) dy1 with φ1 = 0 and φ2 = 12 x12 . Then stationary phase method imply that R1 B1 R2   h−3/2 which easily implies (2.2.37). Remark 2.2.17. (i) Arguments of the above proof imply that if n ≥ 3, condition (2.2.36) is fulfilled and p is diagonal, then singularities propagate along branching bicharacteristics and while switching the branch magnitude acquires factor h1/2 ; so as l (in the condition u = O(h−l )) and s (in the WF) are fixed, only "2(l + s)# switches should be allowed. See V. Kucherenko [1] for the original proof (where n ≥ 2 was considered). (ii) K− is a different matter. Note that

158

CHAPTER 2. PROPAGATION IN THE INTERIOR...

(2.2.40) If J ⊂ {1, ... , n} and gj (z) = 0 ⇐⇒ j ∈ J then  L(z) = αj ∇# gj , α = (αk )k∈J ∈ Ker CJ ∩ R+ #J j∈J

where CJ = ({gj , gk })j,k∈J . If we know that (2.2.41) For each J ⊂ {1, ... , n} L(z) = 0 where L(z) is now defined by (2.2.40) then every generalized bicharacteristics is also a branching bicharacteristics. However, if assumption (2.2.41) 16) fails and one can chose non-vanishing  α ∈ C1 such that α(z) ∈ L(z) then bicharacteristics of g  = j∈J αj gj on ΣJ = {gj = 0 j ∈ J} are generalized bicharacteristics of p but not branching bicharacteristics of {gj }j∈J . Example 2.2.18. Let n = 2, (2.2.38) holds but {g1 , g2 } vanishes at some points. Note that L(z) = K (z) as g1 = g2 = ω = 0. Further, assume that (2.2.42) g1 = g2 = ω = 0 =⇒ {g1 , ω} · {g2 , ω} > 0 with ω = {g1 , g2 }. Then while Σ1 and Σ2 intersect transversally due to (2.2.34), bicharacteristics of gj are tangent to Σ3−j as g1 = g2 = ω = 0. Note due to (2.2.42) one can consider ω as a “time” parameter along bicharacteristics as ω is close to 0; one can see easily that generalized bicharacteristics are ordinary bicharacteristics with branching at Σ1 ∪ Σ2 and that there are no more than two branching on each of bicharacteristics. More precisely, every maximally extended bicharacteristics passing near but not through Λ = Σ1 ∩ Σ2 ∩ {ω = 0} has either two (as on Figure 2.7(a)) or none (as on Figure 2.7(b)) branching points and each time trajectory switches branches the magnitude of solution acquires factor h1/2 . On the other hand, such trajectory passing through Λ has exactly one branching point (as on Figure 2.7(c)). Model: g1 = ξ1 , g2 = ξ1 − ξ2 − x12 /2. Conjecture 2.2.19. As trajectory switches branches on Λ the magnitude of solution acquires factor h1/3 . 16) As n = 3 (2.2.41) fails iff {g1 , g2 } · {g2 , g3 } > 0 and {g2 , g3 } · {g3 , g1 } > 0 (then the third inequality of this sort holds as well).

2.2. GEOMETRIC INTERPRETATION

γ1 (z)

159

γ1 (z)

γ1 (z)

•Λ

•Λ

Σ2

•Λ

Σ2

(a)

Σ2

(b)

(c)

Figure 2.7: Thick lines show K− (z) Proof of this conjecture would require a study of oscillatory integrals with degenerate phase functions. Example 2.2.20. Let n = 2, (2.2.38) holds and (2.2.43) g1 = g2 = ω = 0 =⇒ {g1 , ω} · {g2 , ω} < 0 with ω = {g1 , g2 }. In this case bicharacteristics of gj passing through Λ do not hit Σ3−j again. On the other hand if branching on Σ1 ∩ Σ2 \ Λ occurs then due to Proposition 2.2.16 factor O(h1/2 ) is acquired. Therefore again singularities propagate along branching bicharacteristics.  However again as in Remark 2.2.17(ii) bicharacteristics of g  = j αj gj  (with αj > 0 defined from j αj {gj , ω} = 0) are generalized bicharacteristics of p but not a branching bicharacteristics of {g1 , g2 }. Operators with Completely Factorizing Characteristic Symbol: Involutive Case Example 2.2.21. Assume that (2.2.44) For each J ⊂ {1, ... , n} ∇# gj , j ∈ J are linearly independent on ΣJ = {gj = 0 ∀j ∈ J}. and (2.2.45)

gj = gk = 0 =⇒ {gj , gk } = 0

∀j = k.

160

CHAPTER 2. PROPAGATION IN THE INTERIOR...

Then ΣJ are involutive manifolds of codimensions r = #J: (2.2.46)

∇ # g j ∈ T z ΣJ

∀z ∈ ΣJ ∀j ∈ J.

Therefore due to Frobenius Theorem ΣJ is locally stratified into submanifolds of dimension r :   # ΛJ (z) = e j∈J tj ∇ gj , t ∈ B(0, ) ⊂ Rr . (2.2.47) Let   t ∇# g j j∈J j Λ− (2.2.48) , t ∈ B(0, ) ∩ R− r J (z) = e which is diffeomorphic locally to Rr . Then K− (p, z) = Λ− J(z) (z) with J(z) = {j : gj (z) = 0}. In particular, if  J = {j} and r ≥ 2 we get limiting bicharacteristics γj (z) ⊂ ΣJ . In particular, if ΣJ   z  → z ∈ ΣJ and J  = J(z  )  J then K− (z  ) → − ΛJ  (z). Operators with not Factorizing Characteristic Symbol. I Consider case when g has a double characteristic point but is not completely factorizing in its vicinity. Then due to microhyperbolicity condition (2.2.1) microlocally gj2 (2.2.49) g = −g12 + 2≤j≤n

with smooth functions gj and n = 3, 4 17) . Let Σ = {g = 0} and (2.2.50)

Σ0 = {g = ∇# g = 0} = {gj = 0 ∀j = 1, ... , n}.

Then the eigenvalues of Hess# g calculated at Σ0 play the crucial role in the behavior of bicharacteristics and propagation of singularities18) . As z ∈ Σ0 consider the second order germ g (z; ζ) of g ; then due to (2.2.1) (2.2.51) At each point z ∈ Σ0 germ g (z; ζ) is a hyperbolic polynomial with respect to ∇# x1 17)

n = 3, 4 in the generic case of real or complex matrix symbol p, respectively. Some eigenvalues could be 0; the rest is coming in pairs belonging to (R ∪ iR) \ 0, and quartets belonging to C \ (R ∪ iR) and eigenvalues λ, −λ, λ† and −λ† have the same Jordan structure. 18)

2.2. GEOMETRIC INTERPRETATION

161

and therefore (2.2.52) At each point z ∈ Σ0 germ Hess# (z) either has only purely imaginary eigenvalues or it has two simple real eigenvalues ±λ(z) and all other eigenvalues are purely imaginary; further, all non-zero purely imaginary eigenvalues ±iλj come in pairs (so the multiplicity of iλj and −iλj coincide and associated Jordan cells have height 1 while Jordan cells associated with eigenvalue 0 “normally” have heights at most 2; as there are no real eigenvalues there always is at least one Jordan cell of height 2 associated with eigenvalue 0 and even one cell of the height 4 may appear. One can deduct the proof from Williamson Theorem (see V. Arnold [2]) or find it in V. Ivrii & V. Petkov [1] or L. H¨ormander [6] where the crucial role of this distinction in the Cauchy problem is uncovered. We just note that non-zero eigenvalues of Hess# g coincide with those of the product diag(1, −1, ... , −1)({gj , gk })j,k=1,...,n . Let us assume that (2.2.53) ∇# g1 , ... , ∇# gn are linearly independent on Σ0 , which is equivalent to (2.2.54) Σ0 is C∞ -submanifold and Tz (Σ0 ) = Ker Hess# (z) at each point z ∈ Σ0 . Operators with not Factorizing Characteristic Symbol. II In this subsubsection we assume that (2.2.55) At no point z ∈ Σ0 matrix Hess# (z) has real non-zero eigenvalues. Then without any loss of the generality one can assume that (2.2.56) g (x, ξ) = −ξ12 + B(x, ξ  ) with B(x, ξ  ) ≥ 0, and |∂x1 B| ≤ cB. (2.2.57)

ξ  = (ξ2 , ... , ξd )

Indeed, we can achieve (2.2.56) by multiplying g by disjoint from 0 symbol and symplectic change of variables. Observe that if ∂x21 B > 0 at some point

CHAPTER 2. PROPAGATION IN THE INTERIOR...

162

z ∈ Σ0 then Hess# g (z) would have non-zero real eigenvalues; therefore ∂x21 B = 0 on Σ0 and then in virtue (2.2.53) Σ0 = {ξ1 = h2 (x  , ξ  ) = ... = hn (x  , ξ  ) = 0} which implies (2.2.57). 1 Observe that at level 0 (i.e. as g (x, ξ) = 0), |ξ1 | = B 2 and along bicharacteristics |dξ1 /dx1 | ≤ C and therefore we arrive to Statement (i) below. Then generalized bicharacteristics cannot enter or leave Σ0 and therefore dz/dt ∈ Tz Σ0 along generalized bicharacteristics residing on Σ0 ; further, dz/dt ∈ Ran Hess# (z) = (Tz Σ0 ) at point z ∈ Σ0 . Then we arrive to Statement (ii) below. Proposition 2.2.22. Let conditions (2.1.2), (2.2.1), (2.2.54) and (2.2.55) be fulfilled. Then (i) Let z ∈ Σ = {g = 0} and dist(z, Σ0 ) = ρ > 0; consider z(t) bicharacteristics of g with z(0) = z. Then (2.2.58) c −1 ρ ≤ dist(z(t), Σ0 ) ≤ cρ and

c −1 ρ|t| ≤ dist(z, z(t)) ≤ cρ|t|

as |t| ≤ ρ−1 . (ii) On Σ0 generalized bicharacteristics of p are defined by dz ∈ K # (p, z) ∩ Tz Σ0 ∩ (Tz Σ0 )⊥ dt

(2.2.59) which is (2.2.31). Assume now that

(2.2.60) σ|Σ0 has a constant rank. Then there exists foliation of Σ0 into isotropic submanifolds19) Λ(z): (2.2.61)

Tz Λ(z) = Tz Σ0 ∩ (Tz Σ0 )⊥

∀z ∈ Σ0

and generalized bicharacteristics reside inside the leaves of this foliation. Consider one leaf Λ(z). Without any loss of the generality one can assume that (2.2.62) Λ(z) = {ξ  = ξ  = x  = 0} with ξ  = (ξ  ; ξ  ) = (ξ1 , ... , ξk ; ξk+1 , ... , ξd ). 19)

I.e. manifolds such that σ|Λ = 0.

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163

Let us consider symbol g2 (x  ; ξ  , x  , ξ  ) which is a 2-germ of g (x, ξ) at Σ0 ; obviously g2 (x  ; ξ  , x  , ξ  ) is a second order homogeneous polynomial with respect to (ξ  , x  , ξ  ) and let us set x  = ξ  = 0; we get g  (x  , ξ  ) = g2 (x  ; ξ  , 0, 0) a second order homogeneous polynomial with respect to ξ  . Obviously all these polynomials are strictly hyperbolic. Then (2.2.63) K− (z) ⊂ Λ(z) and is the dependence conoid for g  (x  , ξ  ) 20) Problem 2.2.23. We leave to the reader (i) To prove that the generic case is k = 1, 2; so k = 1 only if n = k + 2m is odd (and thus equals 3). (ii) Investigate limiting bicharacteristics (residing on Σ0 ) in the cases of Σ0 involutive submanifold (i.e. rank σ|Σ0 = 2d − 2n which is equivalent {gj , gk }|Σ0 = 0 ∀j, k) and Σ0 not involutive submanifold (i.e. rank σ|σ0 > 2d − 2n); in particular, consider the case of Σ0 symplectic submanifold (i.e. rank σ|Σ0 = 2d − n). (iii) For each limiting bicharacteristics γ(z) prove existence of solution u with WF(Pu) = ∅, WF (u) = γ(z) (in the vicinity of a given point z ∈ Σ0 ). For detail see V. Ivrii [4]. Example 2.2.24. The toy-model symbol with such properties is  1 2 −ξ1 + ξj2 + ωj2 (xj2 + ξj2 ) (2.2.64) 2 2≤j≤k k+1≤j≤k+m with k ≥ 1, m ≥ 0 and ωj > 0; then n = k + 2m. Operators with not Factorizing Characteristic Symbol. III In this subsubsection in addition to (2.2.53) we assume that (2.2.65) At each point z ∈ Σ0 matrix Hess# (z) has two real non-zero eigenvalues ±λ, λ = 1 21) ; 20)

See any course in partial differential equations; R. Courant and D. Hilbert [2] is my favorite. 21) Obviously the latter could be achieved by multiplying g by λ−1 .

CHAPTER 2. PROPAGATION IN THE INTERIOR...

164

Then we can assume without any loss of the generality that (2.2.66)n

g (x, ξ) = x1 ξ1 + B(x  , ξ  ) + On (Σ0 )

with x  = (x2 , ... , xd ) etc, Σ0 = {(x, xi ) : x1 = ξ1 = 0, (x  , ξ  ) ∈ Σ }, Σ a smooth manifold, (2.2.67)

B(x  ξ  ) = O2 (Σ ),

B(z  )  dist(z  , Σ )2

On (Λ) denotes a space of C∞ -functions vanishing on Λ with all their derivatives of order less than n and so far n = 3. One can prove it easily from microhyperbolicity (in this case in the direction ∇# (x1 + ξ1 )) and conditions (2.2.53) and (2.2.65). Proposition 2.2.25. In appropriate symplectic coordinates (2.2.66)n and (2.2.67) hold with arbitrarily large n = ∞. Proof. Let us make induction with respect to n. Assume that (2.2.66)n has been proven and let us prove it with n replaced by (n + 1). Consider undesirable terms in the right-hand expression of (2.2.66)n : bjk (x  , ξ  )x1j ξ1k , bjk = On−l (Σ ) (2.2.68) j,k:j+k=l

with l = 1, ... , n. We do not care about l = 0 as we include this term in B(x  ξ  ). Let us make almost symplectomorphism xj → xj + ∂ξj S, ξj → ξj − ∂xj S with (2.2.69) S= sjk (x  , ξ  )x1j ξ1k , sjk = On−l (Σ ). j,k:j+k=l

Then canonical relations will be fulfilled modulo O2n−4 and they could be fixed by a symplectomorfism (x, ξ) → (x, ξ) + O2n−3 (Σ0 )) which does not spoil (2.2.66)n+1 as 2n − 3 ≥ n. One can prove easily that by such transformation one can eliminate all terms in (2.2.68) except those with j = k: bjj (x  , ξ  )(x1 ξ1 )j and those  latter 0 terms could be eliminated by multiplying g (x, ξ) by 1 + On−2 (Σ ) . Thus we can escalate l from l = 1 to l = n + 1 but then we get On+1 (Σ0 ). Therefore we can achieve (2.2.66)n for any n and since transition from n to n + 1 is achieved by (x, ξ) → (x, ξ) + On−1 (Σ0 )) we can take an asymptotic limit and achieve (2.2.66)∞ .

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165

Assume for a moment that instead of (2.2.66)∞ we have an exact equality (2.2.70)

g (x, ξ) = x1 ξ1 + B(x  , ξ  ).

Then B  (x  , ξ  ) is a movement integral and we have the following picture Therefore bicharacteristics with B(x  , ξ  ) = 0 miss Σ0 while bicharacterΣ2 Σ1 istics with B(x  , ξ  ) = 0 and thus with x1 ξ1 = 0 have limiting points on Σ0 as t → +∞ (for x1 = 0) and as t → −∞ (for ξ1 = 0) and on {B(x  , ξ  ) = 0} picture of K − (z) is as on Figure 2.6. More precisely: consider a toymodel (2.2.71). One can look at sec• Σ0 tion of the circular cone by planes parallel to its axis (as on Figure 2.8 as k = 2, m = 0); as m ≥ 1 one needs to remember about extra circular motions in (xj , ξj )-planes (j = k + 1, ... , k + m) which was Σ1 Σ2 present in the previous subsubsection as well. Figure 2.8: Bicharacteristics Example 2.2.26. The toy-model symbol with such properties is  1 (2.2.71) λx1 ξ1 + ξj2 + ωj2 (xj2 + ξj2 ) 2 2≤j≤k k+1≤j≤k+m with k ≥ 1, m ≥ 0 and ωj > 0; then n = k + 2m. In this case (2.2.72) Σ0 = {x1 = ξ1 = ξ  = ξ  = x  = 0}, x = (x1 ; x  ; x  ; x  ) = (x1 ; x2 ... , xk ; xk+1 , ... , xk+2m ; xk+2m+1 , ... , xn ), etc Σ0 has codimension k + 2m + 1 and rank σ|Σ0 = 2m. Then ξj (j = 2, ... , k), (xj2 + ξj2 ) (j = k + 1, ... , k + m) and xj , ξj (j ≥ k + m + 1) are motion integrals while x1 and ξ1 are exponents e ±λt . Let us prove that this happens in the general case as well:

CHAPTER 2. PROPAGATION IN THE INTERIOR...

166

Proposition 2.2.27. Let conditions (2.1.2), (2.2.1), (2.2.53), and (2.2.65) be fulfilled. Then (i) There exist (dim Σ0 + 1)-dimensional smooth submanifolds Σj ⊂ Σ = {g = 0} and real valued smooth symbols ηj (j = 1, 2) such that Σ0 = Σ1 ∩ Σ2 , (2.2.73)

dηj ∈ / T z Σ0

for z ∈ Σ0 ,

(2.2.74) and (2.2.75)

{η1 , η2 } = 1

on Σ0

ηj = 0

  and ∇# g = η3−j + O(|B|∞ ) ∇# ηj

on Σj

and moreover, (a) as z ∈ Σj bicharacteristics of g have limiting points on Σ0 as t → (−1)j ∞; (b) These bicharacteristics (modulo parametrization) coincide with bicharacteristics of ηj ; (c) further, for z ∈ Σj K− (z) consists of bicharacteristics of ηj and η3−j with branching on Σ0 exactly as on Figure 2.6. (ii) Let z ∈ {g = 0} \ (Σ1 ∪ Σ2 ). Then K− (z) = γg− (z) and it does not have limiting points on Σ0 . Proof. We sketch the proof; for details see Proposition 3.8 in V. Ivrii [1]. Consider bicharacteristics which start as t = 0 at z ∈ Σ ∩ {x1 + ξ1 = 0}, B(z) = ρ2 . Using (2.2.53) and canonical form (2.2.66)∞ one can prove easily that for 0 < t < | log ρ| − C0 |x1 |  ρe t ,

C0−1 ρe −t − C0 (ρe t )n ≤|ξ1 | ≤ C0 ρe −t + C0 (ρe t )n ,

and C0−1 ρ2 − C0 (ρe t )n ≤B(x  , ξ  ) ≤ C0 ρ2 + C0 (ρe t )n and then that |Dzα Φt (z)| ≤ Cα

∀z ∈ Σ ∩ {x1 + ξ1 = 0} \ Σ0 ,

1 0 < t < | log B(z)| − C . 2

Then as B(z) → 0 we get in the limit γ1± (z) for z ∈ Σ0 with signs ± corresponding two possible signs of x1 . Further, one can prove easily that

2.3. OPERATORS WITH ROUGH SYMBOLS

167

 γ1 (z) = γ1+ (z) ∪ γ2− (z) is a smooth curve and that Σ1 := z∈Σ0 γ1 (z) is a smooth manifold. Similarly, but with t replaced by −t we get γ2± (z), γ2 (z) and Σ2 . The proof of other statements is rather trivial. Remark 2.2.28. Observe that limiting bicharacteristics are of the form γ1± (z) ∪ γ2∓ (z) with z ∈ Σ0 . Problem 2.2.29. It would be interesting to investigate if one always can achieve (2.2.70) rather than just (2.2.66)∞ . Operators with not Factorizing Characteristic Symbol. IV Problem 2.2.30. (i) It would be interesting to consider symbols satisfying condition (2.2.76) At each point z ∈ Σ0 germ Hess# (z) has a Jordan cell of the height 4 associated with eigenvalue 0 with the toy-model (2.2.77)

 1 −2ξ1 ξ2 − αξ22 + x22 + ξj2 + ωj2 (xj2 + ξj2 ) 2 3≤j≤k k+1≤j≤k+m

with α = 0, k ≥ 3. Note that at ξ1 = ξ2 = x2 = 0 non-zero eigenvalues of Hess# (g ) are ±α1/2 and ±iωj so as α < 0 we are in the framework of Proposition 2.2.22 and as α > 0 we are in the framework of Proposition 2.2.27. One can analyze pretty easily bicharacteristics of (2.2.77) in all three cases of α = const. (ii) Further, it would be really interesting to consider variable α f.e. α = kx1 with k = 0 when type is changed on Σ0 .

2.3

Operators with Rough Symbols

Our goal now is to generalize Theorem 2.1.2 and then some of its corollaries to the case when instead of belonging to Ψh operator under consideration

168

CHAPTER 2. PROPAGATION IN THE INTERIOR...

belongs to Ψh,ρ,γ,N or Ψh,ρ,γ,an with ρ ≤ 1, γ ≤ 1 satisfying an appropriate form of the uncertainty principle. Operators with rough symbols serve as proxies (or approximations) for operators with irregular symbols and some rather limited results we derive for those later in Section 27.2.

2.3.1

Main Theorem

Statement of the Main Theorem Let us analyze assumptions of Theorem 2.1.2 and look at its proof and applications. We do not need to have φj “rough”, and smooth and even analytic functions work well for applications. On the other hand, we consider there [P, Qj ] and we need to assume that these operators also belong to Ψh,ρ,γ or Ψh,ρ,γ,an , i.e. in the smooth case we need to assume that γ pm (x, ξ)|| ≤ cmαβ ρ−α γ −β ε−m (2.3.1) ||∂ξα ∂xβ ∂x,ξ

∀m, α, β : |α| + |β| + 2m ≤ N ∀γ : |γ| ≤ 1 with ρ, γ satisfying uncertainty principle (1.1.97): (2.3.2)

ε = min ρj γj ≥ h1−δ . j

In the proof we need also assume that h−1 (P − P ∗ ) belongs to the same class, i.e. in addition to (2.1.2) we need to assume that † )(x, ξ)|| ≤ cmαβ ρ−α γ −β ε1−m (2.3.3) ||∂ξα ∂xβ (pm − pm

∀m, α, β : m ≥ 1, |α| + |β| + 2m ≤ N Then the proof of Theorem 2.1.2 implies Theorem 2.3.1. Suppose P ∈ Ψh,ρ,γ (Rd , H, H) satisfying (2.1.2), (2.3.1) and (2.3.3) with ρ, γ satisfying uncertainty principle (2.3.2). Let Ω  T ∗ Rd and let φ1 , ... , φJ be real-valued CbN functions in T ∗ Rd such that at every point z ∈ Ω the symbol p is microhyperbolic in the directions Hφ1 , ... , HφJ Let u be tempered, i.e., u ≤ h−l . Then (2.1.4) and (2.1.5) imply (2.1.6).

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169

Skipping easy details, instead let us go to a more difficult task: case of Ψh,ρ,γ,an and logarithmic uncertainty principle (1.1.174). Then we need to replace (2.3.1) and (2.3.2) respectively by γ pm (x, ξ)|| ≤ c(m!|α|!|β|!)σ ρ−α γ −β ε−m (2.3.4) ||∂ξα ∂xβ ∂x,ξ

∀m, α, β : |α| + |β| + 2m ≤ N ∀γ : |γ| ≤ 1 and † )(x, ξ)|| ≤ c(m!|α|!|β|!)σ ρ−α−m γ −β ε−m+1 (2.3.5) ||∂ξα ∂xβ (pm − pm

∀m, α, β : m ≥ 1, |α| + |β| + 2m ≤ N with N = −1 ,  = h/ε and (2.3.6)

min ρj γj ≥ ε = Ch| log h|σ j

σ ≥ 1.

Theorem 2.3.2. Suppose P ∈ Ψh,ρ,γ (Rd , H, H) satisfying (2.1.2), (2.3.4) and (2.3.5) with ρ, γ satisfying logarithmic uncertainty principle (2.3.6) with σ ≥ 1 and large enough C 22) . Let Ω  T ∗ Rd and let φ1 , ... , φJ be real-valued CbN functions in T ∗ Rd such that at every point z ∈ Ω the symbol p is microhyperbolic in the directions Hφ1 , ... , HφJ Let u be tempered, i.e., u ≤ h−l . Then (2.1.4), (2.1.5) imply (2.1.6). Proof of the Main Theorem Remark 2.3.3. (i) Recall that assumptions (2.1.4), (2.1.5) mean that for ¯ ∩ {φ1 ≤ appropriate scalar symbols q1 and q2 equal to 1 in the vicinities of Ω } ∩ · · · ∩ {φJ ≤ } and ∂Ω ∩ {φ1 ≤ } ∩ · · · ∩ {φJ ≤ } respectively and for Q1 , Q2 their Weyl quantizations (2.3.7)

Q¯1 Pu ≤ Chs+1 ,

Q¯2 u ≤ Chs .

(ii) Without any loss of the generality one can assume that (2.3.8)

Q¯1 u ≤ Chs−δ

with any fixed exponent δ > 0. Otherwise we will be able to prove Theorem in steps reducing  and shrinking supp(q1 ). 22) C must depend on all other constants including  in the definition of WF when we are looking at “-vicinity.”

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(iii) Without any loss of the generality one can assume J = 1: one can always find φ satisfying microhyperbolicity condition such that ¯ ∩ {φ < 0} ¯ ∩ {φ1 ≤ 0} ∩ · · · ∩ {φJ ≤ 0} ⊂ Ω Ω and ¯ ∩ {φ < } ⊂ Ω ¯ ∩ {φ1 ≤ } ∩ · · · ∩ {φJ ≤ }. Ω (iv) Without any loss of the generality one can assume that φ is analytic and symbols q¯j introduced in (i) satisfy (2.3.9)

|∂xα ∂ξβ φ| ≤ c |α|+|β|+1 (|α + β|)!

(2.3.10)

|∂xα ∂ξβ f | ≤ c(cN)(|α|+|β|−K )+

∀α, β : |α| + |β| ≤ N

with arbitrarily large fixed K 23) 24) . With Q = Q ∗ of the same type let us repeat (2.1.13): (2.3.11)

− Re i([P, Q]u, u) = Re i((Q + Q ∗ )Pu, u) + Re i((P ∗ − P)Qu, u).

As in the proof of Theorem 2.1.2 we pick up Q = q w (x, hD) with   (2.3.12) q = ζ(x, ξ)2 χ φ(x, ξ) where (2.3.13)

   and supp(ζ) ⊂ Ω ∩ φ <  . ζ = 1 in Ω ∩ φ < 2

Let χ(t) = 0 for t ≥ /4, both χ and ζ satisfy (2.3.10). One can prove easily that (2.3.14) Let both χ and ζ satisfy (2.3.10). Then q also satisfies (2.3.10) (with different constants). 23)

I.e. (2.3.10) holds for f = q¯j . Symbols qj satisfying (2.3.10) and supported in arbitrary small constant vicinities of the corresponding sets exist due to Proposition 1.A.2; however as σ = 1 such symbols depend on h but conditions are satisfied in the uniform manner. 24)

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171

Let us consider [P, Q] first. With O(hK ) error in (2.3.11) one can replace [P, Q] by ζ w r w ζ w with   (2.3.15) r = ζ1 p • χ(φ) − χ(φ) • p ζ1 with ζ1 of the same type (2.3.13) as ζ but with  replaced by 3. Really, the error would not exceed C0N+1 N N hN ε¯−N h−l with ε¯ = min min(ρj , γj ) ≥ ε j

and with fixed l and this expression does not exceed C0 (C0 N)N h−l with −1/σ −l  = h/ε and with N = 1 −1/σ it does not exceed e −2  h = O(hK ) −1 as  ≤ C | log h| with large enough constant C which follows from the logarithmic uncertainty principle (2.3.6). Consider (2.3.16) p • q − q • p + ih{p, q} =   h  |α|+|β|  α β 2 (−i)|α|−|β| h Dξ Dx p) · Dxα Dξβ q) α!β! 2 α,β:|α+|β|≥2, |α|−|β|∈2Z+1

and plug q = χ(φ). Note that 1 ν D χ(φ) = (2.3.17) ν!

k≤|ν|

ν1 ,...,νk : |ν1 |≥1,...,|νk |≥1, ν1 +...+νk =ν

χ(k) (φ)

 1 D νj φ ν j! 1≤j≤k

where χ(k) (t) = ∂tk χ(t). Therefore (2.3.18) Let χ satisfy (2.3.10). Then modulo negligible one can rewrite the right hand expression of (2.3.16) as (2.3.19) χ(l+1) (φ)rln (x, ξ)n+l h 1≤l+n≤N

with φ = φ(x, ξ) and (2.3.20) ||∂ξα ∂xβ rln || ≤ c1

l+n+|α|+|β|

(l!)σ−1 (n!α!β!)σ ρ−α γ −β ∀l, n, α, β : l + n + |α| + |β| ≤ N.

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172

We will be very particular choosing χ. Namely, we consider χ satisfying (2.3.10) and then replace in the formulae above χ(t) by  ∞  (j)  2  χ (t ) dt ; (2.3.21) χj (t) = 2j t

one can see easily that (2.3.22) −2j N −2j χj (φ) and N −j χ(j) (φ) satisfy (2.3.10) uniformly with respect to j ≤ N = 1 −1/σ ; therefore every formula above holds uniformly as well. Then (2.3.18) implies that (2.3.23) Modulo negligible one can rewrite the right hand expression of (2.3.16) with q = χj (φ) as (2.3.24) χ(l1 +j) (φ)χ(l2 +j) (φ)rl1 l2 n (x, ξ)2j+l1 +l2 +n h 1≤l1 +l2 +n≤N

with (2.3.25) ||∂ξα ∂xβ rl1 l2 n || ≤ c11

l +l2 +n+|α|+|β|

(l1 !l2 !)σ−1 (n!α!β!)σ · ρ−α γ −β

∀l, n, α, β : l1 + l2 + n + |α| + |β| ≤ N. Now using “revert” formula 2  h  |α|+|β|  α β  i |α|−|β| h Dξ Dx p) • Dxα Dξβ q) (2.3.26) pq ≡ α!β! 2 α,β: one can rewrite (2.3.24) as (2.3.27) l1 +j χ(l1 +j) (φ) • rl1 l2 n (x, ξ)hn • l2 +j χ(l2 +j) (φ) 1≤l1 +l2 +n≤N

rl1 l2 n

where Then

also satisfy (2.3.25).

  (2.3.28) j := h−1 p • χj (φ) − χj (φ) • p + ih{p, χj (φ)} ζ 2 ≡ tl1 +1 • rl1 l2 n (x, ξ)hn • tl2 +j + j 1≤l1 +l2 +n≤N

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173

with tm = χ(m) (φ)ζm and (2.3.29) j = ζ 2



 l1 +j χ(l1 +j) (φ) • rl1 l2 n (x, ξ)hn • l2 +j χ(l2 +j) (φ) −

1≤l1 +l2 +n≤N



ζl1 +j χ(l1 +j) (φ) • rl1 l2 n (x, ξ)hn • ζl2 +j χ(l2 +j) (φ)

1≤l1 +l2 +n≤N

and therefore (2.3.30) |(Lj u, u)| ≤ c2 Tj u · c2



 c2l (l!)σ−1 · Tj+l u +

1≤l≤N−j



c2l (l!)σ−1 · Tj+l u

2

+ C2 Tj u2 + |(Lj u, u)|

1≤l≤N−j

with Tm = Opw (tm ), Lj = Opw (j ) and Lj = Opw (j ). One can prove easily that due to (2.3.8) and (2.3.22)   c3l (l!)σ−1 · Tj+l u + C3 h2s 2j N 2j . (2.3.31) |(Lj u, u)| ≤ C3 hs j N j 0≤l≤N−j

Meanwhile (2.3.32)

{p, χj (φ)} = −χ(j) (φ)2 2j {p, φ}

and due to microhyperbolicity condition (2.3.33)

{p, φ} = b 2 + f † p + pf

where (2.3.34) both b, f satisfy (2.3.4) type estimates and b is Hermitian and elliptic. Therefore modulo (2.3.27)-type expression one can rewrite (2.3.32) as (2.3.35)

j χ(j) (φ) • b • b • j χ(j) + χ(j) • f † • p + p • f • j χ(j) .

Repeating arguments leading to (2.3.30)–(2.3.30) we conclude that (2.3.36)

− ih−1 ([P, Q  ] Opw (ζ 2 )u, u) ≥ 1 Tj u2 − (2.3.30) − (2.3.31).

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174

Here we used the (2.3.8) and the first of estimates (2.3.7) exactly as in the proof of Theorem 2.1.2. Also, without any loss of the generality we can assume that − ih−1 (p1† − p1 ) = b˜2 + f † p + pf

(2.3.37)

where both b, f satisfy (2.3.4) type estimates and b˜ is Hermitian and elliptic. Really, due to (2.3.33) we can achieve this by multiplying operator P by an equivalent operator e kφ Pe −kφ with large enough k ∈ R+ . Then the second term in the right hand expression of (2.3.11) multiplied by h−1 is less than −1 (Qj u, u) + Ch2s+δ due to Ga ˚rding inequality with the semiclassical parameter . Then (2.3.11) implies (2.3.38) 1 Tj u2 +1 (Qj u, u) ≤ (2.3.30)+(2.3.31)+chs Qj u+Ch2s 2j N 2j . Note, that qj2 ≤ (cN)j qj and using to Ga ˚rding inequality with the semiclassical parameter h we conclude that Qj u2 ≤ C (cN)j (Qj u, u) + Ch2s+δ

(2.3.39)

again due to (2.3.8). Then (2.3.38) implies that μj := 1 Tj u + 1 (Qj u, u)1/2

(2.3.40) satisfy inequalities (2.3.41)

μj ≤



μl c3l (l!)σ−1 + fn

with fn = C2 (cN)j hs

1≤l≤N−j

where the last term is greater than Chs+δ . Then25) (2.3.42) μj ≤ C4 c4l (l!)σ−1 fj+l . 0≤l≤N−j

Plugging fj+l and N = 1 1/σ one can see easily that μj ≤ Chs ; picking up j = 0 we conclude that (2.1.6) holds. Theorem 2.3.2 is proven. Really, solving (2.3.41) for fj = 0 and for j < k only one can prove by induction with respect to j = k, ... , 0 that μj ≤ C4 c4k−j ((k − j)!)σ−1 μk , which implies (2.3.42) for the original inhomogeneous system. 25)

2.3. OPERATORS WITH ROUGH SYMBOLS

2.3.2

175

Corollaries

After propagation of singularities Theorem is proven, it implies few corollaries similar to those of Theorem 2.1.2. Remark 2.3.4. In the regularity conditions of Theorem 2.3.2 assertions of Remark 2.1.3 and Proposition 2.1.4 remain true. Theorem 2.3.5. For operator P = hDt + A with the regularity of A as in Theorem 2.3.2 rather than of 2.1.2 (i) Assertion of Theorem 2.1.5 remains true: (2.1.28) and (2.1.29) imply (2.1.30); Remark 2.1.6 also remains true. (ii) Assertions of Corollaries 2.1.7 and 2.1.9 (finite speed of propagation) remain true. (iii) Assertions of Propositions 2.1.10 and 2.1.12 remain true. (iv) Assertions of Proposition 2.1.13 and Corollary 2.1.14 remain true. (v) Assertions of Proposition 2.1.16, Corollary 2.1.17 and Remark 2.1.18 remain true. (vi) Furthermore, as ρ = 1 assertions of Theorem 2.1.19 and Remark 2.1.21 remain true. Moreover, if ρ ≤ 1 then estimate (2.1.59) is replaced by (2.3.43) ||Ft→h−1 τ χT (t)Γx (Q1x U tQ2y )|| ≤ CMh1−d

 h s ρT ∀τ : |τ | ≤ ε0

∀x ∈ Rd

where ρ = minj ρj . (vii) Assertions of Remark 2.1.22 remains true. (viii) Finally, as ρ = 1 assertion of Theorem 2.1.23(ii) with the same modification as in (vi) remains true. Moreover, if ρ ≤ 1 then estimate (2.1.59) is replaced by  h s ρT ∀τ : |τ | ≤ ε0

(2.3.43) Ft→h−1 τ χ±T (t)Γx (Q1x U tQ2y ) ≤ CMhm−d

where ρ = minj ρj .

∀x ∈ Rd

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176

(ix) Further, if φ = φ(x  ) wuth ξ  = (ξ1 , ... , ξk ) then in (vi), (viii) one can take ρ = minj≤k ρj . (x) Meanwhile modified in the same way Statements Theorems 2.1.19(i) and 2.1.23(i) remain true as ρ ≤ γ. Proofs of all these statements just repeats those of those we refer to. Moreover, let us assume that as h → +0 symbol p(x, ξ, h) tends to some limit which we denote also by p. Let us define K (p, z), K # (p, z) and K± (p, z) for this limiting symbol. Then Theorem 2.3.6. For operator A with the regularity as in Theorem 2.3.2 rather than of 2.1.2 (i) Assertions of Lemma 2.2.1, Proposition 2.2.7 remain true; further, Definition 2.2.8 make sense and assertions of Lemma2.2.9 remain true. (ii) Moreover, assertion of Theorem 2.2.10 remains true: (2.2.25) and (2.2.26) imply (2.2.27). Again, proofs of all these statements just repeats those of those we refer to.

2.4

Heisenberg Approach to Propagation of Singularities along Long Bicharacteristics

The results of Section 2.2 imply the well known fact that singularities for operators with scalar principal symbols propagate along bicharacteristics. In this section we improve and generalize this result in two directions: namely, we treat large time intervals depending on h and we replace the description of the singularities by means of wave front sets with a more refined description related to boxes depending on h.

2.4. HEISENBERG APPROACH

2.4.1

177

Evolution of Operators

Heisenberg Approach We start with the Cauchy problem for the operator P = hDt + A

(2.4.1) where

(2.4.2) A = Aw (x, hDx , h) is an operator with a Hermitian Weyl symbol and hence A is a self-adjoint operator. Solution to Pu = 0 is given by ut = e −ih (2.4.3) with (2.4.4)

e −ih

−1 tA

Qu0 = e −ih

−1 tA

Qe ih

Q t = e −ih

−1 tA

−1 tA

−1 tA

u0 where ut = u(., t); then

· e −ih

Q 0 e ih

−1 tA

−1 tA

u0 = Q t ut

.

Therefore one can apply observable Q 0 = Q to the initial state or alternatively apply evolved observable Q t to the evolved state ut and the result would be the same. Equivalently one can apply observable Q 0 = Q to the evolved state ut or alternatively apply evolved observable Q −t to the initial state u0 and the result would be the same. In the latter case physicists would say that we use the Heisenberg representation. Since WF are intimately related to pseudodifferential operators, it is obvious that the investigation of the propagation of singularities of solutions to this Cauchy problem can be reduced to the treatment of the operator Q t where Q 0 = Opw (q 0 ) is a pseudodifferential operator. The crucial question is if Q t is a pseudodifferential operator. We already know that for a standard pseudodifferential operator A such that and (2.4.5) a = A0 is a scalar real-valued symbol. −1

and Q and finite time t the propagators e ±ih tA are Fourier integral operators corresponding to the symplectomorphisms ϕ±t where ϕt : z(0) → z(t) and z(t) is a solution of the Hamiltonian system (2.2.33). This implies that Q t is also a pseudodifferential operator. However, we use these results only as a heuristic preliminary.

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178

Definition (2.4.4) implies that Q t is a solution of the Cauchy problem ∂t Q t = −ih−1 [A, Q t ],  Q t t=0 = Q 0 .

(2.4.6) (2.4.7)

Let us assume that Q t is a pseudodifferential operator with Weyl symbol q where for now we treat symbols as formal power series; we will justify this assumption later. Then the Cauchy problem (2.4.6)–(2.4.7) implies the similar Cauchy problem for symbols; namely, for a term qnt in the decomposition of q t we obtain the equation t

(2.4.8)n ∂t qnt + {a, qnt } + i[as , qnt ] =  1 (α) t(β) t(β) (α)  (−i)|α|−|β| Aj(β) qk(α) − i |α|−|β| qk(α) Aj(β) |α|+|β| 2 α!β! 0≤k≤n−1 j+|α|+|β|=n−k

with initial condition (2.4.9)n

 qnt t=0 = qn0 .

Recall that {., .} are Poisson brackets, [., .] is the commutator in L(H, H), as is the subprincipal symbol of A (i.e., as = A1 ), An are terms in the (α) decomposition of the Weyl symbol of A and b(β) = ∂xβ ∂ξα b. Now we throw of our assumption that Q t is an h-pseudodifferential operator. We needed this assumption only to derive problems (2.4.8)n – (2.4.9)n . So, let us consider this problem, where for now “symbol” means “function” rather than “symbol of an operator.” Let us start with the principal symbol q0t which satisfies the following transport equation (2.4.8)0

∂t q0t + {a, q0t } + i[as , q0t ] = 0

where [., .] means the matrix (i.e., operator in the auxiliary space H) commutator. Moreover, if (2.4.10) as is also a scalar symbol

2.4. HEISENBERG APPROACH

179

we obtain the following equation: (2.4.11)

∂t q0t + {a, q0t } = 0

which can be rewritten in the form ∂t (q0t ◦ Ψt ) = 0

(2.4.12)

where Ψt : T ∗ Rd → T ∗ Rd is a symplectomorphism; namely, Ψt : z(0) → z(t) where z(t) is a solution of the dynamical system dz(t) = Ha (z(t)) dt

(2.4.13) and hence

q0t = q00 ◦ Ψ−t .

(2.4.14)

Definition 2.4.1. The family Ψt is called the Hamiltonian flow generated by the Hamiltonian a. In the general case (i.e. without assumption (2.4.10)) one can rewrite equation (2.4.8)0 in the form (2.4.15)

  ∂t Rt−1 (q0t ◦ Ψt )Rt = 0

where Rt is a differentiable matrix family in H satisfying the Cauchy problem (2.4.16)

∂t Rt = −i(as ◦ Ψt )Rt ,

R0 = I .

These matrices are unitary since as is Hermitian. Therefore Rt† (q0t ◦ Ψt )Rt = q00 and (2.4.17)

q0t = (Rt q00 Rt† ) ◦ Ψ−t .

Note that Rt,t  = Rt Rt†

(2.4.18) satisfies the Cauchy problem (2.4.19)

∂t  Rt,t  = −i(as ◦ Ψt )Rt,t  ,

Rt,t  = I .

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180

Solving Heisenberg Equations. Standard Case (α)

As long as T is bounded, and a(β) ◦ Ψt are bounded at (x, ξ) for ±t ∈ [0, T ] so are derivatives of Ψt (x, ξ) with respect to (x, ξ). Further, since dΨt is a linear symplectomorphism and therefore det dΨt = 1, derivatives of Ψ−t  remain bounded at Ψt (x, ξ) as 0 ≤ ±t  ≤ ±t ≤ T . Assume that these properties hold for all z ∈ Ω = B(¯ z , ε) and that (α) an(β) ◦ Ψt are bounded at (x, ξ) for ±t ∈ [0, T ]. Further, assume that (2.4.20)

q 0 ∈ Sh,ε,ε ,

supp(q 0 ) ⊂ Ω.

Then it immediately follows from (2.4.8)n that (2.4.21)

q t ∈ Sh,ε,ε ,

supp(q 0 ) ⊂ Ωt = Ψt Ω.

Moreover, B(¯ zt , C0−1 ε) ⊂ Ωt ⊂ B(¯ zt , C0 ε) where z¯t = Ψt (¯ z ). Then as ε satisfies microlocal uncertainty principle ε ≥ h(1−δ)/2

(2.4.22)

one can quantize q t . Thus Q t = q t w (x, hD) satisfies (2.4.6)–(2.4.7) modulo O(hN ) 26) operators. Therefore  −1  −1 (2.4.23) ∂t e ih tA Q t e −ih tA ≡ 0 also modulo O(hN ) 26) operators. Then (2.4.24) and finally (2.4.25)

e ih

−1 tA

Q t e −ih

Q t ≡ e −ih

−1 tA

−1 tA

≡ Q0

Q 0 e ih

−1 tA

.

This proves that Q t = q t w differs from Q t defined by (2.4.4) by O(hN ) and therefore Theorem 2.4.2. Let A ∈ Ψh be an operator with the Hermitian and scalar principal symbol. Let Ψt be a corresponding Hamiltonian flow. Furthermore, let Q 0 ∈ Ψh,ε,ε where ε ∈ (0, 1] satisfies (2.4.22). Then 26)

In the operator norm.

2.4. HEISENBERG APPROACH

181

(i) For ±t ∈ [0, T ] with bounded T operator Q t defined by (2.4.4) belongs to Ψh,ε,ε and is defined by the procedure described in the previous section. (ii) In particular, supp(q t ) = Ψt (supp(q 0 )). In the same way one can prove Theorem 2.4.3. Let A ∈ Ψh,an be an operator with the Hermitian and scalar principal symbol. Let Ψt be a corresponding Hamiltonian flow. Furthermore, let Q 0 ∈ Ψh,ε,ε,an where ε ∈ (0, 1] satisfies logarithmic uncertainty principle (2.4.26)

ε ≥ C (h| log h|)1/2 .

Then (i) For ±t ∈ [0, T ] with bounded T operator Q t defined by (2.4.4) belongs to Ψh,ε,ε,an and is defined by the procedure described in the previous section. (ii) In particular supp(q t ) = Ψt (supp(q 0 )). Remark 2.4.4. Actually in the symbol construction we need only that con0 ditions  to q be fulfilled only in Ω and conditions to symbol of A in Ψt (Ω). ±t∈[0,T ]

Discussion Now we want to extend the results of the previous subsubsection to the case of unbounded (but not too large) T . Then derivatives of Ψt may cease to be bounded. Usually in applications they have either exponential (general case) or polynomial (completely integrable case) growth with respect to t. Also in the applications it is more natural to assume that A belongs to Ψh,ρ,ρ with unbounded but not to large ρ. In this theory since different components of (x, ξ) are messed up during Hamiltonian motion, one needs to assume that all components of ρ and γ are equal. For the sake of simplicity we consider t ∈ [0, T ] but exactly the same results hold as −t ∈ [0, T ].

182

CHAPTER 2. PROPAGATION IN THE INTERIOR...

Properties of Hamiltonian Flow First we need to establish the growth properties of the Hamiltonian flow. It appears that everything is defined by the growth of dΨt and by the growth of F along trajectories. Lemma 2.4.5. Let Ψt : Rd → Rd be a flow defined by the dynamical system dz = F (z). dt

(2.4.27)

Let z¯ ∈ Rd , T > 0 and z(t) = Ψt (¯ z ). Let us assume that (2.4.28) and (2.4.29)

|(∂zα F )(z(t))| ≤ cμρ−|α| |dΨt−t  (z(t  ))| ≤ cω

∀t ∈ [0, T ]

∀α : 1 ≤ |α|

∀t, t  ∈ [0, T ]

where dΨt is a differential of Ψt (with respect to z) and μ, ρ, ω are (positive) parameters such that (2.4.30) Then (2.4.31) with (2.4.32)

μρT ω ≥ 1, |(∂zα Ψt )| ≤ Cα ωη 1−|α|

ω ≥ 1. ∀t ∈ [0, T ] ∀α : 1 ≤ |α|

η = ρ2 ω −2 μ−1 T −1 .

Proof. Let us use induction with respect to n ≥ 1; for n = 1 the estimate (2.4.31) coincides with (2.4.29). Let us assume that (2.4.31) holds for all α with |α| < n and let us consider z(t, u) = Ψt (u). Differentiating the initial system n times with respect to u we obtain that (α)

dzk dt

(2.4.33)

=



Fkj (z)zj

(α)

+ fk,α

1≤j≤d

where Fk and zk are components of F and z respectively and Fkj = ∂zj Fk , (α) zk = ∂uα z, (α ) (α ) fk,α = c¯d,I ,α1 ,...,αs FI (z(t))zi1 1 · · · zis s , 2≤|I |≤r

α1 ,...,αs :s=|I |, |αi |≥1,α1 +...+αs =α

2.4. HEISENBERG APPROACH

183

where I = (i1 , ... , is ) ∈ {1, ... , d}s , FI = ∂zi1 ... ∂zis F , and c¯··· are absolute constants. Then (2.4.28), (2.4.30), and the induction assumption imply that (2.4.34)

|fk,α | ≤ C μ(μT )n−2 ω 2n−2 ρ2−2n (α)

On the other hand, (2.4.33) and the equality zk = 0 for t = 0 and |α| ≥ 2 imply that  t (dΨ)t−t  (z(t  ))fα (t  )dt  z (α) (t) = 0 (α)

(α)

where z and fα are vectors with components zk and fk,α respectively; this formula, (2.4.29) and (2.4.34) imply (2.4.31) for |α| = n where we change the constant C if necessary. Remark 2.4.6. (i) For Hamiltonian system (2.4.13) condition (2.4.28) means precisely that  (α)  (2.4.35) |a(β) z(t) | ≤ cμρ1−|α|−|β| ∀t : [0, T ] ∀α, β : 2 ≤ |α| + |β| ≤ n + 1. (ii) If for system (2.4.13) condition (2.4.29) is fulfilled for t  = 0 then this condition is fulfilled for all t  ∈ [0, t] with ω replaced by ω 2d ; in fact,  −1 dΨt−t  (z(t  )) = dΨt (z(0)) dΨt  (z(0)) and since det dΨt = 1 we conclude that (dΨt )−1 is a cofactor matrix. Solving Heisenberg Equations for Long-Time Evolution Let us consider Hamiltonian trajectory z(t) with z(0) = z¯ and the evolution of Ω0 = B(0, ε). Our analysis will deal only with the Hamiltonian tube  Ψ t Ω0 Q := 0≤t≤T

where Ω0 = B(¯ z , ε). We want to estimate qnt and their derivatives as z ∈ Ωt but we are going to deal with rnt := qnt ◦ Ψt in Ω0 . We assume that Hamiltonian flow Ψt satisfies (2.4.27)–(2.4.30). As as is scalar we have r0t = q00 . In the general case however formula is (2.4.18) and

CHAPTER 2. PROPAGATION IN THE INTERIOR...

184

ΩT = ΨT Ω0

Q1

Q2

Ωt = Ψt Ω0

• z¯

• z¯

Ω0 = B(¯ z , ε) (a)

Ω0 = B(¯ z , ε)

(b)

Figure 2.9: Hamiltonian tubes: in (a) for Dt − λ(x, Dx ), and in (b) for Dt − A(x, Dx ) with eigenvalues of a(x, ξ) of constant mutiplicities. we need to estimate derivatives of Rt,t ; Recall that since as is Hermitian, Rt,t  is unitary. Assuming that α s (2.4.36) a || ≤ C μωρ−1−|α| ∀α : 1 ≤ |α| ||Dx,ξ and (2.4.37) μT ≥ 1, ω ≥ 1, ρ ≤ 1 we conclude that α ||Dx,ξ (2.4.38) (as ◦ Ψt )|| ≤ C μρ−2 ω 2 η 1−|α| ∀α : 1 ≤ |α|

and then repeating in the simplified way proof of Lemma 2.4.5 we arrive to Lemma 2.4.7. In the framework of Lemma 2.4.5 assume that (2.4.35) and (2.4.37) are fulfilled. Further, let as be a Hermitian symbol satisfying (2.4.36) (in the Hamiltonian tube Q). Then (2.4.39)

α Rt,t  || ≤ C η −|α| ||Dx,ξ

∀(x, ξ) ∈ Ωt  ∀t, t  ∈ [0, T ] ∀α.

Therefore we arrive to Corollary 2.4.8. In the framework of Lemma 2.4.7 q0t ◦ Ψt satisfies γ (q0t ◦ Ψt )|| ≤ C ε−1 ς 1−|γ| (2.4.40) ||Dx,ξ

∀(x, ξ) ∈ Ω0 ∀t ∈ [0, T ] ∀γ : 1 ≤ |γ|

2.4. HEISENBERG APPROACH

185

with ς = min(ε, η).

(2.4.41)

Consider now equation (2.4.8)n defining qn with n ≥ 1; one can rewrite it as (2.4.42) ∂t rnt + i[as ◦ Ψt , rnt ] = fnt :=  1 (α) t(β) t(β) (α)  (−i)|α|−|β| Aj(β) qk(α) − i |α|−|β| qk(α) Aj(β) ◦ Ψt |α|+|β| 2 α!β! 0≤k≤n−1 j+|α|+|β|=n−k

with the initial condition rn0 = qn0 ; observe that we can rewrite (2.4.42) as   (2.4.43) ∂t Rt−1 (qnt ◦ Ψt )Rt = Rt−1 fnt Rt . 



α ,β Further, we can rewrite qk(β) ◦ Ψt as the sum of bα,β;α ,β  Dx,ξ (qkt ◦ Ψt ) with 1 ≤ |α | + |β  | ≤ |α| + |β| and with the the coefficients satisfying t(α)

(2.4.44) Meanwhile (2.4.45)





γ ||Dx,ξ bα,β;α ,β  || ≤ C ω |α|+|β|+|γ| η −|γ|−|α|−|β|+|α |+|β | .

 (α)  γ ||Dx,ξ Ak(β) ◦ Ψt || ≤ C μω k η −|γ| ρ−k−|α|−|β|

provided we assumed that (2.4.46)

||Ak(β) || ≤ C μω k ρ−k−|α|−|β| (α)

in the respective range of α, β, γ, k. Then the induction with respect to n yields Proposition 2.4.9. Let Ψt : Rd → Rd be a Hamiltonian flow defined by a. Let us assume that in the Hamiltonian tube Q conditions (2.4.35) and (2.4.29) are fulfilled with ρ and ω ≥ 1 satisfying (2.4.30) and (2.4.37). Further, let us assume that in the Hamiltonian tube Q all terms of Weyl symbol of A satisfy (2.4.46) and that q 0 ∈ Sh,ε,ε . Then for all n γ (2.4.47) ||Dx,ξ (qnt ◦ Ψt )|| ≤ C (μT ρ−2 )n ς −2n−|γ|

∀(x, ξ) ∈ Ω0 ∀t ∈ [0, T ] ∀k, γ.

CHAPTER 2. PROPAGATION IN THE INTERIOR...

186

Proof. Assuming that (2.4.47) is fulfilled for k < n we conclude that γ fnt || ≤ C μρ−2 ς −2n−|γ| ||Dx,ξ

(2.4.48)

which due to Lemma 2.4.7 implies (2.4.47). It follows from (2.4.47) and Lemma 2.4.5 that γ qnt || ≤ C (μT ρ−2 )n ς −2n−|γ| ω |γ| (2.4.49) ||Dx,ξ

∀(x, ξ) ∈ Ωt ∀t ∈ [0, T ] ∀k, γ. To have (2.4.50)

 n

qnt hn decaying we need to assume ς ≥ (μT )1/2 ρ−1 h(1−δ)/2

and to perform quantization we need (2.4.51)

ς ≥ ωh(1−δ)/2 ;

plugging ς = ε and ς = η in both we get   (2.4.52) ε ≥ max (μT )1/2 ρ−1 , ω h(1−δ)/2 and   (2.4.53) ρ ≥ (μT )1/2 max ω 2/3 h(1−δ)/6 , ω 3/2 h(1−δ)/4 ; Thus we (almost) proved Theorem 2.4.10. Let A ∈ μρΨh,.,. (Rd , H, H) be a self-adjoint operator with the scalar principal symbol satisfying in the Hamiltonian tube Q conditions (2.4.29), (2.4.35), (2.4.46) with ρ and ω ≥ 1 satisfying (2.4.30), (2.4.37) and (2.4.53). Let q 0 ∈ Sh,ε,ε be supported in Ω0 with ε satisfying (2.4.52). Let q t be a symbol described in Proposition 2.4.9. Then for t ∈ [0, T ] operator Q t = Opw (q t ) satisfies (2.4.54)

Q t − e −ih

−1 tA

Q 0 e ih

−1 tA

 ≤ C μThs .

Remark 2.4.11. As we consider Sh,.,.,N and Ψh,.,N with finite N one should take s = δ  (δ)N where δ  > 0 as δ > 0.

2.4. HEISENBERG APPROACH

187

Proof of Theorem 2.4.10. Obviously ∂t Q t + ih−1 [A, Q t ] ≤ C μhs which is equivalent to ∂t (e ih

−1 tA

Q t e −ih

−1 tA

) ≤ C μhs

because the operator A is self-adjoint. On the other hand, Q 0 − Q ≤ Chs . Then e ih

−1 tA

Q t e −ih

−1 tA

− Q 0  ≤ C μThs

(note that μT ≥ 1) which is equivalent to (2.4.54).

2.4.2

Propagation of Singularities

Scalar Principal Symbol Case Theorem 2.4.10 immediately implies Theorem 2.4.12. Let A ∈ μρΨh,.,. (Rd , H, H) be a self-adjoint operator with the scalar principal symbol satisfying in the Hamiltonian tube Q conditions (2.4.29), (2.4.35), (2.4.46) with ρ and ω ≥ 1 satisfying (2.4.30), (2.4.37) and (2.4.53). Let ε satisfy (2.4.52). Let us assume that either (a) q ∈ Sh,ε,ε is supported in Ω0 , q  ∈ Sh,ς/ω,ς/ω and for some t = t ∗ ∈ [0, T ] (2.4.55)

dist(z  , Ψt (z)) ≥ ς/ω

∀z ∈ supp(q) ∀z  ∈ supp(q  )

or (b) The assumptions of (a) are fulfilled with permuted q and q  and Ψt replaced by Ψ−t . Then (i) For for t = t ∗ ∈ [0, T ], Q = Op(q) and Q  = Op(q  ) the following inequality holds: (2.4.56)

Qe ih

−1 tA

Q   ≤ C μThs −1 tA

(ii) Moreover, let U(x, y , t) be the Schwartz kernel of e ih supp(q  ) ⊂ {|ξ| ≤ c}. Then the following estimate holds: (2.4.57)

||Qx U(., ., t ∗ )t Qy || ≤ C  hs−d T

and supp(q) ∪

∀x, y ∈ Rd .

CHAPTER 2. PROPAGATION IN THE INTERIOR...

188 Diagonalization

We need to generalize this Theorem 2.4.12. Namely, we consider operator whose principal symbol is not scalar but whose spectrum in some interval consists of few rather disjoint eigenvalues (2.4.58)

Spec a(z) ∩ [−τ1 − , −τ2 + ] = {λ1 (z), ... , λm (z)} ∩ [−τ1 , −τ2 ]

∀z ∈ Ω

and for every j (2.4.59)

Spec a(z) ∩ [λj (z) − , λj (z) + ] = λj (z)

∀z ∈ Qj 

where λj are assumed to be uniformly continuous and Qj = t∈[0,T ] Ψλj ,t Ω0 is a Hamiltonian tube generated by λj and the lower bound to  will be defined later. We want to reduce this case to one of Theorem 2.4.12. To do this we need a series of axillary statements. Proposition 2.4.13. Let a ∈ Sh,γ,γ be Hermitian symbol and the conditions (2.4.58) and (2.4.59) be fulfilled. Let Ω be simple connected. Then there exist unitary symbols r (z), r1 (t, z), ... , rm (t, z) defined in Ω, Ω1 (t), ... , Ωm (t) respectively where Ωj (t) = Ψλj ,t Ω such that (i) In Ω (2.4.60)

(2.4.61)

and (2.4.62)

rj (0, z) = r (z)

⎛

λ1 (z)I ⎜ .. ⎜ r † (z)a(z)r (z) = a0 = ⎜ . ⎝ 0 0

(2.4.64)

⎞ ··· 0 0 .. ⎟ .. .. . . ⎟ . ⎟, · · · λm (z)I 0 ⎠ ··· 0 a (z)

  Spec a (z) ∩ [λj (z) − , λj (z) + ] = ∅ 2 2

(ii) and, moreover, for every j in Ωj (t) (2.4.63)

∀j = 1, ... , m,

⎛

bj (t, z) † 0 rj (t, z)a(z)rj (t, z) = aj,t = ⎝ 0 bj (t, z)  Spec aj (t, z) ∩ [λj (z) − , λj (z) + 2

∀j = 1, ... , m

⎞ 0 bj (t, z) ⎠, λj (z) 0 IV 0 bj (t, z)  ]=∅ 2

2.4. HEISENBERG APPROACH with

 aj (t, z)

and (2.4.65)

=

189

bj (t, z) bj (t, z) bj (t, z) bjIV (t, z)



||∂tk rj(β) || ≤ C −|α|−|β|−k γ −|α|−|β|−2k (α)

∀α, β, k.

Proof. Let us define (2.4.66)

1 πj (z) = 2πi

 j

(ζ − a(z))−1 dζ

for z ∈ Ωj (t) where j is a closed smooth curve contained in the ring { 18 γ ≤ |ζ − λj (z)| ≤ 14 γ} ⊂ C and going one time around λj (z) in the counter-clockwise direction; then πj (z) is the orthogonal projector to the eigenspace of a(z) corresponding to the eigenvalue λj (z). Here we used condition (2.4.59). Then in Ωj (t) symbol π(z) satisfies (2.4.65); here we used conditions (2.4.59) again and an assumption that a ∈ Sh,γ,γ . Then one can find r , rj (t) satisfying (2.4.65) with k = 0. It should be noted that the periodicity or approximate periodicity of some trajectories of the Hamiltonian flows would make Qj not simple connected possibly prevent the existence of rj not depending on t. However it would not happen during time T  γ 2 . Indeed, we can replace Ω by its γ-vicinity and note that the speed does not exceed cγ −1 . This implies statement of Proposition. So, we proved that symbols r , rj ∈ Sh,ρ,ρ with ρ = γ; therefore we need to assume that (2.4.67)

ρ = γ ≥ h(1−δ)/2 .

Proposition 2.4.14. In the framework of Proposition 2.4.13 assume that condition (2.4.67) is fulfilled. Further, let a ∈ νSh,γ,γ with (2.4.68)

h(γ)−2 ≤ ν ≤ hδ .

Then there exist symbols r , r1t , ... , rmt ∈ Sh,ρ,ρ (T ∗ Rd , H, H) satisfying (2.4.60) and such that γ 2k k ∂tk rj,t ∈ Sh,ρ,ρ (T ∗ Rd , H, H), ∀k, with ρ = γ and the corresponding norms of these operators do not exceed C , and

CHAPTER 2. PROPAGATION IN THE INTERIOR...

190

(i) Modulo operators with symbols vanishing in Ω (2.4.69)

(2.4.70)

R ∗ R ≡ RR ∗ ≡ I , ⎞ ⎛ A1 I ... 0 0 ⎜ .. . . .. ⎟ .. ⎜ . .⎟ . R ∗ AR ≡ ⎜ . ⎟ ⎝ 0 ... Am I 0 ⎠ 0 ... 0 A

(ii) and modulo operators with symbols vanishing in Ωj (t) (2.4.71) (2.4.72)

∗ ∗ Rj,t Rj,t ≡ Rj,t Rj,t ≡ I, ⎛ ∗ ∗ (∂t Rj,t ) + Rj,t ARj,t −ihRj,t

⎞   0 Bj,t Bj,t ≡ ⎝ 0 Aj,t 0 ⎠  IV 0 Bj,t Bj,t

where Rj,t = Op(rj,t ), R = Op(r ) and  IV  IV , ... , Bj,t are equal to λj I , a , bj,t , ... , bj,t (2.4.73) Symbols of operators Aj,t , A , Bj,t modulo νSh,ρ,ρ .

Proof. Let us use induction with respect to n: (2.4.74)

V ∈ ν n 1−n Sh,ρ,ρ (Ω, H, H)

and (2.4.75)

∂tk Vj,t ∈ γ −2k γ −2k ν n 1−n−k Sh,ρ,ρ (Ωj (t), H, H)

where V and Vj,t are the differences between the symbols of the left-hand and right hand expressions in (2.4.70) and (2.4.72) respectively while (2.4.69) and (2.4.71) are assumed to be fulfilled with much higher precision. Proposition 2.4.13 implies this assertion for n = 1 as long as ν ≥ h/(γ)2 . In order to make the induction step it is sufficient to construct Hermitian symbols l and lj,t satisfying (2.4.60), (2.4.74), (2.4.75) 27) , and (2.4.76) 27)

[l, a0 ] = −iV

So we plug r = l etc in these equations.

in Ω,

2.4. HEISENBERG APPROACH

191

and (2.4.77)

0 ] = −iVj,t [lj,t , aj,t

in Ωj (t)

0 where a0 and aj,t are defined by (2.4.61) and (2.4.63) respectively. Indeed, let us replace Rj,t by Rj,t (I + iLj,t )(I + Lj,t ) with Lj,t = Op(lj,t ),   Lj,t = Op(lj,t ) with Hermitian symbols lj,t satisfying (2.4.63) with n replaced by (n + 1) and providing (2.4.71) after the induction step; then one needs to pick up Lj,t = − 12 L∗j,t Lj,t . Moreover, let us replace R by R(I + iL)(I + L ) with L = − 12 L∗ L; then the induction step will be made obviously. On the other hand, the construction of l and lj,t satisfying (2.4.76) and (2.4.77) is obvious and, moreover, Hermitian symbols l with diagonal blocks equal to 0 and lj,t of the form

⎞ 0 ∗ 0 = ⎝ ∗ 0 ∗⎠ 0 ∗ 0 ⎛

lj,t

are uniquely defined from (2.4.76) and (2.4.77). Heisenberg Evolution for Matrix Principal Symbols Let us introduce the operators R˜j,t = Rj,t πj , R˜j = Rπj where πj are projectors corresponding to the block decomposition in (2.4.61) and j = 1, ... , m + 1. Then (2.4.69) and (2.4.70) imply that (2.4.78)

R˜j∗ R˜k ≡ δjk I

(2.4.79)

∗ ˜ R˜j,t Rj,t

(2.4.80)

AR˜j ≡ Aj R˜j

in Ω (j, k = 1, ... , m + 1),

≡I

in Ωj (t) (j = 1, ... , m + 1), in Ω (j = 1, ... , m + 1),

and (2.4.81)

AR˜j,t − ih−1 [Aj,t , Qj,t ] ≡ R˜j,t Aj,t −1

−1

in Ωj (t) (j = 1, ... , m)

with Am+1 = A and Qj,t = e −ih tAj Q 0 e ih tAj . Let us consider Cauchy problem (2.4.6)–(2.4.7) for Aj . Then in order to fit into theory of the previous Subsection 2.4.1, conditions (2.4.30), (2.4.32),

CHAPTER 2. PROPAGATION IN THE INTERIOR...

192

(2.4.37), (2.4.41), (2.4.52) and (2.4.53) must be fulfilled with μ = ρ−1 and ρ = γ: (2.4.82)

T ≥ max(ω −1 , γ),

(2.4.83)

η = 3 γ 3 ω −2 T −1 , ς = min(ε, η),   ε ≥ max (−3 γ −3 T )1/2 , ω h(1−δ)/2 ,   γ ≥ T 1/3 max ω 4/9 h(1−δ)/9 , ωh(1−δ)/6

(2.4.84) (2.4.85)

ω ≥ 1,

γ ≤ 1,

while ν satisfies (2.4.68) and also the following version of (2.4.38)   (2.4.86) ν ≤ min hδ , ω−1 γ −1 h , ν = h(γ)−2 shouls also fit into this last inequality (2.4.87)

ωγ ≥ 1,

γ 2 3 ≥ h1−δ .

Then Theorem 2.4.10 yields the following Proposition 2.4.15. Let a ∈ Sh,γ,γ 28) be Hermitian symbol and the conditions (2.4.58), (2.4.59) be fulfilled. Let A be a self-adjoint operator and A − Opw (a) ∈ νSh,γ,γ . Let domain Ω be simple connected. Further, let conditions (2.4.29) and (2.4.82)–(2.4.87) be fulfilled. Finally, let (j = 1, ... , m) (2.4.88) Qj = R˜j Q R˜j∗ and let Qj,t be a solution of (2.4.81) with initial data Qj,0 = Qj . Then (2.4.89)

Qjt − e −ih

−1 tA

Qe ih

−1 tA

 ≤ Chs T

∗ for Qjt = R˜j,t Qj,t R˜j,t .

As m = 1 and we are interested in the left-hand expression of (2.4.90) only we get Corollary 2.4.16. Let conditions of Proposition 2.4.15 be fulfilled with m = 1. Then   −1 −1 ∀τ ∈ [τ1 , τ2 ] (2.4.90) ||Ft→h−1 τ χT (t) Q t e −ih tA − e −ih tA Q || ≤ Chs where χ is our standard function supported in either [−1, 1] or [−1, 0] or [0, −1] as conditions of Proposition 2.4.15 are fulfilled either for all t ∈ [−T , T ] or for all t ∈ [−T , 0] or for all t ∈ [0, T ] respectively and ∗ Q t = Q1t = R˜1,t Q1,t R˜1,t . 28)

All conditions need to be fulfilled only in Qj .

2.4. HEISENBERG APPROACH

193

Main Theorem Theorem 2.4.17. Let a ∈ Sh,γ,γ 28) be Hermitian symbol and the conditions (2.4.58), (2.4.59) be fulfilled. Let A be a self-adjoint operator and A − Opw (a) ∈ νSh,γ,γ 28) . Let domain Ω be simple connected. Further, let conditions (2.4.29) and (2.4.82)–(2.4.87) be fulfilled. Furthermore let us assume that either (a) q ∈ Sh,ε,ε is supported in Ω0 , q  ∈ Sh,ς/ω,ς/ω and for t ∈ [t ∗ − κ, t ∗ + κ] ⊂ [0, T ] condition (2.4.91)

dist(z  , Ψt (z)) ≥ ς/ω

∀z ∈ supp(q) ∀z  ∈ supp(q  )

is fulfilled for Hamiltonian flows generated by λj for all j = 1, ... , m or (b) The assumptions of (a) are fulfilled with permuted q and q  and Ψλj ,t replaced by Ψλj ,−t . Finally, let U(x, y , t) be the Schwartz kernel of e ih supp(q) ∪ supp(q  ) ⊂ {|ξ| ≤ c}, χ ∈ C∞ ([−1, 1]) and

−1 tA

; assume that

κ ≥ h1−δ .

(2.4.92)

Then the following estimate holds: (2.4.93) ||Ft→h−1 τ χκ/2 (t − t ∗ )Qx U(t, ., .) tQy || ≤ C  hs ∀x, y ∈ Rd

∀τ ∈ [τ1 , τ2 ]

Proof. Assume that we are in the framework of assumption (a) (the other case is considered similarly). −1 Theorem 2.4.12 microlocalizes e ih tAj Q in Ψλj ,t (supp(q)) and equality (2.4.80) is fulfilled in its vicinity. Therefore modulo negligible (2.4.94)

e ih

−1 tA

−1 R˜j Q  ≡ e ih tAj R˜j Q 

∀j = 1, ... , m

as long as 0 ≤ t ≤ T and Q  has a symbol supported in vicinity of supp(q). Recall that if we are looking for energy levels τ ∈ [τ1 , τ2 ]  (2.4.95) Ψt (z) = Ψλj ,t (z) j=1,...,m

194

CHAPTER 2. PROPAGATION IN THE INTERIOR...

Therefore in virtue of Theorem 2.4.12 estimate (2.4.93) holds for Uj (x, y , t), −1 the Schwartz kernels of e ih tA R˜j with 1 ≤ j ≤ m. Next, using ellipticity of Am+1 on energy levels τ ∈ [τ1 , τ2 ] in the vicinity of supp(q) we conclude that (2.4.93) holds j = m + 1 as well. Finally these results combined with R˜j R˜j∗ Q ≡ Q (2.4.96) j=1,...,m+1

imply (2.4.93).

Comments Theorem 2.1.2 was first proven for wave front sets in V. Ivrii [1]. Theorem 2.3.2 was first proven under slightly more restrictive conditions in M. Bronstein, V. Ivrii [1]. The description of propagation of singularities in terms of generalized bicharacteristics is due to S. Wakabayashi [2]. Similar and also different results for equations with multiple characteristics could be found in V. Ivrii [4], [5]. Pseudoconvexity method based on the second commutators to study propagation of singularities was used in V. Ivrii [9]. For wave front sets there is a paper due to A. Volovoi [1] where a parametrix construction by means of Fourier integral operators for long bicharacteristics was examined very carefully; his approach is essentially more complicated. Moreover in standard microlocal analysis (when there is no exterior parameter such as h or a frequency) these results are senseless. D. Robert [5], D. Robert and M. Combescure [1] used coherent states to describe a long term propagation. The more detailed description of singularities can be made using polarization wave front sets: Definition 2.4.18. Let u = O(hs−δ ). Then ( PWFs (u) = (x, ξ; v ) ∈ T ∗ Rd × H : v ∈ Ker q0 (x, ξ) ∀q ∈ Ψh such that qu = O(hs ) is called polarization wave front set.

)

2.4. HEISENBERG APPROACH

195

Description of polarization wave front sets for systems with constant multiplicities of characteristics follows from our results: their (x, ξ) components travel along bicharacteristics and their v components belongs to Ker P0 (x, ξ) and “rotates”. The definition of polarization wave front set is due to N. Dencker, who also investigated propagation of them not only in the case of characteristics of constant multiplicities, but also in the framework of conical refraction [1] and [3], [4].

Chapter 3 Propagation of Singularities near the Boundary In this chapter we perform the same task as in the previous Chapter 2 but near the smooth boundary. In Section 3.1 we prove Theorem 3.1.7 which is the main theorem of the energy estimates method and is formulated in the terms of some auxiliary functions exactly as Theorem 2.1.2 was. However the phase space T ∗ X is “glued” at the boundary ∂X and we really need several such auxiliary functions and the notion of the microhyperbolicity of the operator in the direction is replaced by the notion of the microhyperbolicity of the boundary value problem in the multidirection. In Section 3.2 we prove Theorem 3.2.4 which is the geometric reincarnation of Theorem 3.1.7 exactly like Theorem 2.2.10 was the the geometric reincarnation of Theorem 2.1.2. However geometric picture becomes much more complicated even in the case of the scalar operators. In Section 3.3 we prove some statements similar to those of the second part of sSection 2.1. Theorem 3.2.4 is not very precise: for example it cannot exclude creeping rays even in the case of wave equation. So, in Section 3.4 we reinvent the energy estimates method and prove (under much more restrictive assumptions) Theorems 3.4.4 and 3.4.5 which improve Theorem 3.1.7 and in combination with it provide in some cases more precise description of the propagation. Finally, in Section 3.5 we study propagation of singularities along long bicharacteristic billiards which are “almost transversal to the boundary” and may contain large number of reflections (even with branching); this section is similar to Section 2.4.

© Springer Nature Switzerland AG 2019 V. Ivrii, Microlocal Analysis, Sharp Spectral Asymptotics and Applications I, https://doi.org/10.1007/978-3-030-30557-4_3

196

3.1. ENERGY ESTIMATES APPROACH

3.1

Energy Estimates Approach

3.1.1

Statement of the Main Theorem

197

Preliminary Notes In this section our geometric domain is X = R+ ×X  = R+ ×Rd−1  (x1 ; x  ) = (x1 ; x2 , ... , xd ) with the boundary ∂X = 0 × X  and interior X˙ = X \ ∂X . We consider the operator (3.1.1) Ak (x, hD  )(hD1 )m−k P= 0≤k≤m

where (3.1.2)

  Cbj R+ , Ψh,N−j (X  , H, H) .

Ak ∈ Ψh,N (X , H, H) = 0≤j≤N

Let ak (x, ξ  ) and (3.1.3)

p(x, ξ) =



ak (x, ξ  )ξ1m−k

0≤k≤m

be the principal symbols of Ak and P respectively. We assume that (3.1.4) p(z) is a Hermitian matrix for every z ∈ T ∗ X , i.e., that the system in question is symmetric in the principal. Let Ω  T ∗ X , Ω  T ∗ ∂X , Ω|∂X = ι−1 Ω 1) and let us assume further that (3.1.5)

||a0 (z)w || ≥ 0 ||w ||

∀w ∈ H ∀z ∈ Ω,

i.e., that the boundary is not characteristic in Ω. Then (3.1.4) yields the equality (3.1.6) p(z, D1 )u, v + − u, p(z, D1 )v + = iT (z)ðu, ðv m ∀z ∈ T ∗ ∂X ∀u, v ∈ S(R+ , H) where ., .+ and ., .m are inner products in L2 (R+ , H) and Hm respectively, ðu = ð(m) u = (u, D1 u, ... , D1m−1 u)|x1 =0 is the m-trace of u at ∂X and T (z) ∈ 1)

Recall that ι : T ∗ X |∂x → T ∗ ∂X is a natural map.

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198

L(Hm , Hm ) is a matrix with blocks * am−j−k−1 for j + k ≤ m + 1 (3.1.7) Tjk (z) = 0 for j + k ≥ m + 2 i.e.,

j, k = 1, ... , m

⎛

⎞ am−1 am−2 ... a1 a0 ⎜am−2 am−3 ... a0 0 ⎟ ⎜ ⎟ ⎜ .. . . . .. .. ⎟ T (z) = ⎜ ... . . . .⎟ ⎜ ⎟ ⎝ a1 a0 ... 0 0 ⎠ a0 0 ... 0 0

Then equality (3.1.6) implies the estimate (3.1.8)

||T (z)w || ≥ ||w ||

∀w ∈ Hm

∀z ∈ Ω

with a constant  = (m, c, 0 , c2 ) > 0, c2 = max1≤k≤m supz∈Ω ||ak (z)||. Without any loss of the generality we may replace  by 0 . Let B = (B1 , ... , Bm ) be a boundary operator, Bk ∈ Ψh,N−k (∂X , Hb , H) where Hb is another auxiliary Hilbert space with an inner product ., .b and a norm ||.||b . Let us assume that there exist symbols βk ∈ CbN (∂X , L(Hb , H)) and β = (β1 , ... , βm ) such that (3.1.9) T (z)w , w m + 2 Re b(z)w , β(z)w b ≤ 0

∀w ∈ Hm

∀z ∈ Ω

where bk and b are the principal symbols of Bk and B respectively; in this case the boundary value problem (P, B) is called dissipative in Ω . Moreover, if (3.1.9)∗ T (z)w , w m + 2 Re b(z)w , β(z)w b ≤ −0 ||w ||2 ∀w ∈ Hm

∀z ∈ Ω

then the boundary value problem (P, B) is called strictly dissipative in Ω . In what follows the most important case is H = CD ; one can easily prove ¯ ± are the numbers of positive (negative) eigenvalues then that if D± and D of the matrices a0 and T respectively then equality (3.1.6) yields that ⎧ 1 ⎪ ⎨ mD for even m, 2 ¯± = (3.1.10) D ⎪ ⎩ 1 (m − 1)D + D± for odd m. 2

3.1. ENERGY ESTIMATES APPROACH

199 ¯

Moreover, the most important case is Hb = CD+ when condition (3.1.9) is equivalent to the pair of conditions (3.1.11) and (3.1.12)

¯+ rank b(z) = D

¯ ∀z ∈ Ω

T (z)w , w m ≤ 0

¯ . ∀w ∈ Ker b(z) ∀z ∈ Ω

Furthermore, condition (3.1.9)∗ is equivalent to (3.1.11) and (3.1.12)∗

¯ . ∀w ∈ Ker b(z) ∀z ∈ Ω

T (z)w , w m ≤ −0 ||w ||2

Let us notice that (3.1.6) implies that (3.1.13) |η|m ||w || ≤ C (||p(z, η)w || + ||w ||)

∀w ∈ H ∀z ∈ Ω

∀η ∈ R.

Let Σρ = Σρ (P) be the set of all the points z ∈ T ∗ X at which the ellipticity condition (3.1.14)

||p(z)w || ≥ ρ||w ||

∀w ∈ H

is violated; here and below ρ > 0 is an arbitrarily small constant. Further, let Σρ = ιΣρ |∂X be the set of all the points z ∈ Ω at which the ellipticity condition (3.1.14)

||p(z, η)w || ≥ ρ||w ||

∀w ∈ H ∀η ∈ R

is violated. For z = (x  , ξ  ) let us introduce the linear space (3.1.15)

H (z) = {v ∈ L2 (R+ , H) : p(z, D1 )v = 0}.

Then for z ∈ / Σρ the following estimates hold: (3.1.16)

||e κx1 D1j v ||+ ≤ Cj ||v ||+

∀j ≥ 0 ∀v ∈ H (z)

with an appropriate exponent κ = κ(m, c, c1 , 0 , ρ) > 0 and constants Cj = Cj (m, c, c1 , 0 , ρ). Furthermore, let us introduce the sets Σb,ρ = Σb,ρ (P, B): z ∈ Σb,ρ if either z ∈ Σρ or the following condition (3.1.17)

||b(z)ðm v ||H ≥ ρ||v ||2+

∀v ∈ H (z)

CHAPTER 3. PROPAGATION NEAR BOUNDARY

200 is violated. Finally, let (3.1.18)

Σρ ,

Σ = Σ(P) =

and Σb = Σb (P, B) =

ρ>0

Σb,ρ ; ρ>0

it is obvious that (3.1.19) Σ and Σb are closed sets and that the space H (z) depends continuously (in the natural sense) on z ∈ Ω \ Σ where Σ = ιΣ|∂X . Definition 3.1.1. A point z ∈ Σ ⊂ T ∗ X (z ∈ T ∗ X \ Σ) is called a characteristic (elliptic respectively) point of the operator P. A point z ∈ Σb ⊂ Ω (z ∈ Ω \ Σb ) is called a characteristic (elliptic respectively) point of the problem (P, B). By means of standard elliptic arguments of Section 1.4 one can easily prove the following semiclassical result Theorem 3.1.2. Let condition (3.1.6) be fulfilled, N = N(d, s, M, l) and let u be temperate in the sense (3.1.20)

D1j uX ≤ h−M

∀j = 1, ... , m + l.

Then (3.1.21)

1

s− 2 (u) ∩ Ω ⊂ WFs,l (Bðh u) ∪ Σb WFs,m+l b b (Pu) ∪ WF

where here and in what follows (3.1.22)

WFs,l b (u) :=



WFs−j  (D1j u)|∂X

0≤j≤l

is called the boundary wave front set of order (s, l) of u and ðh u = ðh(m) u = (u, hD1 u, ... , (hD1 )m−1 u)|∂X . Let us recall that WFt  denotes a partial wave front set (see Definition 2.1.8). Later we will prove this theorem. Let us now assume that (3.1.23)

and

H =CD1 ⊗H01 ⊕ ... ⊕ CDM ⊗H0M ⊕H0 p = p1 ⊗IH01 ⊕ ... ⊕ pM ⊗ IH0M ⊕p0

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(3.1.24) p0 on H0 satisfies condition (3.1.14) with  = 0 where H01 , ... , H0M , H0 are auxiliary Hilbert spaces and p1 , ... , pM , p0 are symbols on CD1 , ... , CDM , H0 respectively. Then at every point z ∈ Ω the estimate (3.1.16) holds with an appropriate exponent κ > 0 and constants Cj depending on p(z, .). Moreover, the space H (z) is a lower semicontinuous function of z in the sense of Definition 2.2.2. Let us introduce the characteristic polynomial (3.1.25)

g (z, η) = det p1 (z, η) · · · det pM (z, η).

Definition 3.1.3. For z ∈ Ω ⊂ T ∗ ∂X let us denote by η1 (z), ... , ηn (z) all the real distinct roots of g (z, η) where also n = n(z). Then T = ( , ν1 , ... , νn ) ∈ Tz T ∗ ∂X × Rn is called multidirection. Definition 3.1.4. Let conditions (3.1.1)–(3.1.6), (3.1.23) and (3.1.9) be fulfilled. Then the problem (P, B) is microhyperbolic at the point z ∈ Ω ⊂ T ∗ ∂X in the multidirection T = ( , ν1 , ... , νn ) if the following two conditions are fulfilled: (i) For every k = 1, ... , n the operator P is microhyperbolic at the point zk = (z, ηk ) in the direction k = (, νk , 0) ∈ Tzk (T ∗ X |∂X ) ⊂ (Tzk T ∗ X )|∂X ; (ii) The following estimate holds (3.1.26)

− Re (T  p)(z, D1 )v , v + + Re i(T  b)(z)ðv , β(z)ðv b ≥ ||v ||2+ − c1 (c1 ||b(z)ðv ||2 − T (z)ðv , ðv m )

∀v ∈ H (z)

(compare with estimate (2.1.1) in Definition 2.1.1). In what follows we will prove the following Proposition 3.1.5. Let the problem (P, B) be microhyperbolic at the point ˜ B) ˜ also satisfy z in the multidirection T . Further, let the problem (P, conditions (3.1.1)–(3.1.6), (3.1.23) and (3.1.9) and suppose that p˜ − p Cb1 ≤ ε,

b˜ − b Cb1 ≤ ε,

|˜ z − z| ≤ ε, |T˜ − T | ≤ ε

with a small enough constant ε > 0. ˜ B) ˜ is also microhyperbolic at the point z˜ in the Then the problem (P, multidirection T˜ .

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202

˜ does not necessarily Remark 3.1.6. Obviously the structure of ι−1 z˜ ∩ Σ(P) −1 coincide with that of ι z ∩ Σ(P) and it can happen that n˜(˜ z ) > n(z). However one can avoid this difficulty by either of two methods: ˜ close to zk then (i) If zki with i = 1, ... , tk (tk ≥ 0) are points of ι−1 z˜ ∩ Σ(P) the condition |T˜ − T | ≤ ε is understood in the sense that |˜ki − k | ≤ ε ∀i = 1, ... , lk ∀k = 1, ... , n 2) ; (ii) The condition that ηk (k = 1, ... , n) are all the real roots of g (z, η) is replaced by the condition that the set {η : (z, η) ∈ Σ } is contained in the union of the intervals (ηk − μ, ηk + μ) (k = 1, ... , n) where |ηk − ηj | ≥ 3μ ∀j = k and 0 < μ ≤ μ ¯=μ ¯(c, 0 , c1 , c3 ) where 0 and c1 are constants from the microhyperbolicity conditions (2.1.1) with z,  replaced by zk , k (with k = 1, ... , n), (3.1.6) and (3.1.26), and c3 = maxj aj  Cb2 + maxj bj  Cb2 . In this case ε = ε(d, m, M, D∗ , c, 0 , c1 , c2 , ), D∗ = m maxj Dj .

2)

I.e. |˜ − | ≤ ε and |˜ νki − νk | ≤ ε ∀i = 1, ... , lk

∀k = 1, ... , n.

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Finally, the Main Theorem Now we can formulate the main result of this section: Theorem 3.1.7. Let conditions (3.1.1)–(3.1.6), (3.1.23), (3.1.9) be fulfilled and let N = N(d, m, s, M, l, D∗ ). Further, let z ∗ ∈ Ω ⊂ T ∗ ∂X and φj ∈ CN (T ∗ ∂X ), φkj ∈ CN (T ∗ X  ×R+ ) with j = 1, ... , J, k = 1, ... , n be real-valued functions such that (3.1.27)

φ1j = ... = φnj = φj

at T ∗ ∂X

∀j

and for every j = 1, ... , J the problem (P, B) is microhyperbolic at the point z ∗ in the multidirection Tj = (∇# φj , −∂x1 φ1j , ... , −∂x1 φnj )(z ∗ ). Furthermore, let u satisfy (3.1.20) and (3.1.28)

  s+1,l 1 WFb (Pu) ∪ WFs+ 2 (Bðh u) ∩ Ω ∩ {φ1 ≤ t1 } ∩ ... ∩ {φJ ≤ tJ } = ∅,

(3.1.29) WFs+1 (Pu)∩Ωk ∩{φk1 ≤ t1 }∩· · ·∩{φkJ ≤ tJ } = ∅ (3.1.30)

∀k = 1, ... , n,

 WFs,0 b (u) ∩ ∂Ω ∩ {φ1 ≤ t1 } ∩ · · · ∩ {φJ ≤ tJ } = ∅

and (3.1.31) WFs (u) ∩ ∂Ωk ∩ {φk1 ≤ t1 } ∩ · · · ∩ {φkJ ≤ tJ } = ∅

∀k = 1, ... , n

where Ω and Ωk are small enough neighborhoods of z and zk in T ∗ ∂X and T ∗ X respectively not depending on u, t1 , ... , tJ such that (3.1.32)

ιΩk |∂X = Ω

∀k = 1, ... , n.

(i) Then (3.1.33)

WFs,m+l (u) ∩ Ω ∩ {φ1 ≤ t1 } ∩ · · · ∩ {φJ ≤ tJ } = ∅ . b

(ii) Moreover, if condition (3.1.9)∗ is fulfilled then one can weaken condition 1 (3.1.28) replacing in it WFs+ 2 (Bðh u) by WFs (Bðh u):   (3.1.28) WFs+1,l (Pu) ∪ WFs (Bðh u) ∩ b Ω ∩ {φ1 ≤ t1 } ∩ · · · ∩ {φJ ≤ tJ } = ∅

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and at same time obtain in addition that (3.1.34)

WFs−i (D1i u|∂X ) ∩ Ω ∩ {φ1 ≤ t1 } ∩ · · · ∩ {φJ ≤ tJ } = ∅ ∀i = 0, ... , m + l − 1.

Remark 3.1.8. (i) Assumption (3.1.29) is referring to points of T ∗ X˙ . (ii) Assumptions (3.1.6), (3.1.20), (3.1.28) and (3.1.30) imply (3.1.30) again with WFs,0 (u) replaced by WFs,m+l (u). (iii) The uniform version of Theorem 3.1.7 (such as was proven in the proof of Theorem 2.1.2) is obvious but very long and tedious to formulate.

3.1.2

Proof of the Main Theorem

In order to prove Theorems 3.1.2 and 3.1.7 and Proposition 3.1.5 we first of all reduce the general case to a special one: first to m = 1 and then to the block-diagonal form with each block corresponding to a single point ι−1 z ∗ and an elliptic block of the special type. Reduction to a Special Case Step 1. Reduction to m = 1. First of all, one can assume that m = 1. Indeed, let us set u = (uj )j=1,...,m with uj = (hD1 )j−1 u. Then in the intersection with the vicinities of ι−1 z ∗ or z ∗ WFs (u) = WFs (u), WFt (Pu) = WFt (P u),

WFs,m+l  (u) = WFs,l+1  (u), WFs,l  (Pu) = WFs,l  (P u)

provided |s − t| ≤ 1 where P = A0 hD1 + A1 , (3.1.35)

A0 = (Am−j−k+1 )j,k=1,...,m

is an operator with principal symbol T (z), A1 = −A0 L, (3.1.36)

L = (Ljk )j,k=1,...,m ,

Ljk = δj+1,k − A0 Am−k+1 δj,m

where A0 is a parametrix of A0 here (condition (3.1.6) guarantees its existence) and δj,k is the Kronecker symbol as usual. Then modulo operators with symbols vanishing in the vicinity of z ∗ (3.1.37)

A1 = (−Am−j−k+2 + Am−j+1 δk,1 + Am−k+1 δj,1 )j,k=1,...,m .

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205

Observe that condition (3.1.6) for P implies the same condition for P. Moreover, condition (3.1.4) for P is equivalent to the same condition for P. Furthermore, condition (3.1.23) for P is equivalent to to the same condition for P with Dj replaced by mDj . One can prove easily that ΣC −1 ρ (P) ⊂ Σρ (P) ⊂ ΣC1 ρ (P) 1

for appropriate C1 = C1 (c, 0 , c1 , m) and arbitrary ρ > 0. Moreover, Bðh(m) u = Bðh(1) u and hence (3.1.38) Condition (3.1.9) or (3.1.9)∗ for (P, B) is equivalent to to the same condition for (P, B); from this one can prove easily that ΣC −1 ρ (P, B) ⊂ Σρ (P, B) ⊂ ΣC1 ρ (P, B) 1

for appropriate C1 = C1 (c, 0 , c1 , c2 , m) and arbitrary ρ > 0. Finally, one can prove easily the following inequalities |(p)(z)w , w  − (p)(z)w , w m | ≤ C || · ||p(z)w ||||w || ∀w ∈ Hm : w = w1 and |( p)(z, D1 )v , v + − ( p)(z, D1 )v , v + | ≤ C | | · ||p(z, D1 )v ||+ · ||v ||+ ∀v ∈ S(R+ , Hm ) : v = v1 where as usual ||.||+ and ., . is a norm and an inner product in L2 (R+ , .). Hence (3.1.39) P is microhyperbolic at z in the direction  if and only if P has the same property; moreover, (P, B) is microhyperbolic at z in the multidirection T if and only if (P, B) has the same property. The reduction step is complete: Proposition 3.1.9. Under reduction (3.1.35)–(3.1.37) conditions (3.1.6), (3.1.4), (3.1.23), (3.1.9) or (3.1.9)∗ , and microhyperbolicity conditions are preserved. Furthermore all wave front sets and Σ∗ are are essentially preserved as described above.

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Step 2. Isolation of characteristic roots. Let us now show that Proposition 3.1.10. (i) In the proofs of Theorem 3.1.7 and Proposition 3.1.5 one can assume without any loss of the generality that (3.1.40)

| det pj (z ∗ , η)| ≥ 

∀η ∈ C : |η − ηj | ≥ μ

where μ > 0 is arbitrarily small,  > 0 depends on μ, ηj are not necessarily distinct anymore and   0 p0 , p0 = ρ(η − α), p0 = (η − α† )ρ† (3.1.41) p0 = p0 0 with the matrix symbols α and ρ such that (3.1.42) ||(η − α)w || ≥ ||w ||,

||ρw || ≥ ||w || ∀w ∈ H

∀η ∈ C : Im η ≤ μ.

(ii) In the proof of Theorem 3.1.2 one can assume without any loss of the generality that   0 p0  , p0 = η − α , p0 = η − α p0 = (3.1.41) p0 0 and (3.1.42) holds for α = α and α = −α . (iii) Also in this reduction D∗ does not increase. In order to make the reduction step let us first consider the separate symbols pj (z, η) with j = 1, ... , J. Let ηjk be all the real distinct roots of det pj (z ∗ , η), k = 1, ... , nj . Then ηjk are all the real distinct eigenvalues of Lj (z ∗ ) where Lj (z) = −a0j (z)−1 a1j (z) 3) . Hence there exist symbols rj (z) defined, smooth and invertible such that in Ω ⎞ ⎛ L˜j1 ... 0 0 ⎜ .. . . .. ⎟ . ⎜ . .. . ⎟ rj−1 Lj rj = L˜j = ⎜ . ⎟, ⎝ 0 ... L˜jnj 0 ⎠ 0 ... 0 L˜j0 where 3) Let us recall that pj (z, η) = a0j (z)η + a1j (z) ) and z ∈ Ω where Ω is a small enough neighborhood of z ∗ in T ∗ X  × R+ which we shrink if necessary.

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207

(3.1.43) For k ≥ 1 all the eigenvalues of L˜jk are equal to ηjk and all the eigenvalues of L˜j0 are non-real. Then

rj† a1j rj = rj† a0j rj L˜j

and taking the Hermitian conjugate we obtain that (rj† a0j rj )L˜j = L˜†j (rj† a0j rj ). Since L˜j is of the block-diagonal type and since η =  ζ † if η and ζ belong to † the spectra of different blocks (where ζ is a complex conjugate number) we conclude that rj† a0j rj is also of the block-diagonal type and hence rj† a1j rj is also of this type. Therefore multiplying p(z, η) on the left and on the right by r † (z) and r (z) respectively we obtain pjk ⊗ IH0j ⊕ p0 (3.1.44) p=⊕ 1≤j≤M,1≤k≤nj

provided

⎞ 0 0 r1 ⊗ IH0j ... ⎟ ⎜ ... r = ⎝ ... rM ⊗ IH0M ⎠ ; 0 ... 0 I ⎛

here det pjk (z ∗ , η) has no complex root except ηjk ∈ R and p0 satisfies (3.1.14) ; after this reduction step the space H0 can be larger than before. Similarly, let us consider L0 (z) = −a00 (z)−1 a10 (z) 4) ; then the spectrum of L0 (z) resides in {η ∈ C, | Im η| ≥ } and hence there exists a smooth and invertible symbol r0 (z) such that in Ω    L0 0 −1 ˜ r0 L 0 r0 = L 0 = 0 L0 where the spectrum of L0 resides in {η ∈ C, Im η ≥ } and the spectrum of L0 resides in {η ∈ C, Im η ≤ −}. Then using the same arguments as before (but taking into account that η = ζ † provided η and ζ belong to the spectrum of the same  we  block) 0 ∗ . obtain that both symbols r0† a00 r0 and r0† a10 r0 are of the form ∗ 0 4)

Let us recall that p0 (z, η) = a00 (z)η + a10 (z).

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Then multiplying (3.1.44) on respectively with ⎛ I ⎜ .. ⎜ r = ⎜. ⎝0 0

the left and on the right by r † and r ... ... ... ...

⎞ 0 0 .. .. ⎟ . .⎟ ⎟, I 0⎠ 0 r0

we obtain p0 of the form (3.1.41). Thus, the second step of the reduction is provided by multiplying P on the left and on the right by Op(r )∗ and Op(r ) respectively where r is now the product of the two matrices constructed in the two sub-steps of this step. Step 3. Reduction of Lower Order Terms. In the final reduction step we obtain ⎛ ⎞ P1 ... 0 0   ⎜ .. . . .. ⎟ .. 0 P0 ⎜. ⎟ . . . (3.1.45) P =⎜ P0 = ⎟, P0 0 ⎝ 0 ... Pn 0 ⎠ 0 ... 0 P0 with P0 = P0 ∗ where for k = 1, ... , n the principal symbols of Pk are of the form pkj ⊗ IH0jk , (3.1.46) pk = ⊕ 1≤k≤Kj

pjk are the same as in (3.1.44), and p0 and p0 (the principal symbols of P0 and P0 respectively) are of the same type as in (3.1.41). In order to make this reduction step we assume that (3.1.45) is fulfilled modulo operators with symbol hn (F0 η + F1 ); we then apply induction on n. For n = 1 these approximate equalities are fulfilled. In order to make the induction step we multiply P on the right and on the left by (I + hn Q ∗ ) and (I + hn Q) respectively; then the step in the first equality (3.1.45) will be made provided for j = k (3.1.47)1

qkj† a0k + a0j qjk = F0jk ,

(3.1.47)2

qkj† a1k + a1j qjk = F1jk

3.1. ENERGY ESTIMATES APPROACH

209

where a0k , a1k are diagonal blocks and qjk , F0jk , F1jk are blocks of the corresponding symbols, Q = Op(q). Since both F0 and F1 are Hermitian matrices we need to fulfill these equalities only for j < k; for j > k they will then be fulfilled automatically. Finding qkj† from the first equality and substituting into the second one we −1 obtain after multiplication by a0j that we need only to solve the equation (3.1.48)

− qjk Lk + Lj qjk = Fjk

−1 for j < k where Lk = −a0k a1k and Fjk is obtained from F0jk and F1jk . This equation is solvable since the spectra of Lj and Lk are disjoint (see Appendix 3.A.1). The choice of qjk for j = k is arbitrary. In order to make the induction step in the second equality in (3.1.45) we need to solve the equations

(3.1.49)1 (3.1.49)2

† q(21) ρ† + ρq(21) =F0(11) , † q(21) α† ρ† + ραq(21) =F1(11)

and similar equations for q12 where ρ = p0 , q(jk) are blocks of q00

  q(11) q(12) , = q(21) q(22)

F0(jk) and F1(jk) are similar blocks of the matrices F000 and F100 , and these blocks are Hermitian for j = k. The solution of the first equation is of the form 1 q(21) = ρ−1 ( F0(11) + iΛ) 2 where Λ is an arbitrary Hermitian symbol. Substituting this equality to the second equation we obtain that (3.1.50)

 Λα† − α Λ = iF(11)

 where α = ρα ρ−1 and F(11) is some given Hermitian symbol. We can solve this equation because the spectra of α and α† are disjoint (see Appendix 3.A.1 again). Moreover, one can choose q(11) and q(22) arbitrarily. The induction step is done. The reduction is complete.

CHAPTER 3. PROPAGATION NEAR BOUNDARY

210 Elliptic Theory

The following series of assertions is crucial in the proof of Theorem 3.1.2 and is important in the proof of Theorem 3.1.7:   Proposition 3.1.11. Let α ∈ Cbl R, Ψh (Rd , H, H) be an operator with the principal symbol α0 such that (3.1.51) ||(η − α0 (z))w || ≥ ||w || ∀w ∈ H

∀η ∈ C : Im η ≤  ∀z ∈ R × T ∗ Rd

and let l ≥ 1. Then for h ∈ (0, h0 ] with h0 = h0 (d, c, , c2 ) > 0, c2 = |||α||| there exists an operator family   G ∈ Cl+1 Δ, L(L2 (Rd , H), L2 (Rd , H)) with Δ = {(x1 , y1 ) ∈ R2 , x1 ≥ y1 } such that (i) The following relations hold: (3.1.52)

G (x1 , x1 ) = I ,

(3.1.53) Dxj 1 Dyk1 G (x1 , y1 ) ≤ Ch−j−k e −h

−1 (x

1 −y1 )

∀j, k : j + k ≤ q + 1

where here and below C = C (d, s, M, l, , c2 ). (ii) If the operators G± : L2 (Rd+1 , H) → L2 (Rd+1 , H) are given by the formulae  x1 −1 (3.1.54)+ G (x1 , y1 )f (y1 ) dy1 , (G+ f )(x1 ) = +ih −∞  ∞ (3.1.54)− (G− f )(x1 ) = −ih−1 G (y1 , x1 )f (y1 ) dy1 x1

then G+∗ = G− and (3.1.55)

G±  ≤ C ,

(3.1.56)+

PG+ = G+ P = I ,

(3.1.56)−

P ∗ G− = G− P ∗ = I

where P = hD1 − α and then P ∗ = hD1 − α∗ . Moreover, if f is tempered (i.e., D1j f  ≤ h−M ∀j = 0, ... , l) then (3.1.57)

WFs,l+1  (G± f ) = WFs,l  (f ).

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211

+ d (iii) If the operator family Gt : L2 (Rd , H) → L2 (R+ t × R , H) with Rt = {x1 , x1 ≥ t} is given by the formula

(Gt f )(x1 ) = G (x1 , t)f ,

(3.1.58) then

1

(3.1.59)

Gt  ≤ Ch 2 ,

(3.1.60)

PGt = 0,

ðt Gt = I

where ðt u = u|x1 =t and, moreover, if f  is tempered then WFs,l+1  (Gt f  ) = {t} × WFs− 2 (f  ). 1

(3.1.61)

Proof. Let us note that for h ∈ (0, h0 ] with an appropriate constant h0 > 0 the following estimate holds: 1 (η − α)v  ≥ v  2

1 ∀v ∈ L2 (Rd , H) ∀η ∈ C : Im η ≤  ∀x1 ∈ R. 2

Then for operators G(1) (x1 , y1 ) = e ih

−1 (x

1 −y1 )αy1

G(2) (x1 , y1 ) = e ih

−1 (x

1 −y1 )αx1

with αy1 = α|x1 =y1

and (3.1.52) and (3.1.53) with j = k = 0 are fulfilled. Hence if we define G± by formulae (3.1.54)± with G(1) and G(2) respectively instead of G then (3.1.55) would be fulfilled while (3.1.56)± would be fulfilled only partially and approximately: PG+ = I + R+ ,

P ∗ G− = I + R− ,

R±  ≤ Ch.

Then for h ∈ (0, h0 ] with an appropriate constant h0 > 0 the operators G± = G± (I + R± )−1 satisfy (3.1.56)± partially (3.1.56)

PG+ = P ∗ G− = I

and are of the form (3.1.54)± with appropriate functions G = G± (x1 , y1 ) satisfying (3.1.52)–(3.1.53); however these functions may be different for different indices ±.

CHAPTER 3. PROPAGATION NEAR BOUNDARY

212

But then (3.1.56) yields that G+∗ P ∗ = G−∗ P = I and these equalities and (3.1.56) again imply that G+ = G−∗ and then (3.1.56)± hold; therefore the functions G± in formulae (3.1.54)± coincide. This implies (3.1.60) and (3.1.53). It remains to prove (3.1.57) and (3.1.61). In fact, it is sufficient to prove that WFs  (G± f ) and WFs  (Gt f  ) are contained in the right-hand sets of (3.1.57) and (3.1.61) respectively; these inclusions and (3.1.56) and (3.1.60) imply (3.1.57) and (3.1.61). Furthermore, one can see easily that it is sufficient to prove these inclusions for G± f and Gt f  where Gt is given by (3.1.58) with G replaced by G(1) . Note that e ih

−1 τ α

Q=

1  −1  −1 ∂λk e ih τ λα Qe ih τ (1−λ)α λ=0 + k! 0≤k≤n   −1  1 1 −1 (1 − λ)n ∂λn+1 e ih τ λα Qe ih τ (1−λ)α dλ = n! 0 1 −1 k (Adih−1 α Q)τ k e ih τ α + k! 0≤k≤n  1 1 n+1 ih−1 τ (1−λ)α (1 − λ)n (Adn+1 e dλ ih−1 α Q)τ n! 0

where AdM is an operator of commutation with M and AdkM are defined by (Ad0M Q) = Q, (Ad1M Q) = [M, Q] and (AdkM Q) = [M, (Adk−1 M Q)] for k ≥ 1. It is easy to deduce from (3.1.53) that if Q is a scalar pseudodifferential 1 −1 operator then the norm of the last term does not exceed Cn hn e − 2 τ h and hence modulo negligible operators e ih

−1 τ α

Q≡

1 −1 (Adkih−1 α Q)τ k e ih τ α k! 0≤k≤n

with large enough n = n(s). This equality obviously implies the inclusions in question. ˜ its -vicinity. Theorem 3.1.12. Let Ω be a domain in R × T ∗ Rd−1 and Ω   ˜ ˜ Further, let Ω = Ω ∩ {x1 = 0}, Ω+ = Ω ∩ {x1 ≥ 0}, Ω = Ω ∩ {x1 = 0} and ˜+ = Ω ˜ ∩ {x1 ≥ 0}. Ω

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213

˜ Furthermore, let α ∈ Cl (R, Ψh (Rd , H, H)) satisfy (3.1.51) for all z ∈ Ω and let l ≥ 1. Then there exist operators   G0 ∈ L L2 (∂X , H), L2 (X , H) with G0  ≤ Ch1/2 and   G ∈ L L2 (X , H), L2 (X , H) with G0  ≤ C with the same type of constant C as before such that for tempered u0 , f (i.e., with u0 ∂X ≤ h−M , D1j f X ≤ h−M ∀j ≤ l) the following relations hold: (3.1.62) (3.1.63)

1 ˜ WFs,l+1  (G0 u0 ) ⊂ WFs− 2 (u0 ) ∩ Ω, ˜ +, WFs,l+1  (Gf ) ⊂ WFs,l  (f ) ∩ Ω

(3.1.64)

WFs,l  (PG0 u0 ) ∩ Ω+ = ∅,

(3.1.65)

WFs− 2 (ðG0 u0 − u0 ) ∩ Ω = ∅,

(3.1.66)

WFs,l  (PGf − f ) ∩ Ω+ = ∅,

(3.1.67)

WFs− 2 (ðGf ) = ∅,

1

1

and (3.1.62)

˜ +, WFs,l+1  (G ∗ f ) ⊂ WFs,l  (f ) ∩ Ω

(3.1.64)

WFs,l  (P ∗ G ∗ f − f ) ∩ Ω+ = ∅.

Moreover, if D1j uX ≤ h−M for all j ≤ l + 1 then the following relations hold: (3.1.68)

WFs,l+1  (u − GPu − G0 ðu) ∩ Ω+ = ∅,

(3.1.68)

WFs,l+1  (u − G ∗ P ∗ u) ∩ Ω+ = ∅.

Proof. Without any loss of the generality one can assume that (3.1.51) is ˜  for a small enough satisfied in R × T ∗ Rd . Indeed, (3.1.51) is satisfied in Ω ¯ (z) = α(z)ϕ(z) + i(1 − ϕ(z))I satisfies (3.1.51) constant  > 0 and hence α in R × T ∗ Rd provided ϕ is a smooth scalar function equal to 1 in the 1 -neighborhood of Ω and vanishing outside of the 23 -neighborhood of Ω 3 and such that 0 ≤ ϕ ≤ 1.

CHAPTER 3. PROPAGATION NEAR BOUNDARY

214

Let us define G¯± and G¯0 according to the Proposition 3.1.115) and let us set G¯ = R G¯+ T where here R and T are the operators of restriction from Rd+1 to X and extension from X to Rd+1 by 0 in Rd+1 \ X respectively. Then G¯∗ = R G¯− T and operators G¯ and G¯0 obviously satisfy (3.1.62)– ˜ = T ∗ Rd and with Ω+ = Ω ˜ + = R+ × T ∗ Rd . Moreover, (3.1.64) with Ω = Ω the obvious equalities PTu =TPu − ih(ðu)δ(x1 ), P ∗ Tu =TP ∗ u − ih(ðu)δ(x1 ) combined with (3.1.56)± and (3.1.54)± imply the equalities u = G¯∗ P ∗ u;

u = G¯Pu + G¯0 ðu,

these equalities combined with (3.1.57) and (3.1.60) imply (3.1.68) and (3.1.68) with Ω+ = R+ × T ∗ Rd . Finally, G = Op(ϕ)G¯ Op(ϕ) and G0 = Op(ϕ)G¯0 Op(ϕ0 ) are the operators in question provided ϕ0 = ϕ|x1 =0 . Proof of Theorem 3.1.2. The simplifying reduction of Subsubsection 3.1.2.1 Reduction to a Special Case implies that without any loss of the generality one can assume that Σ(P) = ∅ and   P1 0 P= , H = H1 ⊕ H 2 0 P2 where Pj = hD1 − αj and α1 and α2† satisfy (3.1.51). Therefore if Ω ⊃ ω and ω does not intersect WFs,l b (Pu) then Theorem 3.1.12 yields that (u2 ) = ∅, ω ∩ WFs,l+1 b

ω ∩ WFs,l+1 (u1 − G0 v ) = ∅, b 1

ω ∩ WFbs,l+1 (u1 ) ⊂ WFs− 2 (v ) where G0 is constructed for P1 and v = ðu1 . Then 1

1

1

ω ∩ WFs− 2 (Bðu) = ω ∩ WFs− 2 (B  v ) ⊃ ω ∩ WFs− 2 (B ∗ B  v ) where B  is the restriction of B to H1 . One can see easily that H (z) = H1 and hence the set of characteristic points of operator B ∗ B  is exactly Σb (P, B). Hence if ω ∩ WFs−1/2 (Bðu) = ∅ then ω ∩ WFs−1/2 (v ) ⊂ Σb (P, B). Therefore ω ∩ WFbs,l+1 (u) ⊂ Σb (P, B). 5) We now use a bar for the operators constructed in that proposition, with t = 0 now in (3.1.60) and (3.1.61).

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215

Main Theorem Refined The rest of this section is almost entirely devoted to the proof of the following assertion which is slightly more precise than Theorem 3.1.7: Theorem 3.1.13. Let P be a reduced operator satisfying the conditions of Theorem 3.1.7, i.e., we suppose that conditions (3.1.1) with m = 1, (3.1.4), (3.1.6), (3.1.41), (3.1.42), (3.1.45) and (3.1.46) are satisfied. Further, let z ∗ ∈ Ω  T ∗ ∂X and let us assume that each polynomial det pk (z ∗ , η) with k = 1, ... , n has only one complex root and this root is real (namely ηk ∈ R). Further, let boundary operator B satisfy condition (3.1.9). Finally, let φj ∈ CN (T ∗ ∂X ), φkj ∈ CN (R+ × T ∗ X  ) be real-valued functions satisfying (3.1.27) (j = 1, ... , J, k = 1, ... , n). Let us assume that for every j = 1, ... , J the problem (P, B) is microhyperbolic at the point z ∗ in the multidirection Tj = (∇# φj , −∂x1 φ1j , ... , −∂x1 φnj )(z ∗ ). Let N = N(d, s, M, l) and l ≥ 2D∗ − 2, D∗ = maxk Dk where Dk are the dimensions of Hk . Then (i) Let u satisfy (3.1.20) and (3.1.69)

 s+1,l  1 WFb (P0 u0 ) ∪ WFs+ 2 (Bðu) ∩ Ω ∩ {φ1 ≤ t1 } ∩ · · · ∩ {φJ ≤ tJ } = ∅,

(3.1.70)

WFs+1,l  (Pk uk ) ∩ Ωk ∩ {φk1 ≤ t1 } ∩ · · · ∩ {φkJ ≤ tJ } = ∅ ∀k = 1, ... , n,

(3.1.71)

WFs− 2 (ðu0 ) ∩ ∂Ω ∩ {φ1 ≤ t1 } ∩ · · · ∩ {φJ ≤ tJ } = ∅

1

and (3.1.72)

WFs  (uk ) ∩ ∂Ωk ∩ {φk1 ≤ t1 } ∩ · · · ∩ {φkJ ≤ tJ } = ∅ ∀k = 1, ... , n

where u = (u1 , ... , un , u0 ), Ω and Ωk are small enough neighborhoods of z ∗ in T ∗ ∂X and R+ × T ∗ X  respectively such that Ω = Ωk |∂X ∀k = 1, ... n. Then (3.1.73)

1

WFs− 2 (ðu0 ) ∩ Ω ∩ {φ1 ≤ t1 } ∩ · · · ∩ {φJ ≤ tJ } = ∅

CHAPTER 3. PROPAGATION NEAR BOUNDARY

216 and (3.1.74)

WFs,l+1  (uk ) ∩ Ωk ∩ {φk1 ≤ t1 } ∩ · · · ∩ {φkJ ≤ tJ } = ∅ ∀k = 1, ... , n.

(ii) Moreover, if condition (3.1.9)∗ is satisfied then one can weaken condition s+1/2,l (3.1.69), replacing in it WFs+1,l (P0 u0 ) and WFs+1/2 (Bðu) by WFb (P0 u0 ) b s and WF (Bðu) respectively: (3.1.69)



s+ 12 ,l

WFb

 (P0 u0 ) ∪ WFs (Bðu) ∩ Ω ∩ {φ1 ≤ t1 } ∩ · · · ∩ {φJ ≤ tJ } = ∅,

and skip condition (3.1.71) completely, while getting statement (3.1.34) in addition to (3.1.73) and (3.1.74). Proof of Theorem 3.1.7. Let conditions (3.1.20) and (3.1.28)–(3.1.31) be fulfilled. Then there obviously exists ε > 0 depending on u, t1 , ... , tJ such that if ψ = ψ(x1 ) is a smooth function equal to 1 for x1 ≤ 12 ε and vanishing for x1 ≥ ε and if Q is a scalar h-pseudodifferential operator with symbol equal to 1 in the 12 ε-neighborhood of Σ(P) and vanishing outside of the ε-neighborhood of Σ(P) then U = ψu + Q(1 − ψ)u satisfies (3.1.69)–(3.1.72) with Ωk = ˜ιΩk where ˜ι : T ∗ X → R+ × T ∗ X  is a natural map. Then Theorem 3.1.13 applied to U yields (3.1.33) and even (3.1.34) (under condition (3.1.9)∗ ) for u. Proof of Theorem 3.1.13. Step 1 In this proof we can assume without any loss of the generality that (3.1.73) and (3.1.74) hold with s replaced by s − 12 , i.e., that (3.1.75)

WFs−1 (ðu0 ) ∩ Ω ∩ {φ1 ≤ t1 } ∩ · · · ∩ {φJ ≤ tJ } = ∅,

(3.1.76)

WFs− 2 ,l+1  (uk ) ∩ Ωk ∩ {φk1 ≤ t1 } ∩ · · · ∩ {φkJ ≤ tJ } = ∅ ∀k = 1, ... , n. 1

Let us construct operators Qk ∈ Ψh,N (X , C) in the same manner as Q3 in the proof of Theorem 2.1.2, i.e., let us set    (3.1.77) Qk = Opw ζk (x, ξ  ) χ(φkj (x, ξ  )) 1≤j≤J

3.1. ENERGY ESTIMATES APPROACH

217

where the real-valued function χ satisfies (2.1.8); for the sake of simplicity we assume that t1 = ... = tJ = 0, φkj are replaced everywhere by φkj − 14 ε, ε > 0 is small enough, ζk ∈ C0N (R+ × T ∗ X  ), 0 ≤ ζk ≤ 1, ζk are supported in the 12 ε-neighborhood of Ωk and ζk = 1 in the 13 ε-neighborhood of Ωk , and ζ1 = ... = ζn = ζ on T ∗ ∂X . Let us consider the following identities similar to (2.1.10): (3.1.78)

∗

− Re i([Pk , Qk ]uk , uk ) = Re i((Qk + Qk )Pk uk , uk )+ ∗

Re i((Pk − Pk∗ )Qk uk , uk ) + Re h(Q  Tk uk , uk )∂X where Tk = A0k |∂X (now T is an operator with the principal symbol T0 given by (3.1.5), so we have changed notation slightly), Qk = Qk2 and the index k in the last term is skipped because all the operators Qk coincide at ∂X due to condition (3.1.27). Repeating the arguments of the proof of Theorem 2.1.2 with the obvious modifications we obtain the following inequalities similar to (2.1.11): ∗

(3.1.79) | − Re i([Pk , Qk ]uk , uk ) − Re(Q  Tk uk , uk )∂X | ≤ C2 Qk uk 2 + Ch2s . Let us note that the principal symbol of the operator −ih−1 [Pk , Qk ] equals  2{pk , φkj }rkj2 − {pk , ζk } χ(φkj )2 1≤j≤J

1≤j≤J

+

where rkj = ζχ1 (φkj )χ2 (φkj ) i =j χ(φki )2 and the functions χ1 , χ2 are defined in (2.1.11). Then similarly to (2.1.17) we obtain the inequalities   ∗ 2(Skj Rkj uk , Rkj uk ) − Re(Q  Tk uk , uk )∂X  ≤ (3.1.80)  1≤j≤J

C2 Qk uk 2 + C2 hQk uk 2∂X + Ch2s where Rkj = Op(rkj ), Skj = Opw ({pk , φkj }) and a boundary term (the second term) in the right-hand expression appears because the symbols {pk , φkj } depend on ξ1 . The microhyperbolicity of the symbols pk in the directions ∇# φkj yields the existence of symbols qk vanishing in the ε-neighborhoods of the supports

CHAPTER 3. PROPAGATION NEAR BOUNDARY

218

of rkj for all k, j such that (3.1.81) ({pk , φkj } + cpk† pk + (qk − 0 )I )w , w  ≥ 0 ∀w ∈ Hk

∀(x, ξ  ) ∈ R+ × T ∗ X 

∀j = 1, ... , J

∀k = 1, ... , n.

Then passing from symbols to operators and applying the Ga ˚rding inequality we obtain an inequality similar to (2.1.19): (3.1.82) Re(Skj v , v )+cPk v 2 +Re(Qk v , v ) ≥ 0 v 2 −C2 hv 2∂X −Chv 2 . More precisely, we obtain this inequality first only for v with ðv = 0; in −1 order to get rid of this restriction we substitute v −e −h x1 ðv for v in (3.1.82) to derive (3.1.82) in the general case. Substituting v = Rkj uk into (3.1.82) and repeating the arguments of the proof of Theorem 2.1.2 we obtain the inequalities (3.1.83)

Re(Skj Rkj uk , Rkj uk ) ≥ 0 Rkj uk 2 − C2 hRj uk 2∂X − Ch2s

where the index k is omitted in the second term in the right-hand expression since Rkj on ∂X do not depend on k. Proof of Theorem 3.1.13. Step 2 So far we have been just repeating the arguments of the proof of Theorem 2.1.2. However now we need some new ideas: Proposition 3.1.14. Let conditions (3.1.1)–(3.1.6) with m = 1 and (3.1.46) be fulfilled and let us assume that det pkj (z ∗ , η) has only one complex root and this root is real (namely, ηk ∈ R). Then (i) For every  > 0 there exists a neighborhood Ω of z ∗ such that if Q is an operator with principal symbol Q0 supported in Ω and satisfying the inequality |Q0 | ≤ 1 then the following estimate holds: (3.1.84) hQv 2∂X ≤ w 2 + C D1i Pk v 2 + Ch D1l v 2 0≤i≤l¯

¯ 0≤i≤l+1

where we omit the subscript “ X ” as before and l¯ = 2D∗ − 1 where D∗ is the maximal dimension of the Jordan cell in the Jordan normal form of Lkj (z ∗ ), −1 Lkj = −a0kj a1kj ; the maximum is taken for all j = 1, ... , Jk and all the Jordan cells.

3.1. ENERGY ESTIMATES APPROACH

219

(ii) Moreover, (3.1.85) ||v (0)||2 ≤ 2||v ||2+ + C



||D1l pk (z, D1 )v ||2+

0≤l≤l¯

∀v ∈ S(R+ , Hk )

∀z ∈ Ω .

This proposition6) will be proven later (see Subsubsection 3.1.2.7 Proof of Proposition 3.1.14). Let us now finish the proof of Theorem 3.1.13. Taking in (3.1.84) Q with symbol equal to 1 in a neighborhood of the support of the symbol of Qk and using (3.1.70), (3.1.75) and (3.1.76) we upgrade (3.1.83) to the estimate 1 Re(Skj Rkj uk , Rkj uk ) ≥ 0 Rkj uk 2 + c4 hRj uk 2∂X − Ch2s 2

(3.1.86)

with an arbitrarily constant c4 which will be chosen later. So the final estimate at this step in the proof is (3.1.87)

− Re ih−1 ([Pk , Qk ]uk , uk ) ≥ 1 0 Rkj uk 2 + c4 h Rj uk 2∂X − Ch2s 2 1≤j≤J 1≤j≤J

∀k = 1, ... , n;

thus we estimated from below the left hand expression of (3.1.78). Let us now consider the fourth term in the right hand expression of (3.1.78) and sum with respect to k = 0, ... , n; we obtain Re(Q ∗ Tk uk , uk )∂X . (3.1.88) − 0≤k≤n

Definitions of Q and Q  and (3.1.76) imply immediately that modulo terms with absolute values not exceeding C2 hQu2∂X + Ch2s (where C2 does not depend on the choice of c3 and c4 ) this expression equals (3.1.89)

−



Re(Tk Quk , Quk )∂X =

0≤k≤n

− Re(TQu, Qu)∂X + Re(T0 Qu0 , Qu0 )∂X . 6) Which implies that the boundary operator enters into microhyperbolicity condition only through nonreal roots.

CHAPTER 3. PROPAGATION NEAR BOUNDARY

220

Let us transform (3.1.89) modulo terms with absolute values not exceeding C2 hQu2∂X +Ch2s 7) ; in these transformations we will use assumption (3.1.76) repeatedly. Let us recall that in a neighborhood of z ∗ S0 = −T0 − β† b − b † β is a non-negatively defined symbol due to assumption (3.1.9). Hence − Re(TQu, Qu)∂X ≡ Re(SQu, Qu)∂X + 2 Re(BQu, BQu)∂X ≡ − Re(SQu, Qu)∂X + 2 Re(Q[B, Q]u, Bu)∂X because (3.1.69) implies Re(QBu, BQu) ≡ 0 where S = Opw (S0 ) and B = Opw (β). On the other hand, the structure of P0 (see (3.1.45)) yields that |(T0 Qu0 , Qu0 )∂X | ≤ C2 Qu0 ∂X · Qu0 ∂X ; moreover, (3.1.69) and Proposition 3.1.11 imply that Qu0 ∂X ≤ Chs+ 2 and therefore this expression is equivalent to 0. Thus (3.1.90) − Re(Tk Quk , Quk )∂X ≡ 1

0≤k≤n

Re(SQu, Qu)∂X + 2 Re(Q[B, Q]u, Bu)∂X . Let S  be an h-pseudodifferential operator on ∂X with a non-negatively defined Weyl symbol coinciding with that of S in the neighborhood of z ∗ ; then the Ga ˚rding inequality yields that (3.1.91) and hence (3.1.92)

Re(S  v , v )∂X ≥ −C2 v 2∂X Re(SQu, Qu)∂X ≥ −C2 Qu2∂X − Ch2s

where we again have used the fact that the symbol of Q is supported in a small neighborhood of z ∗ . Let us now consider the second term in the right-hand expression in (3.1.90). Calculating the principal symbol of the operator Q[B, Q] we see that this term is equivalent to h(Lj Rj u, Rj u)∂X 1≤j≤J 7)

Where C2 does not depend on the choice of c3 and c4 .

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221

where the operators Rj are defined above and Lj are operators with principal symbols Lj0 = iβ† {b, φj } − i{b † , φj }β. Then (3.1.26) implies that in a neighborhood of z ∗ the symbols Lj0 + c(S0 + b † b + Π) − 20 I are non-negative definite where (3.1.93) Π is the projector to the orthogonal complement of H (z ∗ ). Therefore Ga ˚rding inequality again implies that Re(Lj v , v )∂X = c Re(Sv , v )∂X + cBv 2∂X + cΠv 2∂X ≥ (20 − Ch)v 2∂X − Q v 2∂X where the symbol of Q vanishes in some neighborhood of z ∗ . Setting v = Rj u and multiplying by h we obtain that (3.1.94) h Re(Lj Rj u, Rj u)∂X + ch Re(SRj u, Rj u)∂X + chBRj u2∂X + chΠRj u2∂X ≥ 0 hRj u2∂X − Ch2s . Moreover, condition (3.1.69) implies that the third term in the lefthand expression is equivalent to 0 and the second term is equivalent to   ch Re(S  Qu, Rj u) Rj is an operator with principal +∂X where S = S +C2 h and  2 symbol ζχ4 (φj ) k =j χ(φk ); here χ4 = (χ ) /χ = (2χ1 )2 is a smooth function because of (2.1.8). Then (3.1.91) and the Cauchy inequality yield that (3.1.95) h|(S  Qu, Rj u)∂X | ≤ h(S  Qu, Qu)∂X · (S  Rj u, Rj u)∂X ≤ 1/2

1/2

1 0 Re h(SQu, Qu)∂X + C2 hQu2∂X + Ch2s . 2 On the other hand, (3.1.69) and Proposition 3.1.11 yield that Rj u0 2 ≤ Ch and hence ΠRj u2∂X ≤ C2 Rj uk 2∂X + εRj u0 2∂X + Cε h2s 2s

1≤k≤n

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with arbitrary small ε > 0 provided Ω = Ωε is narrow enough. Then (3.1.90), (3.1.92) and (3.1.94) imply that (3.1.96)

−



∗

Re(Q  Tk uk , uk ) ≥

1≤k≤n

0 h





Rj u0 2∂X − C2 h

≤j≤J

Rj uk 2∂X − C2 hQu2∂X − Ch2s .

1≤j≤J, 1≤k≤n

This inequality combined with (3.1.79), (3.1.87) and (3.1.95) yields after an appropriate choice of constant c4 that (3.1.97)



hRj u2∂X +

1≤j≤J

1≤k≤n

 Rjk uk 2 ≤   Qk uk 2 + Ch2s . C2 hQu2∂X + 1≤k≤n

This estimate is similar to estimate (2.1.18) in the proof of Theorem 2.1.2. Repeating the remaining arguments of the proof of Theorem 2.1.2 together with the choice of constant c3 in (2.1.8) we finally obtain that (3.1.73) and (3.1.74) with l = −1 hold. Then (3.1.70) implies (3.1.74) in the general form. Theorem 3.1.13(i) has beens proven. Proof of Theorem 3.1.13. Step 3 Let us assume now that condition (3.1.9)∗ is fulfilled. Then one can replace inequality (3.1.92) by (3.1.92)∗

Re(SQu, Qu) ≥ (0 − Ch)Qu2∂X − Ch2s

(we assume that (3.1.34) holds with (s − 12 ) instead of s). Moreover, we know that Qu0 ∂X ≤ Chs because of Proposition 3.1.11 and we can weaken condition (3.1.69) to (3.1.69) . But in this case with no analysis of the principal symbols of the commutators we obtain instead of (3.1.95) the following stronger inequality: (3.1.96)∗

−



1 ∗ Re(Q  Tk uk , uk ) ≥ 0 Qu2∂X − Ch2s . 2 1≤k≤n

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This inequality combined with (3.1.79) and (3.1.87) implies that Rjk uk 2 ≤ C Qk uk 2 + Ch2s Qu2∂X + 1≤j≤J, 1≤k≤n

1≤k≤n

and repeating the remaining arguments of the proof of Theorem 2.1.2 we obtain (3.1.73), (3.1.74) and (3.1.34). Theorem 3.1.13(ii) has beens proven. Proof of Proposition 3.1.14 Proof of Proposition 3.1.14. Obviously, it is sufficient to prove this proposition for P = hD1 − L with L ∈ Ψh (X , H, H) where H = Cr and L0 (z ∗ ) has exactly one complex eigenvalue (namely, η ∈ R); now L0 (the principal symbol of L) is not necessarily Hermitian. Obviously, estimates (3.1.84) and (3.1.85) do not depend on the lower order terms; moreover, these estimates are stable with respect to small perturbations of L0 for every fixed  > 0. Therefore it is sufficient to prove them only in the case of constant L = L0 (z ∗ ). Then (3.1.84) is reduced to (3.1.85) by a change of variables x1new = h−1 x1 . So we need to prove only (3.1.85) for z = z ∗ . Moreover, one can assume that η = 0; the general case is reduced to this one by the substitution v = vnew e iηx1 . We will apply the following Lemma 3.1.15. Let L be an r × r matrix such that Lr = 0. Then for every μ > 0 there exist r × r matrices ρj with j = 0, ... , 2r − 2 such that (3.1.98) −

(3.1.99)

ρ†j = ρj



∀j,

Re ρj Lj+1 w , w  ≥ ||Lw ||2 − μ||w ||2

0≤j≤2r −2

and (3.1.100)

−



Re iρj Lj−k w , Lk w  ≤ μ||w ||2

∀w ∈ H.

0≤k≤j≤2r −2

This lemma will be proven later; now let us prove Proposition 3.1.14 in the reduced form. Let Q= ρj D1k Lj−k ; P = D1 − L, 0≤k≤j≤2r −2

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224



then PQ =

ρj (D1j+1 − Lj+1 )

0≤j≤2r −2

and hence

Re QPv , v + = −

ρj Lj+1 v , v + +

0≤j≤2r −2



ρj D1j+1 v , v + =

0≤j≤2r −2

1 ρj Lj+1 v , v + + Re iρj (D1j−k v )(0), (D1k v )(0) − 2 0≤j≤2r −2 0≤k≤j≤2r −2

where we have applied (3.1.98). Let us replace D1k and D1j−k in the last term by Lk and Lj−k respectively; then the absolute value of the error will not exceed   C ||(D1j−1 Pv )(0)||2 + ||v (0)||2 ≤ μ||v ||2+ + C ||D1j Pv ||2+ 1≤j≤2r −2

0≤j≤2r −2

where the last inequality is due to the obvious estimates ||v (0)||2 ≤ ||v ||+ · ||D1 v ||+

and

||D1 v ||+ ≤ ||Pv ||+ + C ||v ||+ .

Hence for every μ > 0 the following estimate holds:

Re  − ρj Lj+1 v , v + +

0≤j≤2r −2

1 Re iρj Lj−k v (0), Lk v (0) ≤ 2 0≤k≤j≤2r −2 ||D1j Pv ||2+ . μ||v ||2+ + C 0≤j≤r −2

These estimates combined with (3.1.100) imply (3.1.85). Proof of Lemma 3.1.15. By induction on l = 1, ... , r we will prove that for every l there exist Hermitian matrices ρj with j = 0, ... , 2l − 2 such that (3.1.101)l − Re ρj Lj+1 w , w  ≥ l ||Lw ||2 − μ||w ||2 − Clμ ||Ll w ||2 0≤j≤2l−2

and (3.1.102)l −



Re iρj Lj−k w , Lk w  ≤ μ||w ||2 + Clμ ||Ll w ||2

0≤k≤j≤2l−2

∀w ∈ H

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where l > 0 do not depend on arbitrarily small μ > 0. Then for l = r we obtain the assertion of the lemma. On the other hand, (3.1.101)1 with μ = 0 and (3.1.102)1 with arbitrarily small μ > 0 hold for ρ0 = L − L † . Let l = 1, ... , r −1; let us assume that (3.1.101)l and (3.1.102)l are fulfilled with some l > 0 and matrices ρj (j = 0, ... , 2l − 2). Then (3.1.101)l+1 and (3.1.102)l+1 will be fulfilled with 2μ instead of μ and with l+1 = 12  provided

(3.1.103)l+1

Re  − ρ2l−1 L2l − ρ2l L2l+1 )w , w  ≥ 1  ||L2l+2 w ||2 − l ||Lw ||2 − μ||w ||2 + Cl,μ ||L2l w ||2 − Cl+1,μ 2

and (3.1.104)l+1

 Re (−ρ2l Ll w , Ll w  ≤ μ||w ||2 − Cl,μ ||Ll w ||2 + Cl+1,μ ||Ll+1 w ||2

∀w ∈ H. Indeed, let us consider all the terms in the left hand expression in (3.1.102)l+1 which are both new in comparison with (3.1.102)l and different from (3.1.104)l+1 ; obviously the absolute values of these terms do not exceed  ε||w ||2 + C;+1,μ,ε ||Ll+1 w ||2 where ε > 0 is arbitrarily small, and the constant  Cl,μ is fixed but the constant Cl+1,μ is not fixed yet. Therefore we need to prove only that for every ε > 0 and l < r there exist matrices α = −α† and β = −β† such that − Re αL2l w , w  − Re βL2l+1 w , w  ≥ −ε||w ||2 + ||L2l w ||2 − Cε ||L2l+2 w ||2 and Re iβLl w , Ll w  ≥ −ε||w ||2 + ||Ll w ||2 − Cε ||Ll+1 w ||2 Finally, let us note that matrices α = L2l − L†2l and β = 4ε−1 (L2l+1 − L†2l+1 ) − i(I − L† L) satisfy these inequalities.

∀w ∈ H.

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226

Proof of Proposition 3.1.5 Obviously it is sufficient to prove this proposition for the reduced operator. Moreover, it is sufficient to check only condition (ii) in Definition 3.1.4. The arguments of the proof of Theorem 3.1.7 imply that if the operator Pk is microhyperbolic at (every) point zk ∈ ι−1 z ∗ in the direction ( , νk ) then for every t > 0 the following estimate holds − Re ( pk )(z ∗ , D1 )v , v + ≥ 0 ||v ||2+ + t||v (0)||2 − Ct ||D1j pk (z ∗ , D1 )v ||2+

∀v ∈ S(R+ , H).

0≤j≤l

Hence the same estimate holds for all z : |z − z ∗ | ≤ ε(t). Then the estimate − Re ( pk )(z ∗ , D1 )v , v + ≥ 0 ||v ||2+ + t||v (0)||2

∀v ∈ Hk (z)

holds where Hk (z) is the space of all solutions of the equation pk (z, D1 )v = 0 decreasing at +∞. Therefore for large enough t > 0 and small enough ε > 0 estimate (3.1.26) at the point z = z ∗ implies the same estimate at points z : |z − z ∗ | ≤ ε. Furthermore, H (z ∗ ) = H0 in our decomposition and H (z) = H0 ⊕ H1 (z) ⊕ · · · ⊕ Hn (z). Hence condition (ii) is stable with respect to perturbations of z and hence it is stable with respect to perturbations of P, B also. Moreover, this condition is obviously stable with respect to  as well.

3.2 3.2.1

Geometric Interpretation Main Theorem

Notations and Definitions This section is similar to Section 2.2. Let us remind that the set  (3.2.1) N(p, b) = (z  , T ) : z  ∈ T ∗ ∂X , and the problem (P, B) is microhyperbolic at the point z  in the multidirection T



is open in the sense of Proposition 3.1.5 and Remark 3.1.6. Moreover, obviously

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227

(3.2.2) The set K(p, b, z  ) = {T : (z  , T ) ∈ N(p, b)} is convex for any z  ∈ T ∗ ∂X . Let us introduce the set (3.2.3) K  (p, b, z  ) = ιK(p, b, z  ) =    , ∃(ν1 , ... , νn ) such that ( , ν1 , ... , νM ) ∈ K(p, b, z  ) ⊂ Tz  T ∗ ∂X where ι : T = (, ν1 , ... , νn ) →  is a natural map. Since microhyperbolicity condition in Definition 3.1.4 consists of two conditions (i) and (ii) and the first condition does not include B while the second one does not contain (ν1 , ... , νn ) we conclude that (3.2.4)

K  (p, b, z  ) = K  (p, z  ) ∩ K  (p, b, z  )

where (3.2.5) K  (p, z  ) = ιK(p, z  ) and K(p, z  ) is defined by (3.2.1) and (3.2.2) with the full microhyperbolicity condition replaced by condition (i) in Definition 3.1.4 while (3.2.6) K  (p, b, z  ) is the set of all directions  ∈ Tz  T ∗ ∂X satisfying condition (3.1.26). Then (3.2.7) K  (p, b, z  ) is an open convex cone and (3.2.8)

K  (p, z  ) =

πK (p, zk ) zk ∈ι−1 z  ∩Σ

(3.2.9)

K  (p, b, z  ) =

πK (p, zk ) ∩ K  (p, b, z  ) zk

∈ι−1 z  ∩Σ

where π : Tι−1 z  T ∗ X → Tz  T ∗ ∂X is a natural map. Thus, K  (p, b, z  ), K  (p, z  ) and K  (p, b, z  ) are subsets of Tz  T ∗ ∂X while K (p, zk ) ⊂ Tzk T ∗ X .

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Now we can introduce the dual (with respect to bilinear form σ) cones K # (p, b, z  ), K # (p, z  ) and K # (p, b, z  ), all subsets of Tz  T ∗ ∂X as well. Obviously (3.2.10) (K # (p, zk ) ∩ Tz  T ∗ ∂X ), K # (p, z  ) = ⊕ zk ∈ι−1 z  ∩Σ

(3.2.11) K # (p, b, z  ) = ⊕



(K # (p, zk ) ∩ Tz  T ∗ ∂X ) ⊕ K # (p, b, z  ).

zk ∈ι−1 ιz  ∩Σ

Finally, let (3.2.12) K # (p, b, z  ) = ⊕



K # (p, zk ) ⊕ K # (p, b, z  )

zk ∈ι−1 z  ∩Σ

(and one can introduce K # (p, z  ) in the same way but without the last term). Remark 3.2.1. (i) K # (p, zk ), K # (p, z  ) and K # (p, b, z  ) are subsets of Tι−1 z  T ∗ X while K # (p, z  ) and K # (p, z  ) are subsets of Tz  T ∗ ∂X . (ii) Obviously these sets have the properties described in Lemma 2.2.3. Without any loss of generality we can assume that X = {x1 ≥ 0} locally. Further, let us assume that ¯ such that z ∈ / T ∗ X |∂X and (3.2.13) ∇# x2 ∈ K # (p, z) ∀z ∈ Ω # #   ∗ ¯ ∩ T ∂X where Ω ⊂ T ∗ X such that ∇ x2 ∈ K (p, b, z ) ∀z ∈ ιΩ (3.2.14)

z ∈ Ω ∩ T ∗ X |∂X =⇒ ι−1 ιz ∈ Ω

This assumption is similar to (2.2.1). Definition 3.2.2. Let Ω ⊂ T ∗ X be a domain satisfying (3.2.14). A gener¯ ∩ T ∗ X (where Ω ¯ is alized bicharacteristic billiard is a curve {z = z(t)} in Ω a closure of Ω) such that (i) If z(t) ∈ / T ∗ X |∂X for some t then |z(t  ) − z(t  )| ≤ C0 |t  − t  | for all    t , t : |t − t| + |t  − t| ≤ 0 dist(z(t), T ∗ X |∂X ) where the constants 0 > 0 and C0 do not depend on t, z(t);

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(ii) In the general case |z  (t  ) − z  (t  )| ≤ C0 |t  − t  | for all t  , t  where z  = (x, ξ  ), ξ  = (ξ2 , ... , ξd ); / T ∗ X |∂X then the function z(t) is differentiable at t (iii) For a.e. t if z(t) ∈ and (3.2.15)

dz ∈ K # (p, z); dt

(iv) For a.e. t if z(t) ∈ T ∗ X |∂X then the function z  (t) is differentiable at t and (3.2.16)

dz ∈ K # (p, b, z  ). dt

We consider examples later. Here let us notice that generalized bicharacteristic billiards are the generalized bicharacteristics inside the domain (equation (3.2.15) coincides with (2.2.23)) but on the boundary they can jump along ι−1 ιz and also experience other effects. Equivalent definition would be if we considered T ∗ X with points z1 , z2 glued together iff z1 , z2 ∈ T ∗ X |∂X and ιz1 = ιz2 . Then one could rewrite (3.2.15), (3.2.16) as (3.2.17)

dz ∈ K # a(p, b, z). dt

where K # (p, b, z) = K # (p, z) as z ∈ / T ∗ X |∂X and K # (p, b, z) = K # (p, b, ιz) ∗ as z ∈ T X |∂X . Definition 3.2.3. Let K± (p, b, z, Ω) be the union of all the generalized ¯ and going out from the point z ∈ Ω ¯ bicharacteristic billiards residing in Ω in the direction of increasing ±t. Finally, Main Theorem Theorem 3.2.4. Let Ω  T ∗ X , Ω  T ∗ ∂X and Ω|∂X = ι−1 Ω . Let (P, B) satisfy the conditions of Theorem 3.1.7 in Ω and let p be symmetric in Ω. Further, let condition (3.2.13) be fulfilled.

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CHAPTER 3. PROPAGATION NEAR BOUNDARY

(i) Let u be admissible (i.e., satisfy (3.1.20)) and (3.2.18) (3.2.19) and (3.2.20) (3.2.21)

WFs+1 (Pu) ∩ K− (p, b, z, Ω) = ∅,   s+1,l 1 WFb (Pu) ∪ WFs+ 2 (Bðh u) ∩ ιK− (p, b, z, Ω) = ∅, WFs (u) ∩ ∂Ω ∩ K− (p, b, z, Ω) = ∅,  − WFs,0 b (u) ∩ ∂Ω ∩ ιK (p, b, z, Ω) = ∅.

Then (3.2.22)

WFs (u) ∩ K− (p, b, z, Ω) = ∅,

(3.2.23)

WFs,l+m (u) ∩ ιK− (p, b, z, Ω) = ∅. b

(ii) Moreover, let condition (3.1.9)∗ be fulfilled in Ω . Then one can weaken 1 the condition (3.2.19) replacing WFs+ 2 (Bðh u) by WFs (Bðh u):  s+1,l  1 WFb (Pu) ∪ WFs+ 2 (Bðh u) ∩ ιK− (p, b, z, Ω) = ∅ (3.2.19) and skip condition (3.2.21) completely and at the same time obtain in addition that (3.2.24)

WFs−j (D1j u|∂X ) ∩ ιK− (p, b, z, Ω) = ∅

∀j = 0, ... , m + l − 1.

Proof. The proof repeats without any essential modifications the proof of ... Theorem 2.2.10: first, we introduce Kρε (... ) versions of all sets K ... (... ) introduced before and study their properties. Further, using Theorem 3.1.7 we prove the counterpart of Proposition 2.2.7: namely, we consider point z  ∈ Ω, and for arbitrarily small ρ, ε we establish Statements (i) and (ii) similar to those of Theorem 3.1.7 but with Ω and Ω replaced by narrow vicinities of z  and ι−1 z  respectively and with {φ1 ≤ t1 } ∩ · · · ∩ {φJ ≤ k1 ≤ t1 }∩ · · · ∩ {φkJ ≤ tJ } replaced  tJ } and {φ # # by z  − Kρε (p, b, z  ) and ι−1 z  − Kρε (p, b, z  ) respectively. ± Furthermore, we introduce Kρε (p, b, z  ) and study their properties. We prove the counterpart of Lemma 2.2.9 and based on the above counterpart of Proposition 2.2.7 and on Proposition 2.2.7 itself we establish Theorem 3.2.4 − but with K− (p, b, z) replaced by Kρε (p, b, z). − Finally, taking ρ → +0, ε → +0 we prove that Kρε (p, b, z, Ω) tends to − K (p, b, z, Ω) which concludes the proof. We leave all obvious but tedious details to the reader.

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231

Discussion and Examples

General Remarks Propagation of singularities near boundary is very complicated. Even analysis of the scalar operators of the principal type becomes really complicated due to jumping along ι−1 (z  ) and especially due to bicharacteristics tangent to the boundary. Even case of elliptic P is as complicated as the general boundaryless case case (see Subsubsection 3.2.2.4 Elliptic Points below). Remark 3.2.5. (i) Under condition (3.1.9)∗ K  (p, b, z) = R2d−2 and hence there is no propagation linked with the boundary conditions. One can prove it easily following the proof of Theorem 3.1.13. This would make results marginally simpler. However in this book we are interested in the symmetric, not strictly dissipative boundary conditions. In Section 3.4 we will show that one can replace (3.1.9)∗ by a weaker condition (3.4.18) which dallows symmetric boundary conditions and has the same effect. (ii) To ensure that boundary value problem is not overdetermined, we assume that ¯ + × D-matrix operator. (3.2.25) B is D Constant Coefficients Case Example 3.2.6. As X = {x ∈ Rd , x1 ≥ 0} and p = p(ξ), b = b(ξ) we have that   (3.2.26) K− (p, b, z) = z − K # (p, b, z) ∩ T ∗ X as z ∈ T ∗ X |∂X    (3.2.27) K− (p, b, z) = z − K σ (p, z) ∩ T ∗ X ∪ K− (p, b, z  ) z  ∈(z−K σ (p,z))

as z ∈ T ∗ X |X \∂X where K (p, b, z) ⊂ 0 × Rdξ and K # (p, b, z) ⊂ Rdx × ι−1 ιz. Transversal Points Let z  ∈ T ∗ ∂X and zk ∈ ι−1 z  . Consider two very special cases: (3.2.28)±

K σ (p, zk ) \ 0 ⊂ {±x1 > 0}.

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232

Definition 3.2.7. (i) As (3.2.28)+ is fulfilled zk is outgoing point ; (ii) As (3.2.28)− is fulfilled zk is incoming point . (iii) As m = 1 (i.e. P = A0 (x, hD  )hD1 + A1 (x, hD  )) and P has a blockdiagonal form (3.1.45), the corresponding component of u is called outgoing and incoming components respectively. Obviously, (3.2.29) As m = 1 and P has a block-diagonal form (3.1.45), condition (3.2.28)± is equivalent to (3.2.30)±

± ak (z  ) is positive definite.

Let us denote by u  , u  and u  collections of all outgoing, incoming and other components respectively. Let D , D , D be dimensions of u , u  , u  .    Let us rewrite boundary operator B in the form B = B B B  . Then (3.2.10) and (3.2.30)+ imply that (3.2.31)

rank b  = D

and therefore one can assume without any loss of the generality that   I B1 B1 (3.2.32) B= 0 B2 B2 (otherwise we multiply it by an elliptic operator from the left). Then microlocally one can exclude outgoing components, consider reduced operator P and boundary operator B: ⎞ ⎛    0 P 0   I B B  1 1 0 ⎠, B= (3.2.33) P =⎝0 P 0 B2 B2 0 0 P  where removed components are dimmed. One can see easily that all conditions are fulfilled including (3.1.11), (3.2.25) and (3.1.9)8) . As we determine (u  , u  ) from the reduced problem, 8)

∗

Which may even become (3.1.9) ; we leave to the reader to find out when it happens.

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233

we can find ðu  from the remaining boundary conditions and then find u  for x1 > 0 solving Cauchy problem for P  with the initial data as {x1 = 0}. Consider now incoming component u  . Assume that we know u  as {x2 ≤ t − ε, x1 ≥ ε} while x2 (z  ) = t; recall that we assume that P is microhyperbolic in direction ∇# x2 where  > 0 and ε > 0 are small enough constants. Then u  is determined in the small vicinity of ι−1 z  from the “Cauchy” problem with data at {x2 ≤ t − ε}. Thus, we can exclude incoming trajectories as well:      P 0 (3.2.34) P= B = B2 B2  , 0 P (but due to the different reason). One can see easily that all conditions are fulfilled including (3.1.11), (3.2.25) and (3.1.9). Elliptic Points Consider now elliptic component u0 of u; in accordance with (3.1.45) we denote it by (u  , u  ) where p0 = (p0 )† and without any loss of the generality one can assume that     0 I 0 −α   , a1 (z ) = (3.2.35) a0 (z ) = I 0 −α† 0 where α satisfies (3.1.51). Then due to Theorem 3.1.12 the equation P0 u  ≡ f  alone determines  u (modulo O(hs )). In a certain sense u  is similar to incoming component of the previous subsubsection. However, while we can remove it from B, we cannot remove it from P without removing u  because  then assumption  p † = p would be violated. Let us rewrite B as B = B  B  B  again where again u  denotes remaining components and D , D , D are dimensions of components. Assuming that (3.2.36)

rank b  = D

we can assume without any loss of the generality that B has form (3.2.32) and then we can determine ðu  from ðu  and then we can remove both u  and u  from P and from B reducing them to P  and B2 . One can see easily that all conditions are fulfilled including (3.1.11), (3.2.25) and (3.1.9).

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Knowing u  we can find ðu  from the boundary conditions and then u  according to Theorem 3.1.12 solving Cauchy problem. However, this reduction requires condition (3.2.36). Assume now that D = 0. Then what we have is B  ðu  = 0. Due to Theorem 3.1.12 equation P  u ≡ f  has no effect on ðu  and therefore in this case analysis of propagation for (P, B) is reduced to the analysis of propagation for B  . Condition (3.3.17) below implies that β b  is symmetric while (3.1.9) implies that β b  is dissipative9) ; here β is an elliptic symbol. Further, condition (3.1.9)∗ clearly implies that β B  is strictly dissipative and thus elliptic. Finally microhyperbolicity condition for (P, B) is equivalent to microhyperbolicity condition for β b  . Remark 3.2.8. The boundary waves are often called surface waves and they are O(hs ) on the distances ≥ Cs h−1 | log h| from the boundary. Example 3.2.9. The most famous example of the boundary waves in the elliptic zone are Rayleigh waves in the elasticity; see f.e. M. Taylor [4]. Bicharacteristics Tangent to the Boundary Let us consider the simplest and the most interesting case. Assume that D = 2, D+ = 1 (may be after removal transversal and removable elliptic components) . One can prove easily that in this case one can reduce p to     I 0 0 1 (3.2.37) ξ ± 0 r (x, ξ  ) 1 0 1 with a scalar real-valued symbol r (x, ξ  ) and then (3.2.38)

g (x, ξ) = −ξ12 + r (x, ξ  ).

Further, let us assume that g (x, ξ) = det p(x, ξ) is of the real principal type (i.e. g = 0 =⇒ |∇g | ≥ ) which now is equivalent to (3.2.39)

|r (x, ξ  ) = 0| ≤ 0 =⇒ |∇x,ξ r (x, ξ  )| ≥ 0

9) We call symbol ε dissipative if i(ε − ε† ) is nonnegative matrix and strictly dissipative if it is strictly positive matrix.

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235

and also microhyperbolic with respect ∇# x2 which is equivalent to (3.2.40)±

|r (x, ξ  ) = 0| ≤ 0 =⇒ ±∂ξ2 r (x, ξ  ) ≥ 0 .

Then (3.2.41) T ∗ ∂X = {(0, x  , ξ  )} = H ∪ G ∪ G := {(x  , ξ  ) : r (0, x  , ξ  ) > 0} ∪ {(x  , ξ  ) : r (0, x  , ξ  ) = 0}∪ {(x  , ξ  ) : r (0, x  , ξ  ) < 0} where H, E and G are called respectively hyperbolic, elliptic, glancing zones.

Analysis in the Hyperbolic Zone. In the hyperbolic zone H we have two transversal trajectories: an incoming trajectory with ξ2 = ±r 1/2 (x  , ξ  ) and an outgoing trajectory with ξ2 = ∓r 1/2 (x  , ξ  ) (in the framework of (3.2.40)± ); reflection goes according to “incident angle = reflection angle” law. Analysis in the Elliptic Zone. In the elliptic zone E propagation can happen only along boundary bicharacteristics. Namely, assuming that the boundary problem satisfies conditions (3.2.25) and (3.3.17) we have (after multiplication by an elliptic symbol) either   (3.2.42) b(x  ξ  ) = i β(x  , ξ  ) or (3.2.43)

  ˜ , ξ) 1 b(x  ξ  ) = i β(x

˜  , ξ  ) are scalar real-valued symbols and the second where β(x  , ξ  ) and β(x ˜ = 0}. case is not reduced to the first one near {β Then in the elliptic zone the decaying (for x1 > 0) solution of p(0, x  , D1 , ξ  )u = 1/2 0 is u = e −(−r ) x1 (−i(−r )1/2 , 1)T (up to a constant factor) with r = r (0, x  , ξ  ); so a prorogation of singularities for (P, B) is reduced there to a propagation for operator Q(x  , hD  ) on the boundary (applied to v = ðu2 ) and the principal symbol of Q is (3.2.44)

q(x  , ξ  ) = −(−r (0, x  , ξ  ))1/2 − β(x  , ξ  )

236

CHAPTER 3. PROPAGATION NEAR BOUNDARY

in the case (3.2.42). In the case (3.2.43) ˜  , ξ  )(−r (0, x  , ξ  ))1/2 + 1 q(x  , ξ  ) = β(x ˜ = 0}, operator Q is elliptic. and since we consider this case only near {β Therefore, in the case (3.2.42) the characteristic manifold of Q is (3.2.45) Σb := {(x  , ξ) : λ(x  , ξ  ) := r (0, ξ, ξ) − β2 (x  , ξ) = 0, β(x  , ξ) > 0} and the propagation is going in the elliptic zone along the boundary along trajectories of λ(x  , ξ  ). Analysis in the Glancing Zone. Consider now (x  , ξ  ) ∈ G. One can see easily that at such points (3.2.46)

K # (p, b, (x  , ξ  )) ∩ T ∗ X = {∇# r (0, x  , ξ  )}cone ;

so we get generalized bicharacteristics going along the boundary: namely, bicharacteristics of r (0, x  , ξ  ) going along G. We call them boundary bicharacteristics The generalized bicharacteristic billiards of (P, B) thus should consist of those bicharacteristics and bicharacteristics of r (x, ξ) − ξ12 and here we have two principally different cases: Bicharacteristically Strongly Convex Domain means that (3.2.47)

x1 = 0, g = 0, {g , x1 } = 0 =⇒ {g , {g , x1 }} < 0

which is equivalent to (3.2.48)

x1 = 0, r (0, x  , ξ  ) = 0 =⇒ ∂x1 r (0, x  , ξ  ) < 0.

In this case no bicharacteristic of g can pass through points of G and thus only boundary bicharacteristics are passing through G. Further, along bicharacteristic billiards passing through points of H close to G symbol (3.2.49)

ρ(x, ξ  ) := r (x, ξ  ) − x1 · ∂x1 r (0, x  , ξ) = ξ12 − x1 · ∂x1 r (0, x  , ξ)

preserves its magnitude (at any finite time interval); so these billiards make short jumps; the height (maximal value of the jump) is  ρ, the length of

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237

each jump and the incidence/reflection angles are  ρ1/2 ; thus the number of jumps during the finite time is  ρ−1/2 . There is a huge literature (see Comments). Let us mention only that it is referred as whispering gallery case and boundary bicharacteristics are called gliding rays. Example 3.2.10. Consider g (x, ξ) = ξ22 − ξ12 − x1 .

(3.2.50)

x1

∂X Figure 3.1: Bicharacteristically convex case: Example 3.2.10.

Bicharacteristically Strongly Concave Domain means that (3.2.51)

x1 = 0, g = 0, {g , x1 } = 0 =⇒ {g , {g , x1 }} > 0

which is equivalent to (3.2.52)

x1 = 0, r (0, x  , ξ  ) = 0 =⇒ ∂x1 r (0, x  , ξ  ) > 0.

In this case through each point (x  , ξ  ) ∈ G passes both bicharacteristics of g and boundary bicharacteristics as well. The x-projection of bicharacteristic of g is tangent to ∂X ; such bicharacteristics are called grazing rays. Since boundary bicharacteristics resides in G, we get that at every its point the generalized bicharacteristic billiard can leave the boundary along grazing ray. Such trajectories consisting of boundary bicharacteristics and may be grazing rays are called creeping rays. Remark 3.2.11. (i) The truth is however that our results in this case are not sharp: singularities as we understand them (which are equivalent to smooth singularities) do not propagate along creeping rays. However analytic singularities do propagate along boundary bicharacteristics and creeping rays.

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238

(ii) Further, Gevrey singularities with index κ ≤ 3 propagate along creeping rays while Gevrey singularities with index κ > 3 don’t. Namely, consider propagator U(x, y , t) and assume that no other ray but a creeping ray only reaches x from y for time t; then |U(x, y , t) = O(h∞ ) and moreover | log u|  sh−1/3 where s ≤ t is a time spent by this ray creeping along boundary bicharacteristic. For more details see 3.5.3. (iii) We will improve Theorems 3.1.7, 3.1.13 and thus 3.2.4 to exclude creeping rays. Example 3.2.12. Consider g (x, ξ) = ξ22 − ξ12 + x1 .

(3.2.53)

grazing ray

creeping rays

x1

∂X Figure 3.2: Bicharacteristically concave case: Example 3.2.10. Dotted lines show creeping rays and the dashed line shows the front of them at some moment

Remark 3.2.13. Consider P = −τ 2 + |ξ|2 in X × Rt where X ⊂ Rd is a domain with the smooth boundary and (ξ, τ ) are dual variables. Then notions of (strong) convexity and concavity coincide with ordinary geometric notions. On Figure 3.3 we consider interior (a) and exterior (b) of the circle; on (b) we also show front of the creeping rays “in the development”. We, however, distinguish strong and strict convexity and concavity; strong is a bit more restrictive as the boundaries of strictly convex and concave domains may be flat at some points (like x2 = x14 ).

3.3. COROLLARIES OF THEOREM 3.1.7

239

grazing ray

(a) Convex case

(b) Concave case

Figure 3.3: Straight rays and non-straight boundary

Domain which is neither Bicharacteristically Concave nor Bicharacteristically Convex. This is when the real troubles come. In this case bicharacteristic billiards can branch even if domain is just convex and the curvature {g , {g , x1 }} vanishes at some points with all its derivatives.

3.3

Corollaries of Theorem 3.1.7

In this section we consider the following issues: the immediate corollaries of Theorem 3.1.7 like the finite speed of the propagation of singularities and the decomposition of a solution into two components (the free space solution and the reflected wave) and the structure in a neighborhood of t = 0 of the restriction of the fundamental solution to the diagonal and the structure of some related functions (we are mainly interested in the normality of the singularity.

CHAPTER 3. PROPAGATION NEAR BOUNDARY

240

3.3.1

Finite Speed of Propagation and Related Topics

Finite Speed of Propagation In this section we consider the operators P = hDt + A, m A ∈ Ψ h (X , H, H),

(3.3.1)

B = (Bj )j=1,...,m , Bj ∈ Ψh (∂X , H , H)

where Ψ h,N (X , H, H) is the class of operators of the form (3.1.1) with Ak ∈ Ψ h,N (X , H, H). Surely Theorem 3.1.7 immediately yields an assertion about the finite speed of the propagation of singularities, but condition (3.1.23) of that theorem is not necessary for this special consequence, so we want to get rid of it and to prove the assertion directly. m

Theorem 3.3.1. Let P, B be operators of type (3.3.1) and let conditions (3.1.4), (3.1.6) and (3.1.9) be fulfilled in the domains Ω+ ⊂ T ∗ (X  ×Rt )×R+ x1 and Ω ⊂ T ∗ (∂X × Rt ) respectively where Ω = Ω+ ∩ {x1 = 0}; the matrix T is now defined by the operator A. Let us assume that (3.3.2)1 ||ak (x, ξ  )|| + ||∇x,ξ ak (x, ξ  )|| ≤ c ∀(t, x, τ , ξ  ) ∈ Ω+

∀k = 0, ... , m,

(3.3.2)2 ||βk (x  , ξ  )|| + ||bk (x, ξ  )|| + ||∇x  ,ξ bk (x, ξ  )|| ≤ c ∀(t, x  , τ , ξ  ) ∈ Ω ∀k = 1, ... , m and (3.3.2)3

|τ | ≤ c

in Ω+

where ak (x, ξ  ) are the principal symbols of Ak , A = Σ0≤k≤m Ak (hD1 )k . Let N = N(d, m, s, M, l) and let u be temperate (i.e. satisfy condition (3.1.20)). Then (i) If z¯ = (t¯, x¯, τ¯, ξ¯ ) ∈ Ω+ , t¯ > 0 and (3.3.3)

WFs+1,l  (Pu) ∩ K− (¯ z ) ∩ Ω+ = ∅,

(3.3.4)

WFs+ 2 (Bðh u) ∩ K− (¯ z ) ∩ Ω = ∅,

(3.3.5)

WFs,m+l  (u) ∩ K− (¯ z ) ∩ ∂Ω+ = ∅

1

3.3. COROLLARIES OF THEOREM 3.1.7

241

z ) ∩ Ω+ is a bounded domain where and if K− (¯ z ) = {(t, x, τ , ξ  ) ∈ T ∗ (X  × R) × R+ : (3.3.6) K± (¯ τ = τ¯, |x − x¯| + |ξ  − ξ¯ | ≤ ±C0 (t − t¯)}, C0 = C0 (m, c), then z ) ∩ Ω+ = ∅. WFs,l  ∩K− (¯

(3.3.7)

(ii) Moreover, if condition (3.1.9)∗ is fulfilled then one can replace superscript s + 12 in (3.3.4) by s (see f.e. (3.1.28) ) and obtain at the same time that (3.3.8)

z) ∩ Ω = ∅ WFs−j (D1j u|∂X ) ∩ K− (¯

∀j = 0, ... , q + m − 1.

z ) replaced by {φ ≤ 0} Proof. It is sufficient to prove this theorem with K− (¯ for φ = t + ψ(x, τ , ξ  ) for every smooth real-valued function ψ with |∇x,ξ ψ| ≤ ε = ε(m, c) > 0;

(3.3.9)

the assertion modified in this way will be indicated by a prime. Moreover, it is sufficient to consider the case l = 0 and one can assume that WFs− 2 ,m  ∩{φ ≤ 0} ∩ Ω+ = ∅ 1

(3.3.10)

and in the proof of (3.3.8) one can assume in addition that (3.3.11)

1

WFs−j− 2 (D1j u|∂X ) ∩ {φ ≤ 0} ∩ Ω = ∅

∀j = 0, ... , m − 1.

Let q(t, x, τ , ξ  ) = ζχ(φ − δ) where ζ ∈ C0∞ (Ω+ ) equals 1 in Ω+ outside of the μ-neighborhood of ∂Ω+ , χ is a function satisfying (2.1.8) and μ > 0 is small enough. Let Q = Opw (q) and Q  = Q 2 . Let us apply the identity (3.3.12)

∗

− Re i([P, Q  ]u, u) = Re i((Q  + Q  )Pu, u)+ ∗

Re i((P ∗ − P)Q  u, u) + Re h(Q  T ðh u, ðh u)∂X ×R

CHAPTER 3. PROPAGATION NEAR BOUNDARY

242

(compare with (3.1.70)) where now the inner product (., .) is taken on X × R and the matrix T is defined by (3.1.5); then after routine estimates we obtain the following inequality: (3.3.13) Q  u2 ≤ Re([A, Ψ]Q  u, Q  u)− Re ih(B[B, Ψ]Q  ðh u, Q  ðh u)∂X ×R + Ch2s where Q  and Ψ are pseudodifferentials operators with the principal symbols q  = ζχ1 (φ − δ)χ2 (φ − δ) and ψ respectively, χ1 and χ2 are defined in (2.1.8); moreover, under condition (3.1.9)∗ one can add the term 1 Q  γh u2∂X ×R to the left-hand expression in (3.3.13) where here and below 1 = 1 (m, c) and C1 = C1 (m, c). On the other hand, (3.3.9) yields that the right-hand expression in (3.3.13) does not exceed (hD1 )j Q  u2 + Ch2s Cε 0≤j≤m

hence the theorem in question will be proven provided we prove the inequality (hD1 )j Q  u ≤ C1 Q  u + Chs . 0≤j≤m

This inequality follows from (3.3.2)3 and the estimate

(3.3.14)

  (hD1 )j v  ≤ C1 v  + Av  + Qv 

0≤j≤m

where Q = Q(x, Dx ) ∈ Ψ h (X , H, H) is some operator with symbol vanishing in πΩ+ × Rξ1 where π is the map (t, x, τ , ξ) → (x, ξ) and the norms are taken on X instead of X × R. Estimate (3.3.14) is stable with respect to small variations of the principal symbol of A and arbitrary variations of the lower order terms and hence it is sufficient to prove the estimate m

(3.3.15)



 ||(hD1 )j v ||+ ≤ C1 ||v ||+ + ||a(z, hD1 )v ||+ )

0≤j≤m

∀z ∈ πΩ+

∀v ∈ S(R+ , H);

3.3. COROLLARIES OF THEOREM 3.1.7

243

the previous estimate follows from this one by a partition of unity in πΩ+ . In order to prove (3.3.15) we use the well known inequality j

(3.3.16)

1− mj

||D1j v ||+ ≤ Cmj ||D1m v ||+m · ||v ||+

∀j = 0, ... , m;

this inequality yields that for every δ > 0   || a(z, hD1 ) − a0 (z)(hD1 )m v ||+ ≤ δ||(hD1 )m v ||+ + Cδ ||v ||+ and hence condition (3.1.6) implies that ||(hD1 )m v ||+ ≤ C1 δ||(hD1 )m v ||+ + Cδ ||v ||+ + Cδ ||a(z, hD1 )v ||+ . This inequality and (3.3.16) obviously imply (3.3.15). Let us replace condition (3.1.9) by the condition (3.3.17)

T (z)w , w m + 2 Re β(z)b(z)w , w  = 0

∀w ∈ Hm .

Such boundary problems are called symmetric. Then one can replace t by −t, P by −P and K− (¯ z ) by K+ (¯ z ). Let us apply Theorem 3.3.1 to u ± = θ(±t)u instead of u and let us use the obvious analogue of Proposition 2.1.10; we then obtain Theorem 3.3.2. Let conditions (3.3.1), (3.1.4), (3.1.6), (3.3.2) and (3.3.17) be fulfilled and let u be temperate. Then if z¯ = (t¯, x¯, τ¯, ξ¯ ) ∈ Ω+ and t¯ > 0, (3.3.18)

z ) ∩ {t ≥ 0} ∩ Ω+ = ∅, WFs+1,l  (Pu) ∩ K− (¯

(3.3.19)

WFs+ 2 (Bðh u) ∩ K− (¯ z ) ∩ {t ≥ 0} ∩ Ω = ∅,

(3.3.20) and (3.3.21)

WFs,m+l  (u) ∩ K− (¯ z ) ∩ {t ≥ 0} ∩ ∂Ω+ = ∅ WFs+1,l  (Pu) ∩ π(K− (¯ z ) ∩ {t = 0} ∩ Ω+ ) = ∅,

(3.3.22)

WFs+ 2  (Bðh u) ∩ (πK− (¯ z ) ∩ {t = 0} ∩ Ω) = ∅,

(3.3.23)

WFs,m+l  (u|t=0 ) ∩ π(K− (¯ z ) ∩ {t = 0} ∩ ∂Ω+ ) = ∅

1

1

where in (3.3.21) and (3.3.22) the wave front sets are calculated as wave front sets of functions on X  × (Rt × R+ ) and ∂X × Rt respectively; then (3.3.24)

z ) ∩ {t ≥ 0} ∩ Ω+ = ∅ WFs,m+l  (u) ∩ K− (¯

This assertion remains true if we take t¯ < 0 and replace K− (¯ z ) by K+ (¯ z) and {t ≥ 0} by {t ≤ 0}.

CHAPTER 3. PROPAGATION NEAR BOUNDARY

244

Decomposition of the Fundamental Solution At this point we assume that conditions (3.1.6) and (3.3.2)1−3 are fulfilled for all z ∈ T ∗ Rd . Let us assume that (3.3.25) The operator A with domain  D(A) = u : D1j u ∈ L2 (X , H) ∀j = 0, ... , m and Bðh u = 0 is a self-adjoint operator in L2 (X , H). Let U(t, x, y ) be the Schwartz kernel of the operator e −ith

. Then

Px U = 0,

(3.3.26)

Bx ðhx U = 0,

(3.3.27) (3.3.26)

−1 A

†

U tPy = 0,

(3.3.27)†

U t(By ðhy ) = 0,

(3.3.28)

U|t=0 = δ(x − y )I ,

(3.3.29)

U(t, x, y ) = U † (−t, y , x).

Then Theorem 3.3.2 immediately implies Theorem 3.3.3. Let conditions (3.3.1), (3.1.4), (3.1.6), (3.3.17) and (3.3.25) be fulfilled and N = N(d, m, M, s, l). Then (3.3.30)

WFs,l  (U) ∩ {|τ | ≤ c} ⊂ {|x − y | + |ξ  + η  | ≤ C0 |t|}

where in the calculation of the wave front set U is treated as a distribution on (Rt × X  × X  ) × R+2 x1 ,y1 . Let us now assume that (3.3.31) The operator A0 with domain  D(A0 ) = u : D1j u ∈ L2 (X 0 , H) ∀j = 0, ... , m is a self-adjoint operator10) in L2 (X 0 , H), X 0 = Rd and coincides with A on X. 10) If condition (3.1.6) is fulfilled in the whole space X 0 then the operator is self-adjoint provided it is symmetric, i.e., provided its Weyl symbol is Hermitian.

3.3. COROLLARIES OF THEOREM 3.1.7

245 −1

Let U 0 (t, x, y ) be the Schwartz kernel of the operator e −ith A ; then U 0 satisfies (3.3.26), (3.3.26)† , (3.3.28) and (3.3.29) with X replaced by X 0 . Then U 1 = U − U 0 satisfies equations (3.3.26), (3.3.26)† , (3.3.29) and (3.3.32)

Bx ðhx U 1 = −Bx ðhx U 0 ,

(3.3.32)†

U 1 ( tBy ðhy ) = −U 0 ( t(By ðhy ),

(3.3.33)

U 1 |t=0 = 0.

0

Hence U 1± = θ(±t)U 1 satisfies (3.3.26), (3.3.26)† and (3.3.32), (3.3.32)† with U 0 replaced by U 0± . In particular, (3.3.26) and (3.3.26)† imply that (3.3.34) U 1± is many times differentiable with respect to x1 , y1 . Theorem 2.1.2 implies that for y1 > 0 (3.3.35) WFs,l  (ðhx U 0± ) ∩ {|τ | ≤ c} ⊂ {|x  − y  | + |ξ  + η  | + y1 ≤ ±C0 t} and hence Theorem 3.3.2 implies that (3.3.36) WFs,l  (U 1± ) ∩ {|τ | ≤ c} ⊂ {|x  − y  | + |ξ  + η  | + x1 + y1 ≤ ±C0 t}; moreover (3.3.37) (3.3.36) holds as y1 ≥ 0. Really, the only reason why we need this assumption is that otherwise D1m−1 U 0± has a jump as x1 = y1 and thus ðhx U 0± is not defined; however we can consider (3.3.28) with δ(x1 − y1 ) replaced by θ(x1 − y1 ) and get for modified U 1± (3.3.36) with l = 0; then for modified U 1± we get (3.3.36) with arbitrarily large l due to (3.3.26) and (3.3.26)† . Differentiating with respect to y1 we return to original U 1± .

3.3.2

Normal Singularity Property

Preliminary Analysis In this subsection we investigate the structure near t = 0 of the functions Γx Q1x U tQ2y and ΓQ1x U tQ2y 11) and some related functions where 11)

Recall that (Γx V )(x, t) = V (x, t) and (ΓV )(t) =



V (x, x, t)dx.

CHAPTER 3. PROPAGATION NEAR BOUNDARY

246

Q1 , Q2 ∈ Ψh m−1 (X × R, H, H) are operators with symbols supported in a small neighborhood of z ∗ ∈ T ∗ (∂X × 0). We now assume that condition (3.1.23) is fulfilled. Let R ∈ Ψm−1 (X × R, Hm , H) be an operator implementing all the h reductions of Subsubsection 3.1.2.1 Reduction to a Special Case. Let us introduce U = Rx U tRy and let us consider its blocks U jk with j, k = 0, ... , n. Moreover, let us subdivide U 00 into four subblocks ,

(11)

(12)

U 00 U 00 (21) (22) U 00 U 00

U 00 =

,

and subdivide U 0k and U j0 with j, k = 1, ... , n into two subblocks , U 0k =

(1)

-

U 0k (2) U 0k

,

  (2) U j0 = U (1) . U j0 j0

Then (3.3.38)

Γx Q1x U tQ2y ≡



Γx Q1,jx U jk tQ2,ky

0≤j,k≤n

where Q1,j , Q2,j ∈ Ψh (X × R, H, Hj ) are operators with symbols supported in a small neighborhood of z ∗ . Theorem 3.1.2, (3.3.26) and (3.3.26)† imply that (3.3.39)

WFs,l  (U 0k )T ∩ z ∗ × Xy × R+2 x1 ,y1 = ∅,

(3.3.39)∗

WFs,l  (U j0 )T ∩ Xx × z ∗ × R+2 x1 ,y1 = ∅

(2)

(1)

where T means that η is replaced by −η. Therefore (3.3.40) Assertion (3.3.36) remains true for Q1,0x U 0k , U j0 tQ2,0y and Q1,0x U 00 tQ2,0y (1) (2) (2) (1) (12) (2) replaced by Q1,0x U 0k , U j0 tQ2,0y and Q1,0x U 00 tQ2,0y respectively (where we use the obvious notation). We need

3.3. COROLLARIES OF THEOREM 3.1.7

247

Proposition 3.3.4. The following formula holds: Γ Q1,jx U jj tQ2,jy + hΓx  Qjk (Vjk ), (3.3.41) Γ Q1,x U tQ2,y ≡ 1≤j≤n

0≤j,k≤n

where V = U|x1 =y1 =0 ,

(3.3.42) (3.3.43)

(Γx  V )(x  , t)





= V (x , x , t),







(Γ V )(x , t) =

V (x, x, t)dx1 . R+

  and Qjk ∈ Ψh ∂X ×∂X ×R+ , L(H, H), L(Hj , Hk ) are operators with symbols supported in a small neighborhood of point z ∗∗ = (x ∗ , y ∗ , t ∗ , ξ ∗ , −η ∗ , τ ∗ ). Proof. It is sufficient to prove that for j = k (3.3.44)

Γ Q1,jx U jk tQ2,ky ≡ hQjk (Vjk );

the equality (3.3.45)

Γ Q1,jx U 00 tQ2,ky ≡ hQ00 (V00 ) (1)

(12)

(12)

is proven in the same way. Let us note that modulo functions negligible in the vicinity of z ∗∗ the following equalities are due to (3.3.26) and (3.3.26)† : (3.3.46)

(hDx1 − Ljx )U jk ≡ 0,

(3.3.47)

U jk (−hDy1 − tLky ) ≡ 0

where Ljx = Lj (x, hDx , hDt ) = −A−1 0j A1j , P j = A0j hD1 + A1j are the diagonal blocks of the reduced operator and here M −1 means a parametrix of M in the neighborhood of z ∗ or z ∗∗ . Hence (3.3.48)

h(Dx1 + Dy1 )U jk ≡ Ljx U jk − U jk tLky

and setting W = U jk |x1 =y1 we obtain that (3.3.49) Ψh

hD1 W ≡ Njk (W ) := Ljx W − W tLky .

operator Njk = N (x1 , x  , y  , hDx  , hDy  ) as an element of Let+ us consider   R × X × X , L(Hj , Hk ), L(Hj , Hk ) .

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CHAPTER 3. PROPAGATION NEAR BOUNDARY

Let us note that if λj and λk belong to the spectra of the principal symbols of Lj and Lk respectively with j = k then |λj − λk | ≥  > 0 at z ∗∗ . Therefore due to Appendix 3.A.1 the principal symbol of the operator Njk is invertible at z ∗∗ and thus this operator is elliptic in a neighborhood of the point in question12) . Then (3.3.49) yields that (3.3.50)

W ≡ Njk −1 (hD1 W ).

Hence Q1,jx W tQ2,ky ≡ −iQjk (hD1 W ).   Let us permute Qjk ∈ Ψh X  × X  × R+ , L(H, H), L(Hj , Hk ) with hD1 and substitute the right-hand expression of (3.3.40) instead of W into the term with the commutator. Moreover, let us repeat this procedure many times; we then obtain Q1,jx W tQ2,ky ≡ −ihD1 Qjk (W ). Integration on x1 ∈ R+ and substitution y  = x  gives us (3.3.44) with Qjk = Qjk |x1 =0 . Main Results Theorem 3.3.5. Let P and B satisfy conditions (3.3.1), (3.1.4), (3.1.6), (3.1.23), (3.3.17), (3.3.25) and (3.3.31). Let us assume that the problem (P, B) is microhyperbolic at the point z ∗ ∈ T ∗ (∂X × R), t(z ∗ ) = 0 in the multidirection T = (x , ξ , ν1 , ... , νn ). Further, let Q1 , Q2 ∈ Ψh (X × R, H, H) have symbols supported in a small enough neighborhood of point z ∗ . Finally, let χ ∈ C0N ([ 12 , 1]) and T > 0 be a small enough constant, N = N(d, m, l, s, D∗ ) and let |α| ≤ l, |β| ≤ l. Then (i) If x = 0, ν1 = · · · = νn = ν then (3.3.51)

||Γx χT (t)Dxα Dyβ Q1,x U tQ2,y || ≤ C  hs

where here and below in this theorem C  depends on T . (1)

(2)

12) The same remains true if we replace Lj and Lk by L0 and L0 respectively and hence all the forthcoming analysis also remains true.

3.3. COROLLARIES OF THEOREM 3.1.7

249

(ii) If x = 0 then ||Γ χT (t)Dxα Dyβ Q1,x U tQ2,y || ≤ C  hs

(3.3.52) and (3.3.53)

||Γx  ðx ðy χT (t)Dxα Dyβ Q1,x U tQ2,y || ≤ C  hs .

(iii) If ν1 = ... = νn = ν then ||Γ χT (t)Dxα Dyβ Q1,x U tQ2,y || ≤ C  hs  where (Γ V )(x1 , t) := X  (Γx V )(x1 , x  , t)dx  . (3.3.54)

(iv) If x = ξ = 0 then ||ðx ðy χT (t)Dxα Dyβ Q1,x U tQ2,y || ≤ C  hs .

(3.3.55)

(v) In the general case ||ΓχT (t)Dxα Dyβ Q1,x U tQ2,y || ≤ C  hs

(3.3.56) and (3.3.57)

||Γ ðx ðy χT (t)Dxα Dyβ Q1,x U tQ2,y || ≤ C  hs .

Proof. Let us note that (iii) and (v) follow from (i) and (ii) respectively by means of the Fourier integral operator on x  with linear canonical transformation Ψ : (x  , ξ  ) → Ψ(x  , ξ  ) such that (Ψ )ξ = 0. Let us prove (i), (ii) and (iv). Let P be a reduced operator and R an operator implementing this reduction. Let U = Rx U tRy ,

U 0 = Rx U 0 tRy

and U 1 = Rx U 1 tRy .

Then (3.3.26), (3.3.26)† and the ellipticity of P0 yield that (3.3.58)

WFs,l  (U 00k ) ∩ π(Ω+ × ΩT+ ) = ∅

∀k = 0, ... , n,

(3.3.58)† and (3.3.59)

WFs,l  (U 0j0 ) ∩ π(Ω+ × ΩT+ ) = ∅

∀j = 0, ... , n,

WFs,l  (U 10k ) ∩ π(Ω+ × ΩT+ ) ⊂ {x1 = 0}

∀k = 0, ... , n,

(3.3.59)

†

WF

s,l 

(U 1j0 )

∩ π(Ω+ ×

ΩT+ )

⊂ {y1 = 0}

∀j = 0, ... , n

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250

where Ω and Ω+ are small enough neighborhoods of z ∗ in T ∗ (∂X × R) and T ∗ (X  × R) × R+ respectively and π is the map deleting the second copy of (t, τ ) in Ω+ × ΩT+ , etc. Moreover, Theorem 3.1.2 yields that (3.3.60)

WFs,l  (U 0k ) ∩ π(Ω+ × ΩT+ ) = ∅

∀k = 0, ... , n,

(3.3.60)†

WFs,l  (U j0 ) ∩ π(Ω+ × ΩT+ ) = ∅

∀j = 0, ... , n.

1(2) 1(1)

Let us note that Corollary 2.1.14 implies that the set (3.3.61)

WFs (U 0jk ) ∩ π(Ω+ × ΩT+ ) × R2ξ1 ,η1 ∩ {|t| ≤ T }

is contained in the C0 T -neighborhood of ξ1 = −η1 . Furthermore, (3.3.26) and (3.3.26)† imply that this set is contained in a small neighborhood of {ξ1 = ζj , η1 = −ζk } (recall that ζj are characteristic roots) and hence this set(3.3.61) is empty for small T > 0: (3.3.62)

WFs (U 0jk ) ∩ π(Ω+ × ΩT+ ) × R2ξ1 ,η1 ∩ {|t| ≤ T } = ∅

∀j = k.

Moreover, Theorem 2.1.2 with J = 2 and φ1 = t and (3.3.63)

φ2 = ξ , x  − y   + νk (x1 − y1 ) − εt

with a small enough constant ε > 0 and Corollary 2.1.14 imply that (3.3.64)

WFs (U 0kk ) ∩ π(Ω+ × ΩT+ ) × R2ξ1 ,η1 ∩  0 ≤ t ≤ T , ξ , x  − y   + νk (x1 − y1 ) < εt = ∅ ∀k = 1, ... , n.

Furthermore, (3.3.26), (3.3.26)† and the ellipticity of P jx and tP ky for |ξ1 | ≥ c, |η1 | ≥ c 13) combined with (3.3.62) and (3.3.64) imply that (3.3.62)

WFs,l  (U 0jk ) ∩ π(Ω+ × ΩT+ ) ∩ {|t| ≤ T } = ∅

∀j = k

and (3.3.64)

13)

WFs,l  (U 0kk ) ∩ π(Ω+ × ΩT+ )∩  0 ≤ t ≤ T , ξ , x  − y   + νk (x1 − y1 ) < εt = ∅ ∀k = 1, ... , n.

Which is due to (3.1.6).

3.3. COROLLARIES OF THEOREM 3.1.7

251

Therefore (3.3.65)

WFs,l  (ðx U 0jk ) ∩ π(Ω × ΩT+ ) ∩ {|t| ≤ T } = ∅

∀j = k,

and (3.3.66)

WFs,l  (ðx U 0kk ) ∩ Ω × ΩT+ ∩  0 ≤ t ≤ T , ξ , x  − y   − νk y1 < εt = ∅

∀k = 1, ... , n.

Let us apply Theorem 3.1.13 with J = 2, φj1 = φ1 = t and (3.3.67)

φj2 = ξ , x  − y   + νj x1 − εt,

φ2 = ξ , x  − y   − εt.

Then (3.3.65), (3.3.66), (3.3.26), (3.3.27) and (3.3.36) yield that (3.3.68) WFs,l  (U 1jk ) ∩ π(Ω+ × ΩT+ )∩  {0 ≤ t ≤ T , ξ , x  − y   − νk y1 + νj x1 < εt = ∅ ∀j, k = 1, ... , n and (3.3.69)

WFs,l  (U 10k ) ∩ π(Ω+ × ΩT+ )∩  0 ≤ t ≤ T , ξ , x  − y   − νk y1 < εt} = ∅

∀k = 1, ... , n.

Similarly, (3.3.58)† , (3.3.26), (3.3.27) and (3.3.36) imply that (3.3.70)

WFs,l  (U 1j0 ) ∩ π(Ω+ × ΩT+ )∩  0 ≤ t ≤ T , ξ , x  − y   + νj x1 < εt = ∅

∀j = 1, ... , n

and  (3.3.71) WFs,l  (U 100 ) ∩ π(Ω+ × ΩT+ ) ∩ 0 ≤ t ≤ T , ξ , x  − y   < εt = ∅. These four relations combined with relations (3.3.62), (3.3.64) and (3.3.58) yield Statements (i), (ii) and (iv); moreover, in the proof of (3.3.52) we also use Proposition 3.3.4 which reduces contribution of the non-diagonal blocks to (3.3.53)-type expression. The arguments used in the proof of Theorem 2.1.19 yield immediately

CHAPTER 3. PROPAGATION NEAR BOUNDARY

252

Theorem 3.3.6. Let conditions of assertions (i), (ii), (iii), (iv) or (v) of Theorem 3.3.5 be fulfilled. Then the left-hand expressions in (3.3.51), (3.3.52) and (3.3.53), (3.3.54), (3.3.55), (3.3.56) and (3.3.57) respectively do not exceed (3.3.72)

Ch1−d−|α|−|β|

 h s T

with constant C not depending on T ∈ [h, T0 ], where T0 > 0 is a small constant. For the forthcoming applications inequality (3.3.56) and its sidekick in Theorem 3.3.6 are the most important. Note that according to Proposition 3.3.4 the contributions of non-diagonal blocks to (3.3.52) and (3.3.56) is given by (3.3.53) and (3.3.57)-type expressions respectively with an extra factor h. On the other hand, we do not need to integrate with respect to x1 the contributions of the diagonal blocks but if we plug into these expressions U 1 instead of U and integrate we recover an extra factor T due to (3.3.36). Then we get immediately Theorem 3.3.7. Let conditions of assertions (ii) or (v) of Theorem 3.3.5 be fulfilled. Then the left-hand expressions in (3.3.52), (3.3.56) respectively with U replaced by U 1 do not exceed (3.3.73)

Ch2−d−|α|−|β|

 h s T

with constant C not depending on T ∈ [h, T0 ], where T0 > 0 is a small constant. In particular we get Corollary 3.3.8. Let conditions of assertions (ii) or (v) of Theorem 3.3.5 be fulfilled. Assume that P is elliptic at ι−1 z ∗ . Then the left-hand expressions in (3.3.52), (3.3.56) respectively do not exceed (3.3.73) with constant C not depending on T ∈ [h, T0 ], where T0 > 0 is a small constant. Really, due to the ellipticity of P U 0 is negligible in the vicinity in question. However it is not true for U 1 unless (P, B) is elliptic as well.

3.4. ENERGY ESTIMATES APPROACH REVISITED

3.4

253

Energy Estimates Approach Revisited

The main purpose of this section is to prove theorem similar to Theorem 3.1.7 but showing that there is no propagation along creeping ways. In such theorem one needs to have function φ depending on ξ1 ; in this case having separate function φj for each component does not make sense. Then χ(φ) would also depend on ξ1 ; to avoid dealing with pseudodifferential operators containing hD1 we however use Weierstrass theorem and rewrite this and other symbols as polynomials with respect to ξ1 modulo g (x, ξ); namely χ(φ) = q(x, ξ) + g (x, ξ)μ(x, ξ) where g (x, ξ) is a determinant as usual and q(x, ξ) is a polynomial with respect to ξ1 . Since g (x, ξ)I = ν(x, ξ)p(x, ξ) then after quantization term an operator which applied to u we can properly bound. So we need to consider case when operator Q(x, hD  ) we used before is replaced by (3.4.1) Q(x, hD) = Qk (x, hD  )(hD1 )j 0≤j≤m

which is a scalar symmetric operator as before. Actually to a certain degree we did it before in Proposition 3.1.14. Remark 3.4.1. Assume that in the vicinity of (¯ x  , ξ¯ ) ∈ T ∗ ∂X operator P has a block form with different blocks corresponding to disjoint groups of characteristic roots ξ1 = ζ∗ (x, ξ  ) of g (x, ξ1 , ξ  ); let gj (x, ξ) be corresponding characteristic polynomials; j = 1, ... , J. Then  Gj (x, ξ)φj (x, ξ), Gj (x, ξ) := gk (x, ξ) (3.4.2) φ(x, ξ) = 1≤j≤J

k =j

would deliver for each group its own function φj and therefore, at least in principle, we need just one function φ if we allow it to depend on ξ1 ; conversely, if each group has just one real characteristic root ξ1 = ζj (x, ξ  ) then our new approach does not deliver anything ne. On the contrary, if gj (x, ξ) has several characteristic roots which cannot be separated (so, they coincide at (¯ x  , ξ¯ )) our new approach separates them “mildly”.

3.4.1

Theorem 3.1.7(ii) Revisited

To test this method we first prove statements (ii) of Theorems 3.1.7 and 3.1.13 under more general condition than (3.1.9)∗ .

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Energy Estimates To do this we replace our old Qk (x, hD  ) by Qk,new = αk (x, hD)Qk (x, hD  ) where an extra factor α(x, hD) is a polynomial with respect to hD1   δj0 + αkj (x, hD  ) (hD1 )j (3.4.3) αk (x, hD) = 0≤j≤m

and αkj (x, ξ  ) are real-valued and Sh,N -norms of αkj do not exceed ε (which is a sufficiently small constant).  Then Qkj,new ≡ δj0 + αkj Qk mod O(h) (with Qk as in the proof of Theorem 3.1.13) and one can see easily that all the terms along the proof of Theorem 3.1.13 produce exactly the same result as in the proof of Statement (i) with the singular exception of the boundary term (3.4.4)

h(A0k Qk,new uk , uk )∂X

where index k indicate diagonal blocks of the corresponding operators and their symbols or components of u. Really, the principal symbol of the commutator term is still −{pk , qk } which is now replaced by (3.4.5)

− {pk , αk qk } = −αk {pk , qk } − {pk , αk }qk .

Let us recall that in this proof −{pk , qk } was basically replaced by χ22 (φk ){pk , φk } with χ2 = (−χ )1/2 (all other terms were properly estimated) and then {pk , φk } was replaced by a positive symbol plus a symbol of the form ρ†k pk + pk ρk . This still is the case after multiplication by αk because we can rewrite (3.4.6) (3.4.7)

αk (x, ξ) = αk (x, ξ  ) + τk (x, ξ)pk (x, ξ), −1 αkj (−a0k a1k )j αk (x  , ξ  ) = 0≤j≤m

and τk and (αk −I ) are both small. On the other hand, rewriting −{pk , αk } in the form (3.4.6) we see that the only suspicious term is −αk qk = −αk χ21 (φk ) where αk = α (x, ξ  ), χ1 = χ1/2 and we already know how to suppress such terms. Further, boundary terms appearing from expression (3.4.8)

∗ (Qk,new uk , Pk u) − (uk , Qk,new Puk )

3.4. ENERGY ESTIMATES APPROACH REVISITED

255

are easy to estimate as well. Therefore let us consider the mentioned above boundary term (3.4.4) and plug (3.4.6), (3.4.7) there; then obviously we need to consider only h(Tk Qk uk , uk )∂X where Tk = Tk (x  , hD  ) is an operator with the symbol (3.4.9)

Tk (x  , ξ  ) =



 −1 a1k )j . δj0 + αkj a0k (−a0k

0≤j≤m

Further we can replace Qk by Qk ∗ Qk where Qk is an operator with the symbol χ1 (φ). Then the boundary term becomes h(Tk Q  uk , Q  uk )∂X (modulo terms we can easily estimate); we skip index k because we assume that functions φk (and thus operators Qk , Qk ) as well) restricted to ∂X do not depend on k. In virtue of arguments of Subsubsection 3.1.2.6 Proof of Theorem 3.1.13. Step 3 we conclude that (3.4.10) Statements (ii) of Theorems 3.1.7 and 3.1.13 remain true provided (3.4.11)

Tw , w  ≤ −||w ||2

∀w ∈ Ker b

where T is a block-diagonal symbol with blocks defined by (3.4.9). Obviously (3.1.9)∗ coincides with (3.4.11) with αkj = 0 ∀j ≥ 1. Let us assume that the weaker condition (3.1.9) holds; then (3.4.12)

a0 = a0− + β† b + b † β

where a0− is non-positive definite matrix. So we want to construct small αkj such that (3.4.13)

T  w , w  ≤ −||w ||2

∀w ∈ Ker b ∩ K

where T  is a block-diagonal symbol with blocks (3.4.14)

Tk (x  , ξ  ) =



−1 αkj a0k (−a0k a1k )j .

0≤j≤m

and K is an invariant subspace of a0− corresponding to eigenvalues in [−, 0].

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Condition (3.4.13) Obviously it is sufficient to establish (3.4.13) at one point; then it will be fulfilled in its vicinity. Further, one can skip assumption that αkj are small since we can always can replace αkj by εαkj . Note that (3.4.15) One needs to satisfy (3.4.13) only for w with all incoming and outgoing components of w equal 0. Indeed, for such components ∓a0k > 0 and one can replace α0k by α0k ±  without changing anything else.   Further, note that (3.4.13) implies (3.2.36) where b = b  b  b  with the reference to (3.2.35), (3.1.51). Really, if (3.2.36) is not satisfied one can T  pick up w = w  0 0 ∈ Ker b and then Tw , w  = 0. On the other hand, if condition (3.2.36) is satisfied we can exclude both components u  , u  as described in Subsubsection 3.2.2 and condition (3.4.13) will be fulfilled for the reduced problem (P, B) if it was fulfilled for the original one but only under extra assumption with w  = w  = 0. Further, one can see easily that Statement (ii) of Theorem 3.1.13 holds for original problem (P, B) provided it holds for the reduced one. Meanwhile, (3.4.13) for the original system fails unless in addition to (3.2.36) rank b  = D (where we recall that D = D ) due to the same arguments as above. However, since our purpose is Statement (ii) of Theorem 3.1.13 rather than (3.4.13) itself, we do not need to add this twin of (3.2.36). Thus (3.4.16) Only w with components corresponding to roots with | Im ζ| ≤  equal 0 need to be considered. Furthermore, −1 −1 (3.4.17) One can replace in (3.4.9) a0k a1k by (a0k a1k − ηk ) with any real (bounded) ηk .

Then adjusting αkj one can fulfil (3.4.13). Thus, what we need is (3.4.18) There exist real αkj such that (3.4.11) is fulfilled for all w ∈ Ker b ∩ K ∩ Ker π where π is the a spectral projector for a0−1 a1 to subspace corresponding to non-real eigenvalues.

3.4. ENERGY ESTIMATES APPROACH REVISITED

257

Examples Example 3.4.2. Let us ⎛ 0 ⎜0 ⎜ ⎜ (3.4.19) a0 = − ⎜ ... ⎜ ⎝0 1

consider D-matrices of the form ⎞ ⎛ 0 ... 0 1 0 ⎜0 0 ... 1 0⎟ ⎟ ⎜ .. . . .. .. ⎟ and a = ⎜ .. ⎟ ⎜. . 1 . . .⎟ ⎜ ⎠ ⎝1 1 ... 0 0 0 ... 0 0 0

... ... ... ... ...

⎞ 0 0⎟ ⎟ .. ⎟ ; .⎟ ⎟ 0 0 0⎠ 0 0 0

0 1 .. .

1 0 .. .

i.e. a0kl = δD+1−k−l , a1kl = δD−k−l 14) . One can see easily that elements of a0 (−a0−1 a1 )j are ajkl = −δD+1−j−k−l . Then (i) Let D = 2n. Then both positive and negative spectral subspaces of a0 have dimension n and thus b should be n × D matrix. Let us prove that   (3.4.20) (3.1.14) is fulfilled provided rank bII = n with b = bI bII . Indeed, selecting recurrently α2j+1 (from j = 0) and picking up α2j = 0 one can achieve (3.4.21)

T  w , w  ≤ −||wI ||2 − ε||wII ||2

with arbitrarily small ε > 0 where wI = (w1 , ... , wn ), wII = (wn+1 , ... , wD ) and thus we achieve (3.1.14) under condition rank bII = n. On the other hand, since for any choice of αj (3.4.22)

Tw , w  = 0

∀w : wI = 0

condition rank bII = n is also necessary for (3.1.14). (ii) Let D = 2n + 1. Then positive and negative spectral subspaces of a0 have dimensions n and (n + 1) respectively and thus b should be n × D matrix. Selecting recurrently α2j (from j = 0) and picking up α2j+1 = 0 one can achieve (3.4.21) where now wI = (w1 , ... , wn+1 ), wII = (wn+2 , ... , wD ). On the other hand, (3.4.22) holds. Therefore claim (3.4.20) holds in this case as well. 14)

We use notation δl := δl,0 .

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CHAPTER 3. PROPAGATION NEAR BOUNDARY

(iii) Let D = 2n + 1 but a0 is replaced by −a0 . Then positive and negative spectral subspaces of a0 have dimensions (n + 1) and n respectively and thus b should be (n + 1) × D matrix. Selecting recurrently α2j (from j = 0) and picking up α2j+1 = 0 one can achieve (3.4.21) in the same notations as in (ii). On the other hand, (3.4.22) holds. Therefore claim (3.4.20) holds in this case as well. Example 3.4.3. In particular let n = 1, D = 2. Then (3.1.9) and (3.4.18) together are equivalent to b = (b1 1) (or could be reduced to this form) with Re b1 ≤ 0. One can easily consider the case when a0 and a1 are block-diagonal with blocks a0k and (a1k + ηk a0k ) of the types covered by example 3.4.2(i)–(iii). Theorems 3.1.13(ii), 3.1.7(ii) and 3.2.4(ii) Generalized So, we arrive to following generalization of Theorems 3.1.13(ii) and 3.1.7(ii): Theorem 3.4.4. Let conditions of Theorem 3.1.13 be fulfilled. Further, let boundary operator B satisfy (3.1.9), (3.2.36) and (3.4.18). Finally, let φj ∈ CN (T ∗ ∂X ), φkj ∈ CN (R+ × T ∗ X  ) be real-valued functions satisfying (3.1.27) (j = 1, ... , J, k = 1, ... , n). Let us assume that for every j = 1, ... , J the problem (P, B) is microhyperbolic at the point z ∗ in the multidirection Tj = (∇# φj , −∂x1 φ1j , ... , −∂x1 φnj )(z ∗ ). Let N = N(d, M, s, l) and l ≥ 2D∗ − 2, D∗ = maxk Dk where Dk are the dimensions of Hk . Let u satisfy (3.1.20), (3.1.69) , (3.1.70), (3.1.71), (3.1.72) where u = (u1 , ... , un , u0 ), Ω and Ωk are small enough neighborhoods of z ∗ in T ∗ ∂X and R+ × T ∗ X  respectively such that Ω = Ωk |∂X ∀k = 1, ... , n. Then (3.1.34), (3.1.73) and (3.1.74) hold. Theorem 3.4.5. Let conditions of Theorem 3.1.13 be fulfilled. Further, let boundary operator B satisfy (3.1.9), (3.2.36) and (3.4.18). Furthermore, let u satisfy (3.1.20), (3.1.28) (3.1.29), (3.1.30) and (3.1.31) where Ω and Ωk are small enough neighborhoods of z and zk in T ∗ ∂X and T ∗ X respectively not depending on u, t1 , ... , tJ such that Ω = Ωk |∂X ∀k = 1, ... n. Then (3.1.33) and (3.1.34) hold.

3.4. ENERGY ESTIMATES APPROACH REVISITED

259

Theorem 3.4.6. Let conditions of Theorem 3.2.4 be fulfilled. Further, let boundary operator B satisfy (3.1.9), (3.2.36) and (3.4.18). Furthermore, let u satisfy (3.2.18), (3.2.19) (3.2.20), and (3.2.21). Then (3.2.22) and (3.2.24) hold.

3.4.2

Main Theorem

Discussion We are interested in te case when g = det p is of the real principal type or more generally when p is block-diagonal with pk ⊗ I blocks and gk = det pk of the real principal type. Then points where ±{gk , x1 }(zk ) > 0 are outgoing and incoming points respectively and the corresponding components could be excluded from the analysis, we are interested in the tangent points zk −1 where {gk , x1 } vanishes. Then (a0k a1k )(z) has Jordan cells of the height Dk as z = ιzk . We will prove (see Appendix 3.A.2) then that symbols a0k and a1k calculated at such points could be reduced to the types covered by example 3.4.2(i)–(iii). This is still pretty complicated due to the interaction of the different components; so we start from the simplest case n = 1, D = 2. Then modulo positive factor (3.4.23)

2  g (x, ξ) = − ξ1 − η(x, ξ  ) + r (x, ξ  );

we have sign “−” since a0 has two eigenvalues of different signs. Without any loss of the generality one can assume that η = 0 i.e. (3.4.24)

g (x, ξ) = −ξ12 + r (x, ξ  ).

Anyway, points (x  ξ  ) ∈ T ∗ ∂X with r (0, x  , ξ  ) > 0, r (0, x  , ξ  ) = 0 and r (0, x  , ξ  ) < 0 are called hyperbolic, elliptic and glancing points respectively, and we are interested in the small vicinities of the glancing points. Microhyperbolicity with respect to ∇# x2 means exactly that (3.4.25)

ς∂ξ2 r (x, ξ  ) > 0

with ς = ±1 (depending on the structure of the system; see more precisely in Appendix 3.A.2.

CHAPTER 3. PROPAGATION NEAR BOUNDARY

260

Further, glancing points (i.e. those with g = x1 = {g , x1 } = 0) could be of three types: gliding with Hg2 x1 = −2∂x1 r < 0 (recall that Hg f = {g , f }, see footnote 13) from Chapter 2), grazing or diffractive with (3.4.26)

Hg2 x1 = 2∂x1 r > 0,

and degenerate with Hg2 x1 = −2∂x1 r = 0. We do not need any extra analysis in the gliding points but for diffractive points we need to prove that there is no propagation along boundary bicharacteristics. The results for degenerate points will follow. So, we consider now only diffractive points. To prove that singularities do not propagate along boundary bicharacteristics one needs to x1 consider incoming bicharacteristics to a boundary point and chose function φ(x, ξ) such that {φ < 0} would be a narrow tongue around it and in the same time ς{g , φ} > 0. We already noticed that this is impossible if φ(x, ξ) does not depend on ξ1 15) ∂X So we need to take φ(x, ξ) depending on ξ1 . Due to Weierstrass theFigure 3.4: Diffractive case: {φ ≤ 0} orem one needs to check φ as a linear function with respect to ξ1 and from the heuristic point of view φ should not decrease at reflections from the boundary. However ςξ1 > 0 at incoming and ςξ1 < 0 at outgoing bicharacteristics; so we need to have (3.4.27)

ς∂ξ1 φ < 0;

therefore adding φ1 (x, ξ  )ξ1 to φ(x, ξ) with negative ςφ0 helps to inequality ς{g , φ} > 0 by increasing ς{g , φ} iff {g , ξ1 } < 0 which is exactly (3.4.26). What actually we would like to prove in general is the following Conjecture 3.4.7. Let P be an operator with the Hermitean principal symbol p microhyperbolic in direction ∇# ξ2 . Assume that    p 0 (3.4.28) p= 0 p  15)

Or under weaker condition “φ(0, x  , ξ) does not depend on ξ1 ”.

3.4. ENERGY ESTIMATES APPROACH REVISITED

261

Let z  ∈ T ∗ ∂X and let zk ∈ ι−1 z  ∩ Char(p  ), k = 1, ... , J. Assume that (3.4.29)

K− (p, zk ) ∩ T ∗ X |∂X ∩ Ωk = {zk }

∀k = 1, ... J.

Further, let boundary operator B satisfy (3.1.9), (3.2.36) and (3.4.18). Furthermore, let u satisfy (3.1.20) and (3.4.30) (3.4.31) (3.4.32) (3.4.33)

/ WFs+1,l  (Pu) ∪ WFs (Bðh u), z ∈   −  K (p , zk ) \ zk ∩ WFs (u) ∩ Ωk = ∅ −





∗

∀k = 1, ... , J,

K (p , z ) ∩ T X |X \∂X ∩ WF (u) ∩ Ω = ∅,     −   K (p , z ) \ ι−1 z ∩ WFs,0  (u) ∪ WFs (u|∂X ) ∩ Ω = ∅ s

where Ω and Ωk are small enough neighborhoods of z and zk in T ∗ ∂X and T ∗ X respectively such that Ω = Ωk |∂X ∀k = 1, ... , J. Then (3.4.34)

    −   K (p , z ) \ ι−1 z ∩ WFs,0  (u) ∪ WFs (u|∂X ) ∩ Ω = ∅.

We will be able to prove this conjecture under certain restrictions. Symbol q: General Construction In this subsubsection we consider the case n = 1 and Jordan cell of an arbitrary order m. Let us consider φ(x, ξ) which is the smooth and realvalued function as usual. We are interested only in φ on Σ = {g (x, ξ) = 0}; then due to Weierstrass theorem we can assume without any loss of the generality that (3.4.35) φ(x, ξ) is a polynomial of degree (m − 1) with respect to ξ1 .   Let consider q(x, ξ) = χ φ(x, ξ) . Again due to Weierstrass theorem we rewrite it as polynomial of degree m − 1 with respect to ξ1 (modulo g (x, ξ)). Then slightly redefining q(x, ξ) we get (3.4.36)

q(x, ξ) =

0≤j≤m−1

  qj (x, ξ  )ξ1j = χ φ(x, ξ) + μ(x, ξ)g (x, ξ).

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262

Equality (3.4.36) reflects two “faces” of q(x, ξ): the second one will be used to calculate commutator {p, q}.   (3.4.37) − {p, χ(φ)}(x, ξ) = χ22 φ(x, ξ) {p, φ}(x, ξ) + μ (x, ξ)g (x, ξ) =   2 q (x, ξ) {p, φ}(x, ξ) + μ (x, ξ)g (x, ξ) where χ2 = (−χ )1/2 , χ1 = χ1/2 and here and below q  (x, ξ), q  (x, ξ) are polynomials of degree (m − 1) according to  with respect to  ξ1 constructed  Weierstrass theorem from χ1 φ(x, ξ) and χ2 φ(x, ξ) respectively. On the other, hand g = Gp = pG where G is a matrix of cofactors and therefore   (3.4.38) p (j) Gp(j) − p(j) Gp (j) {p, g } = {p, G }p + j

and (3.4.39)

{p, g } =

 1 {p, G }p + p{p, G } 2

with matrix Poisson brackets are defined by usual formula {p, G } =    the (j) (j) (in this multiplication order; as usual (j) and (j) denote j p G(j) − p(j) G derivatives with respect to ξj and xj respectively) and thus (3.4.40)

†  − {p, q} = q  (x, ξ  ) {p, φ}(x, ξ  )q  (x, ξ  )+ ν(x, ξ)p(x, ξ) + p(x, ξ)ν † (x, ξ)

with matrix symbols q  and μ . Energy Estimates. I Actually we will need to make some minor adjustments multiplying q by cut-off functions ζ 2 (x, ξ  ) equal 1 in vicinity z  ∈ T ∗ ∂X and slightly shifting argument of χ; so finally we take  (3.4.41) q = χ φ(x, ξ) − ε)ζ 2 (x, ξ  ) with arbitrarily small constant ε > 0. Then we arrive to the following inequality provided p is microhyperbolic in the direction ∇# φ: (3.4.42)

− h−1 Re i([P, Q]u, u)X ≥

0 Q  u2 − ChQ IV u2 − C Q IV Pu2 − C Q V u2 − ChQ  u2∂X

3.4. ENERGY ESTIMATES APPROACH REVISITED

263

where Q IV and Q V are operators with symbols q IV (x, ξ) and q V (x, ξ) which are produced by the Weierstrass theorem from χ(φ − 2ε) and χ(φ − 2ε)ζ1 respectively with ζ1 supported in ε-vicinity of supp(∇ζ). We can also replace here Q IV and Q V by matrix operators not containing hD1 . Let us assume that p is microhyperbolic in direction ∇# x2 and that g (x, ξ) is of principal type in ι−1 z  ∩ {g = 0} = z (which consists of just one point). Then (3.4.25) holds in vicinity of z with ς = ±1. Let us consider incoming bicharacteristics γ − (z); this bicharacteristics is a bicharacteristics of ςg issued from z into direction of t < 0. Assume that (3.4.29) holds, i.e. γ − (z) ∩ T ∗ X |∂X ∩ Ω = z

(3.4.43)

where Ω is vicinity of z. This assumption follows from concavity condition (3.4.26) but is definitely weaker; however this assumption would be incompatible with the convexity condition. Also assume that (3.4.30)–(3.4.33) and a weaker assumption of (3.4.34) hold (3.4.44)



   K− (p  , z  ) \ ι−1 z ∩ WFs−δ,0  (u) ∪ WFs−δ (u|∂X ) ∩ Ω = ∅

with small exponent δ > 0. Let φ be a real-valued function such that p is microhyperbolic in direction of ∇# φ and (3.4.45) {φ < 0} ∩ {g = 0} ∩ Ω contains γ − (z) ∩ Ω and is contained in the small enough vicinity of γ −1 (z) ∩ Ω Proposition 3.4.8. Let p be microhyperbolic in direction ∇# φ at point zk and q be defined by (3.4.41). Then under assumptions (3.4.30)–(3.4.33) and (3.4.44) all the terms but the first one in the right-hand expression of (3.4.42) do not exceed Ch2s . Proof. Proof is obvious. Actually we do not need conditions to Bðu or ðu here.

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CHAPTER 3. PROPAGATION NEAR BOUNDARY

Energy Estimates. II Now we need to consider boundary term (3.4.4) which is after division by h (3.4.46)

Re(A0 Qu, u)∂X = Re



A0 (Qj (hD1 )j u, u)∂X ≡

0≤j≤m−1



Re

j (Qj A0 (A−1 0 A1 ) u, u)∂X

0≤j≤m−1

modulo terms not exceeding Q VI D1j Pu∂X · QVI ∂X ≤ Ch2s 0≤j≤m−

where Q VI = q VI (x  , hD  ) with q VI supported in the small vicinity of z  and the second inequality is due to z  ∈ / WFs+1/2−j (ðD1j Pu) (which is due to (3.4.30)) and (3.4.44). Energy Estimates. III Let us consider the most important case m = 2 first. Without any loss of the generality one can assume that g is given by (3.4.24). Note that under condition (3.4.25) microhyperbolicity condition is equivalent to ς{g , φ} > 0.

(3.4.47)

Then there are only two terms in (3.4.46). Let us assume that in vicinity of z  ς j qj ≥ 0

(3.4.48)j

where qj are defined in (3.4.36) and also that (3.4.49)j

ς j aj w , w  ≤ −||w ||2

∀w ∈ Ker b.

Then Ga ˚rding inequality and (3.4.48)j , (3.4.49)j imply that (3.4.50)

− Re(Qj Aj u, u)∂X ≥  Re ς j (Qj u, u)∂X − μQj u2∂X − C μ−1 Q VI Bu2∂X − Ch2s

3.4. ENERGY ESTIMATES APPROACH REVISITED

265

with arbitrarily small constant μ > 0. We can skip the third term in the right-hand expression due to z  ∈ / s WF (Bðu). Further, since ς j qj ≥ 2εqj2 Ga ˚rding inequality and (3.4.48)j imply that ς j (Qj u, u)∂X ≥ 2εQj u2 − Ch2s and and we can skip the second term in the right-hand expression of (3.4.50) as well. Then the arguments of the proof of Theorem 3.1.13(ii) imply that (3.4.51)

Q  u ≤ Chs

and

ς j (Qj u, u)∂X ≤ Ch2s

which implies z  ∈ / WFs (ðu) since at least one of operators Qj is elliptic in z  . The latter statement is trivial because q0 (z  ) = χ(φ(z) − ε) and q1 = χ (φ(z))(∂ξ1 φ(z)) with z = ι−1 z  ∩ {g = 0}. Then Pu 0 = −ihδ(x1 )A0 u|∂X + (Pu)0 where u 0 is a continuation of u by 0 to {x1 < 0} and ellipticity of the “scalar version” of Q  (which is different from “matrix version” by multiple of P and has a symbol χ2 (φ(z) − ε)) implies that z  ∈ / WFs,0  (u). Let us consider (3.4.49)j . Obviously (3.4.49)0 is exactly (3.1.9)∗ . On the other hand, without any loss of the generality under (3.4.24) one can assume that at z      0 1 ς 0 , a1 = (3.4.52) a0 = 1 0 0 0 and condition (3.4.49)1 is nothing but (3.4.18) (assuming that (3.1.9) holds). Furthermore, we actually do not need (3.1.9)∗ : as (3.1.9) and (3.4.18) are fulfilled we can add to Q operator α1 Q0 hD1 which would lead to an extra boundary term as in the proof of Theorem 3.4.4. Let us prove (3.4.48)j . As R > 0 (3.4.53) (3.4.54) where (3.4.55)

 1 χ(φ+ − ε) + χ(φ+ − ε) , 2  1  χ(φ+ − ε) − χ(φ+ − ε) q1 = 1/2 2R

q0 =

φ± = φ0 ∓ ςφ1 R 1/2

CHAPTER 3. PROPAGATION NEAR BOUNDARY

266

and we assume that − ςφ1 (z  ) > 0.

(3.4.56)

Moreover, as R < 0 one can prove easily that for all n |q0 − χ(φ0 − ε)| ≤ Cn |R|n ,

|q1 − χ (φ0 − ε)φ1 | ≤ Cn |R|n

provided |χ(m) | ≤ Cmδ χ1−δ for any m and δ > 0 and −χ has the same property. Then redefining q0 := q0 + C |r |n , q1 := q1 + C ς|r |n for r < 0 we achieve (3.4.48)j without violating (3.4.36). Here functions q0 , q1 and μ in (3.4.36) are very smooth and we do not infinite smoothness (but actually we can construct infinitely smooth correction). Note that χ(t) = e 1/t as t < 0 and χ(t) = 0 as t > 0 has all the properties we need. Results: Single Glancing Point So, we have proven Theorem 3.4.9. Under assumptions16) D = 2 and (3.4.26) Conjecture 3.4.7 holds. Note that for systems with D = 2 satisfying (3.4.23) and (3.4.25) generalized bicharacteristic billiards are a.e. described by (3.4.57)

dz = ∇# g + κ∇# x1 dt

with κ = 0 as x1 > 0.

Then along boundary bicharacteristics (3.4.58)

 −1 κ = Hg2 x1 · Hx21 g

(this formula is invariant if we replace g by eg with non-vanishing e). Combing Theorems 3.4.9 and 3.4.6 we arrive to 16)

And all other assumptions listed in Conjecture 3.4.7.

3.4. ENERGY ESTIMATES APPROACH REVISITED

267

Theorem 3.4.10. Let D = 2 and (3.4.23), (3.4.25) be fulfilled. Then Theorem 3.4.4 remains true if in the definition (3.4.57) of the generalized bicharacteristic billiard we add an extra condition κ ≤ 0 thus replacing it by (3.4.57)∗

dz = ∇# g + κ∇# x1 dt with κ = 0 as x1 > 0 and κ ≤ 0 as x1 = 0.

Remark 3.4.11. Theorem 3.4.10 remain true with the obvious modification: p has a block diagonal form with the single block described there and all other blocks corresponding to transversal or elliptic components. Further Discussion Unfortunately Theorem 3.4.9 is almost everything we managed to prove by this energy estimate method. We were unable to cover multiple tangent points belonging to ι−1 z or higher multiplicity of ξ1 -root (with the exception of the special case). Neither it works for Neumann boundary conditions for wave operator and one needs to use direct parametrix construction (see Comments). The real power of Theorem 3.4.9 comes when it works together with Theorem 3.2.4. Let us consider several examples: Violation of Assumptions (3.2.36), (3.4.18). Assume that z  is a grazing point with D = 2 but (3.2.36) and (3.4.18) do not necessarily hold. Consider in the vicinity of z  manifold G = {z  , r (z  ) = 0} where we assume that g (z) is of the form (3.4.24) and (3.4.25), (3.4.26) are fulfilled. Further, assume that in the vicinity of z  (3.4.59)

  0 1 , a0 = 1 0

 a1 =

ς 0 0 ςr

 .

  Then boundary operator could be reduced to form 1 b2 with real-valued b2 which vanishes in z  (otherwise we can apply Theorem 3.4.10 directly). Then condition (3.2.15) is fulfilled at points of G with b2 = 0. Therefore Theorem 3.4.9 and thus Theorem 3.4.10 could be directly used there. On the other hand, Theorem 3.2.4 does allow propagation along boundary

268

CHAPTER 3. PROPAGATION NEAR BOUNDARY

bicharacteristic of g (i.e. bicharacteristic  propagation in the  of r ) and also elliptic zone along bicharacteristics of (−r )1/2 + b2 or those of (3.4.60)

ς(r + b22 ),

b2 < 0.

However if we assume that γςr− (z  ) which is (boundary) bicharacteristics of ςr issued from z  in the direction t < 0 leaves Λ = {b2 = 0} immediately, which is the case provided (3.4.61)

x1 = r = b2 = 0 =⇒ {r , b2 }(z  ) = 0

then there is no propagation along it and thus Theorem 3.4.9 could be extended to this case: Theorem 3.4.12. Let in vicinity of z  (3.4.59) and (3.4.24)–(3.4.26) be  fulfilled. Let boundary operator b = 1 b2 with real-valued b2 , b2 (z  ) = 0. Let   −  (3.4.62) γςr (z ) \ z  ∩ {b2 = 0} ∩ Ω = ∅, (3.4.63) (3.4.64) (3.4.65) Then (3.4.66)

z ∈ / WFs+1,l  (Pu) ∪ WFs+1/2 (Bðu),   − z = ι−1 z  ∩ {g = 0} γςg (z) \ z ∩ WFs (u) ∩ Ω = ∅,  −   γβ (z ) \ z  ∩ WFs−1/2 (ðu) ∩ Ω = ∅. z ∈ / WFs,l+1  (u) ∪ WFs−1/2 (ðu).

Further one can prove more global statement where (3.4.61) ensures (3.4.62) at every point of Λ. Several Grazing Ppoints. Let us assume that ι−1 z  consists of n grazing points, n ≥ 2, each of them satisfying (3.4.23), (3.4.25), (3.4.26). Then we have ςk = ± and gk , rk , Gk with k = 1, ... , n. Let    (3.4.67) Λ(m) = Gk . . {J⊂{1,...,n},#J=m} k∈J

Let us assume that (3.4.63), (3.4.64) are replaced by (3.4.63)∗

/ WFs+1,l  (Pu) ∪ WFs (Bðu), z ∈

3.4. ENERGY ESTIMATES APPROACH REVISITED and (3.4.64)∗

  − γςk gk (zk ) \ zk ∩ WFs (u) ∩ Ωk = ∅,

269

zk = ι−1 z  ∩ {gk = 0} ∀k = 1, ... , n.

respectively. Assuming (3.2.36), (3.4.18) we conclude from Theorem 3.4.9 that   (m)   (3.4.68)m Λ \ Λ(m+1) ∩ WFs ∪ WFs (ðu) ∩ Ω = ∅. holds with m = 1. Let us apply Theorem 3.4.4. Let us consider a generalized bicharacteristics of p issued from z  in the direction t < 0. It consists of a boundary bicharacteristics defined by (3.4.69)

dz  ∈ Kb (z  ) dt

where  (3.4.70) Kb (z  ) = convex hall ∇# ςj rj (z  ), j ∈ J as z  ∈ Λ(m) \ Λ(m+1) and rj (z  ) = 0 ⇐⇒ j ∈ J.   Then z  ∈ / WFs ∪ WFs (ðu) for every z  ∈ Ω such that z  has a vicinity Ω such that Kb− (z  ) ∩ Ω ∩ Λ(m) \ Λ(m+1) = ∅ with m = 1. But then (3.4.68)1 implies (3.4.68)2 provided rj = rk = 0 =⇒ {rj , rk } =  0 etc until (3.4.68)n under condition (2.2.41) for gk replaced by ςk rk . Meet a Gliding Component. Let situation described in the previous paragraph hold for all points zk ∈ ι−1 z  with the exception of z1 such that D1 = 2 but it is gliding point i.e. Hg1 x1 < 0 at z1 . Let g1 be of the form (3.2.20) and let conditions (3.4.63)∗ and (3.4.64)∗ with k = 2, ... , n be fulfilled. Let us assume that  −     (3.4.71) γς1 r1 (z ) \ z  ∩ WFs,l  (u) ∪ WFs (ðu) ∩ Ω = ∅. Then the previous arguments yield that   (3.4.72) {r1 = 0} ∩ WFs,l  (u) ∪ WFs (ðu) ∩ Ω = ∅

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CHAPTER 3. PROPAGATION NEAR BOUNDARY

where Ω is small enough vicinity of z  . Applying Theorem 3.2.4 again we conclude that z  ∈ / WFs,l  (u) ∪ WFs (ðu) under condition (2.2.41) for gk replaced by ςk rk (k = 1, ... , n). On the other hand, two gliding components create a real mess in billiards due to jumps in different directions and after jump to some direction the length of the jump to other direction can change drastically.

3.5

Propagation of Singularities along Long Transversally Reflecting Bicharacteristics

In this section we obtain results similar to those of Section 2.4. However, the presence of the boundary makes things much more complicated and we need to assume that in the zone under analysis not only a(x, ξ) has eigenvalues of the constant multiplicities but also that bicharacteristic billiards are transversal to the boundary. Further, we need to assume not only that symbol A(x, ξ) is regular but it is also true for the function defining the boundary and coefficients of the boundary operator. More precisely, we assume that operator “incoming components → outgoing components” is a pseudodifferential operator with a regular symbol.

3.5.1

Evolution of Operators

So, as in Subsection 2.4.1 we are interested in operator (2.4.4) Q t = e −ih

(3.5.1)

−1 tA

Q 0 e ih

−1 tA

where we assume that (3.5.2) A = AB i.e. operator with domain (3.5.3)

D(AB ) = {u : u ∈ L2 (X , H), Au ∈ L2 (X , H), Bðm−1 u = 0}

is a self-adjoint in L2 (X , H) operator.

3.5. LONG BICHARACTERISTICS

271

Standard Theory Here we need to assume that Q = Q(x, hD) is a pseudodifferential operator with the symbol q(x, ξ) and to avoid boundary operators in domain with the boundary we assume that (3.5.4)

dist(πx supp(q), ∂X ) ≥ .

To describe a classical evolution of A we need to consider generalized bicharacteristic billiards; we need to assume that the free space evolution is according to Section 2.4; so we repeat assumptions (2.4.58) and (2.4.59): (3.5.5)

Spec a(z) = {λ1 (z), ... , λm (z)}

∀z ∈ Ω

and for every j = 1, ... , m (3.5.6)

Spec a(z) ∩ [λj (z) − , λj (z) + ] = λj (z)

∀z ∈ Ω

+ where λj are assumed to be uniformly continuous and Ω = K[0,T ] (p, Ω0 ) with Ω0 a -vicinity of supp(q). Here we parametrize bicharacteristics billiards by t and subscript means the range of t. Moreover we need to assume that the reflection is transversal:

(3.5.7)

|{λj , φ}(z)| ≥ ρ

∀z = (x, ξ) : φ(x) ≤ 

where (3.5.8) X = {x, φ(x) > 0},

∂X = {x, φ(x) = 0}, φ(x) ≤  =⇒ |∇φ| ≥ ρ.

If there is no branching of billiards i.e. if  (3.5.9) # k : ∃zk ∈ ι−1 z  , λk (zk ) = 0, {λk , φ}(zk ) > 0 = 1 + ∗ ∀z  ∈ ιK[0,T ] (p, Ω0 ) ∩ T X |∂X

then (3.5.10)

Kt+ (p, z) = Ψt (z) for z ∈ Ω0

where Ψt is a Hamiltonian dynamics with the reflections at ∂X . Otherwise this formula holds but now Ψt is a Hamiltonian dynamics with the reflections and branching at ∂X : (3.5.11)

Ψt (z) = {Ψt,j (z), j = 1, ... , N(t)}

272

CHAPTER 3. PROPAGATION NEAR BOUNDARY

where Ψt,j are Hamiltonian maps. We expect (correctly) that Q t is pseudodifferential operators with the symbol supported in Kt+ but again we need to assume that (3.5.12)

dist(πx Kt+ ¯ (p, supp(q)), ∂X ) ≥ 

for this particular t = t¯; however we cannot assume that this is true on the whole interval [0, t¯]. Therefore the evolution would consist of alternating free space evolution along Hamiltonian trajectories where we apply results of Section 2.4 and reflections. These reflections happen when φ ◦ Ψt (z) ≤ . If billiards branch then different branches have different reflection times and we consider different branches separately replacing operator Q t by the sum of operators supported in different components of Ψt (Ω0 ). Reflections Let us consider vicinity of the boundary: {x, φ(x) ≤ }. Without any loss of the generality one can introduce coordinate system with x1 = φ(x). Let us consider point (z  , 0, τ ) ∈ T ∗ (∂X ×Rt ) and zj = ι−1 z  ∩{λj = −τ }. Let us consider a single incoming component with index j ignoring all other.   So we consider trajectory of τ + λj (z) with {λj , φ} ≤ −ρ. Let us consider  all outgoing trajectories (those of τ + λk (z) with {λk , φ} ≥ ρ where k = 1, ... , m). Let us consider point (z − , −T  , τ ) belonging to the incoming trajectory such that x1   at z − and let us consider operator Q − = Q − (x, hDx ) with the symbol q − supported in -vicinity of z − . Our goal is to consider evolution of this operator during time T   /ρ; this will be a short time evolution in the case of small parameters ρ and   ρ. There are three steps: (i) We construct Gj+ which transforms Cauchy data for uj as t = −T  to solution uj of Pj uj ≡ 0 as t > 0 and to prove that (3.5.13) ðGj+ Q − = Q 0 ðGj+ where Q 0 = Q 0 (x  , t, hDx , hDt ) is an operator with symbol q 0 and  (3.5.14) supp(q 0 ) = (z  , t, τ ) : z  ∈ ιΨt+T  ,j (supp(q − ) ∩ {λj = −τ }) ;

3.5. LONG BICHARACTERISTICS

273

(ii) Then we construct operators Qk0 such that Bðu ≡ 0 implies Q 0 uj ≡  0 k Qk uk where uk are outgoing components of u as k ≥ 1 and u0 is an elliptic component and  (3.5.15) supp(qk0 ) = supp(q 0 ).

supp(q 0 ) (a) Simple reflection

∂X

supp(q2+ ) supp(q1+ )

x1

supp(q − )

supp(q + )

supp(q − )

k

x1

∂X

supp(q 0 ) (b) Branching reflection

Figure 3.5: Reflections as ðGj+ Q − = Q 0 ðGj+ and Gk+ Qk0 =

 k

Qk+ Gk+

(iii) Finally, we construct parametrices Gk+ corresponding to outgoing components and solving Cauchy problems Pk uk ≡ 0 with Cauchy data on ∂X (with x1 considered as “time”) and prove that as t ≥ T  (3.5.16) γT  Gk+ Qk0 = Qk+ γT  Gk+ where γt is operation of fixing t, and Qk+ = Qk+ (x, hDx ) is an operator with the symbol qk+ and (3.5.17)

supp(qk+ ) ∩ {t = T  } = {z : ∃t  ∈ [0, T  ], τ = −λk (z), z  ∈ supp(qk0 ) : z  ∈ ιΨ−T  +t  ,k (z)}.

To implement Steps (i) and (iii) we can consider Cauchy problems with respect to x1 and note that one can always rewrite operator Qj± (x, hDx ) as Qj± (x, ±T  , hDx ) where Qj± (x, t, hDx ) commutes with Pj and one can always rewrite Qj± (x, t, hDx ) as Qj0 (x, t, hDx , hDt ) + Rj (x, t, hDx , hDt )Pj (x, hDx , hDt ). Obviously

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CHAPTER 3. PROPAGATION NEAR BOUNDARY

(3.5.18) As ρ, , T  are (small) constants, both operators Gj+ and γt Gk+ (with t ∈ [T  , T  ]) are Fourier integral operators with the symplectomorphisms z ∈ T ∗ X → (z  , t, τ ) ∈ T ∗ (∂X ×Rt ) and (z  , t, τ ) ∈ T ∗ (∂X ×Rt ) → z ∈ T ∗ X described above. Note also that due to symmetry assumption (3.5.19) Operator B: “incoming components → outgoing components” is unitary modulo lower order terms. This statement is true even if there are elliptic components. Now combining free propagation and reflection we arrive to Theorem 3.5.1. Let conditions (3.5.2), (3.5.4)–(3.5.8) be fulfilled with constants ρ, , T . Let t ∈ (0, T ] and (3.5.12) be fulfilled as well. Also assume that + (3.5.20) K[0,T ] (p, z) consists of one non-branching billiard.

Then for any τ1 < τ2 with small enough |τ2 −τ1 | there exists pseudodifferential operator Q t such that   −1 −1 (3.5.21) Ft→h−1 τ χ¯T  (t − t¯) e −ih tA Q − Q t e −ih tA  ≤ Chs ∀τ ∈ [τ1 , τ2 ] with small enough constant T  and symbol of Q t is supported in  Kt+ (a + τ , supp(q)) : (3.5.22) τ ∈[τ1 ,τ2 ]

where τ1 < τ1 < τ2 < τ2 are arbitrary constants. Since we did not assume that A has a scalar principal symbol we cannot use the same operator Q t for all τ but since λj are separated according to (3.5.5)–(3.5.6) we can exclude hDt from Q t in (3.5.21). Remark 3.5.2. One can consider problems when operator is defined as x1 > 0 and as x1 < 0 with transition conditions at x1 = 0. We refer to it as reflectionrefraction. We show it on Figure 3.6 below. The numbers of reflected waves and refracted waves do not necessarily coincide.

supp(q )

∂X

(a) Simple reflection-refraction

supp(q2+ ) supp(q1+ )

x1

supp(q 0 )

∂X supp(q3+ ) supp(q4+ )

0

275

supp(q − )

supp(q1+ )

x1

supp(q2+ )

supp(q − )

3.5. LONG BICHARACTERISTICS

(b) Branching reflection-refraction

Figure 3.6: Reflections and refractions

3.5.2

Long Time Evolution

Main Theorem Now we want to take ρ,  to be small parameters and T be a large parameter. Actually there are plenty of parameters here: large parameters time T , the bounds to derivatives of coefficients of operators A, B (which is natural to assume to be h-differential operators) and the bounds to number of reflections along one billiard, the total number of billiards (due to branching) and the bounds to elements of DΨt,j . Further, there are also upper bounds to derivatives of φ defining ∂X and the lower bounds to |∇φ|, incidence and reflection angles. Furthermore, here are bounds to derivatives of the symbol of pseudodifferential operator B: “incoming components → outgoing components”. Finally, there is a scaling parameter ε for operator Q. To avoid an orgy of parameters we just assume (with no implications to applications) that ε ≥ h1/2−δ with an arbitrarily small exponent δ > 0 while  all other small parameters are not less than hδ and all large parameters are  not larger than h−δ with δ  = δ  (δ) > 0. So, we assume that all the coefficients of A and B satisfy the inequalities

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CHAPTER 3. PROPAGATION NEAR BOUNDARY

+ in K[0,T ] (p, supp(q))

(3.5.23)1

||D α Aβ || ≤ cρ−|α|−1

(3.5.23)2

||D α Bβ || ≤ cα ρ−|α|−1

∀α,

and (3.5.24)1 (3.5.24)2 (3.5.24)3

∀α, |D α φ| ≤ cα ρ−|α|−1 |φ(x)| ≤ 0 ρ =⇒ |∇x φ(x)| ≥ 0 , d |φ(x)| ≤ 0 ρ =⇒ | φ(x)| ≥ 0 ρ, dt

+ along Hamiltonian trajectories comprising K[0,T ] (p, supp(q)), symbol of B satisfies

(3.5.23)3

||Dxα ,ξ ,τ Bn || ≤ cα ρ−2n−|α|−1

∀α.

Finally we assume that (3.5.25)

|DΨt,j | ≤ cρ−1

+ again along trajectories comprising K[0,T ] (p, supp(q)). Our goal is to prove

Theorem 3.5.3. Let A, B be differential operators of orders m and not exceeding (m − 1) respectively. Let AB be a self-adjoint operator. Let Q ∈ Ψh,ε,ε (X , H, H) with ε ≥ h1/2−δ . Let us assume that T ≤ ρ−1 + and in K[0,T ] (p, supp(q)) conditions (3.5.5), (3.5.6), (3.5.23)1−3 , (3.5.24)1−3 ,  (3.5.25) be fulfilled with ρ = hδ , δ  = δ  (δ) > 0. Further, assume that (3.5.26)1 (3.5.26)2

dist(πx supp(q), ∂X ) ≥ 0 ε, dist(πx Kt+ ¯ (p, supp(q)), ∂X ) ≥ 0 ε.

Finally, assume that (3.5.20) is fulfilled. Then for any τ1 < τ2 with |τ2 − τ1 | ≤ ρ there exists operator Q t ∈ Ψh,ε ,ε (X , H, H) with ε = ρr ε such that (3.5.21) holds with T  = 0 ρr where r is large enough (but not depending  on δ). Symbol of Q t is supported in τ ∈[τ1 ,τ2 ] Kt+ (a + τ , supp(q)).

3.5. LONG BICHARACTERISTICS

277

Proof of the Main Theorem First of all we note that the case of billiard not hitting the boundary is covered by Theorem 2.4.10. So we need to consider the case of the billiard hitting boundary at least once and possibly branching there. Each branch of such billiard consists of hops between boundary points but the first and the last hops are not complete: the former does not start at the boundary and the latter does not end here. Note that the length of every hop could be rather small: if everything but the angle with the boundary is bounded by a constant rather than a parameter, and angle is still bounded from below by ρ then the length of the hop could be as small as  ρ (for the bicharacteristically strongly convex domain as in the Figure 3.1. So the number of hops in such case would be  T ρ−1 . In the general case the length of the hop is bounded from below by ρ3 and the number of hops is bounded from above by T ρ−1  ρ4 (here exact fixed power is of no importance). Let us consider first the Hamiltonian map Ψt along such billiard. Let 0 < t1 < ... < tN < t be moments when billiard hits the boundary and let us set tn = tn − ς, tn = tn + ς be moments “just before” and “just after” the hit. Then near z (3.5.27)

 ◦ · · · Ψt1 ,t1 ◦ Ψt0 ,0 Ψt = Ψt,tN ◦ ΨtN ,tN ◦ ΨtN ,tN−1

where we use Ψt,t  instead of Ψt−t  to emphasize the moments on the billiards and (3.5.28)

 · · · DΨt1 ,t1 · DΨt0 ,0 DΨt = DΨt,tN · DΨtN ,tN · DΨtN ,tN−1

and we claim that Proposition 3.5.4. (m + 1)-th order derivatives of Ψt,0 do not exceed C ρ−1−9m . Proof. We apply induction with respect to m; as m = 0 this just assumption (3.5.25). Let us consider m ≥ 1. Differentiating (m + 1)-times Ψt,0 = Ψt,s ◦ Ψs,0 we get the following terms: (i) Just one term when Ψt,s is differentiated (m + 1) times.

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CHAPTER 3. PROPAGATION NEAR BOUNDARY

(ii) No more than cm terms when Ψt,s is differentiated k times, 2 ≤ k ≤ m. Such terms do not exceed |D k Ψt,s | · |D r1 Ψs,0 | · |D r2 Ψs,0 | · · · |D rk Ψs,0 | with r1 + ... + rk = m + 1, ri ≥ 1. However as time interval between s and t contains no hits the first factor due to Lemma 2.4.5 does not exceed cρ2−3k and by induction assumption all (i + 1)-th factor does not exceed C ρ−8+9ri so the whole product does not exceed C ρ2−3k−(9r1 −8)−...−(9rk −8) = C ρ2+5k−9(m+1) = C ρ3−9m . (iii) One term when Ψt,s is differentiated just one time and Ψs,0 is differentiated (m + 1) times. Repeating this procedure again and again we realize that there will be no more 2n + 2 ≤ ρ−4 (ii)-type terms and their total contribution would not exceed C ρ−1−9m . Therefore only terms when one of the factors gets all (m + 1) derivatives and all other factors get one derivative each need to be considered. So we get n terms of the type (3.5.29)

|DΨt,s | · |D m+1 Ψs,t | · |DΨs,0 |

where we used DΨt,s = DΨt,τ · DΨτ ,s . Note that in (3.5.29) the first and the third factors do not exceed cρ−1 each and in the second factor either   t = tn , s = tn or t = tn+1 , s = tn and we set tN+1 = t, t0 = 0. So, we need to prove only that in this framework |D m+1 Ψs,t | does not exceed C ρ4−9m .  As t = tn+1 , s = tn the time interval between them contains no hits and due to Lemma 2.4.5 |D m+1 Ψs,t | does not exceed C ρ−1−3m . Consider t = tn , s = tn . We can take ς > 0 arbitrarily small here; − + thus tn = tn + 0, tn = tn − 0. Then Ψtn +0,tn −0 = Ψ+ k Ψj where Ψk :   (x  , ξ  , t, τ ) → (x, ξ) and Ψ− j : (x, ξ) → (x , ξ , t, τ ) correspond to outgoing and incoming bicharacteristic respectively. So we can study both of these symplectomorphisms separately. Scaling ρ times with respect to x and ξ and changing coordinates so X locally is defined as {x1 > 0} we arrive to the standard situation with the sole exception that −{λj , x1 }  ζj and {λk , x1 }  ζk with ζj , ζk ∈ [ρ, 1]. Then rescaling with respect to x1 , ξ1 only we arrive to the standard situation. One can prove easily that the inverse change of the coordinates provide the required estimate.

3.5. LONG BICHARACTERISTICS

279

Now consider point z¯ and operator Q with the symbol supported in ε-vicinity of z¯. By assumption τ1,2 = τ¯ ∓ ρ for some τ¯. Without any loss of the generality one can assume that |λj (¯ z ) + τ¯| ≤ 2ρ for some j 17) . Let us repeat arguments of Subsubsection Heisenberg Evolution for Matrix Principal Symbols.3 Heisenberg Evolution for Matrix Principal Symbols. In ρ3 -vicinity of z¯ one can reduce A to the form     Qjj Qjˆj Aj 0 , Q≡ (3.5.30) A≡ 0 Aˆj Qˆjj Qˆjˆj where the principal symbol of Aj is λj and eigenvalues of the principal symbol of Ajˆ differ from τ¯ by 3ρ at least. Then one can redefine Q replacing two lower blocks by 0 18) and then one can redefine A replacing Aˆj by whatever we want19) (and we want it to be equal to λwj ):     Aj 0 Qjj Qjˆj , . , Q := Q(j) = (3.5.31) A := A(j) = 0 λwj 0 0 So we replaced A, Q by (3.5.32)

A(j) = πj A + (I − πj )λwj ,

Q(j) = πj Q

where πj is the corresponding projector. While the global20) reduction of A, Q to form (3.5.30) and definition (3.5.31) is a murky business, pseudodifferential operator πj and thus A(j) , Q(j) are well defined globally20) . However since A(j) has a scalar principal symbol, operator Q t defined by (2.4.4) with A, Q replaced by A(j) and Q(j) respectively is a pseudodifferential operator as long as we do not hit the boundary. Therefore, Theorem 3.5.3 is proven as long there are no reflection points. Now consider the act of reflection. Since we can push the free propagation almost until the boundary we can rescale as we did in the proof of Proposition 3.5.4. After this rescaling we have bicharacteristic transversal to the boundary and therefore we arrive to  −1   −1  (3.5.33) ð∂X e −ih (t−t )A(j) Qjt ≡ Qj ð∂X e −ih (t−t )A(j) for t ∈ [tn − ρ2 , tn + ρ2 ] 17) 18) 19) 20)

Otherwise Ft→h−1 τ χ ¯T (t)UQ ≡ 0 for τ ∈ [τ1 , τ2 ]. Then Ft→h−1 τ U(t)Q changes negligibly as τ ∈ [τ1 , τ2 ]. −1 Then e −ih tA Q changes negligibly. In the zone Σj = {|λj + τ¯| ≤ 3ρ}.

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CHAPTER 3. PROPAGATION NEAR BOUNDARY

where Qj = Qj (x  , hD  , t, hDt ).    −1 As Dj = 1 its principal symbol is qjt ◦ Ψ− and all other symbols are j given by  t  −1  α Qjm (3.5.34) Qj ◦ Ψ− = m−m ,α Dx,ξ j |α|+2l≤2m

with (3.5.35)

β bl,α || ≤ cl,β ρ−6l−3|β| . ||Dx,ξ

However as Dj ≥ 2 j are linear   maps−1from  the space of Dj × Dj -matrices to Tj with unitary matrix Tj . itself; in particular qj = Tj† qjt ◦ Ψ− j In the same manner solving Cauchy problem with the initial data on ∂X we get the similar statement with the reverse time and j corresponding to the incoming bicharacteristics replaced by k corresponding to the outgoing one (which is an incoming bicharacteristics in the reverse time). Finally, as B mapping incoming component uj to outgoing one uk is a unitary operator (modulo negligible) we can rewrite BQj ≡ Qkj B and therefore as t  = tn − ρ2 , supp(χ) ⊂ [tn + ρ2 , tn + ρ2 ] (3.5.21) holds. Combining with free space propagation we arrive to Theorem 3.5.1 as long as there is just one reflection point. However combining with Proposition 3.5.4 we arrive to Theorem 3.5.1 in the general case.

3.5.3

Branching Billiards and Propagation

Theorem 3.5.5. Let A, B be differential operators of orders m and not exceeding (m − 1) respectively. Let AB be a self-adjoint operator. Let Q ∈ Ψh,ε,ε (X , H, H) with ε ≥ h1/2−δ . Let us assume that T ≤ ρ−1 + and in K[0,T ] (p, supp(q)) conditions (3.5.5), (3.5.6), (3.5.23)1−3 , (3.5.24)1−3  and (3.5.25) be fulfilled with ρ = hδ , δ  = δ  (δ) > 0. Further assume that conditions (3.5.26)1 , (3.5.26)2 are fulfilled and + −l non-branching billiards. (3.5.36) K[0,T ] (p, z) consists of no more than C ρ

Let Q ∈ Ψh,ε,ε with the symbol supported in V which is ε-vicinity of z and Q  ∈ Ψh,ε ,ε with the symbol equal 1 in Ψt¯(V  ) where V  is (1 + ρ3 )ε-vicinity of z. Then (3.5.37)

|Ft→h−1 τ χT  (t − t¯(1 − Q  )e −ih

−1 tA

Q| ≤ Chs

∀t ∈ [τ1 , τ2 ].

3.A. APPENDICES

281

Proof. Since each non-branching billiard has no more than C ρ−2 reflection points it is sufficient to prove theorem as + (3.5.38) Each of non-branching billiards comprising K[0,T ] (p, z) has no more than 1 reflection point each.

Indeed, in the general case we can select points tj such that condition (3.5.26)2 is fulfilled with tj instead of t¯ and on interval [tj , tj ] each billiard has no more than 1 reflection point. Then statement of the theorem in the general case follows from it under restriction (3.5.38). Then due to finite speed of propagation it is sufficient to prove this theorem as t1 < t¯ and t1 , t¯ − t1  ρ3 . But then the rescaling brings us to the standard case.

3.A 3.A.1

Appendices On Solutions to Linear Matrix Equations

Let H and H be Hilbert spaces and let us consider the equation (3.A.1)

AX − XB = C

with respect to X ∈ L(H, H ) with given A ∈ L(H, H), B ∈ L(H , H ), C ∈ L(H, H ). Let us assume that the (3.A.2) spectra of A and B are disjoint and let C ⊃ γ be a closed counter-clockwise oriented curve, γ = ∂G where G is a bounded domain in C, Spec(A) ⊂ G , Spec(B) ⊂ G¯. Then  1 (3.A.3) X = (z − A)−1 C (z − B)−1 dz 2πi γ is the (unique) solution of equation (3.A.1). Further, if C ⊃ γ  is a closed counter-clockwise oriented curve, γ  = ∂G  ,  G is a bounded domain in C, Spec(B) ⊂ G  , Spec(A) ⊂ G¯ then  1  (3.A.3) (z − A)−1 C (z − B)−1 dz. X =− 2πi γ  Moreover,

CHAPTER 3. PROPAGATION NEAR BOUNDARY

282

(3.A.4) formula (3.A.3) ((3.A.3) respectively) remains true also in the case when B (A respectively) is an unbounded operator. If A ≥ I , B ≤ −I with  > 0 then the (unique) solution of (3.A.1) is also given by the formula  +∞ (3.A.5) X = e −At Ce Bt dt 0

and if B ≥ I , A ≤ −I then (3.A.5)





0

e −At Ce Bt dt; .

X = −∞

Furthermore, (3.A.6) formula (3.A.5) ((3.A.5) respectively) remains true if −A and B (A and −B respectively) are unbounded operators which generate semigroups tending to 0 quickly enough as t tends to +∞. Let us note that if X is a solution of (3.A.1) then X † is a solution of (3.A.1)†

X † A† − B † X † = C † .

and hence if H = H and A† = B, C † = −C then the uniqueness of solutions implies that X † = X . Similarly, if H and H are complexifications of the real spaces then the operation of complex conjugation is defined and if A, B, C are real then X is also real; moreover, if H = H and AT = B, C T = −C then X T = X . Let us emphasize that the fundamental condition is (3.A.2).

3.A.2

Structure of Gliding Points

Assume that a and b are two Hermitean m × m-matrices, a is non-degenerate and a−1 b has the only eigenvalue 0 and rank a−1 b = m − 1. Let fk (k = 1, ... , m) be the corresponding (associated) eigenvectors: (3.A.7)

bfk = afk−1

k = 1, ... , m; f0 = 0.

Then afk , fj  = bfk+1 , fj  = fk+1 , bfj  = afk+1 , fj−1 

3.A. APPENDICES

283

which implies that (3.A.8)

afk , fj  = ρj+k

with ρl = 0 as l = 2, ... , m;

however  since a is non-degenerate ρ = ρm+1 = 0. Then replacing fj by λfj + k 0. On such small time interval we can construct u(x, y , t) by the successive approximation method picking up as unperturbed operator A(y , hDx , 0). This construction is much more robust and universal than construction as an oscillatory integral. Actually this construction does not use microhyperbolicity at all and 1 we need to assume only that T ≤ h 2 +δ with an arbitrarily small exponent

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CHAPTER 4. GENERAL THEORY IN THE INTERIOR

δ > 0. However without microhyperbolicity it is useful only for Nϕ,L and 1 Nx,ϕ,L and only as L ≥ h 2 −δ . Plugging u(x, y , t) into Tauberian approximations N∗T we will be able to prove that with a certain precision (4.1.6)

NTx ∼ NW x :=



κnx (τ )h−d+n ,

0≤n≤m

or (4.1.7)

NT ∼ N W x :=



κn (τ )h−d+n

0≤n≤m

with (4.1.8)

 κn (τ ) =

κnx (τ ) dx

and the corresponding equalities for NTx,ϕ,L and NTϕ,L . Here κnx and κn depend on Q1 , Q2 , ψ1 , ψ2 ; in particular  (4.1.9) κ0x (τ ) = (2π)−d q10 (x, ξ)ψ1 (x)e(x, ξ, τ )ψ2 (x)q20 (x, ξ) dξ where qj are principal symbols of Qj and (4.1.10)

  e(x, ξ, τ ) = θ τ − a(x, ξ)

is the spectral projector of the principal symbol a(x, ξ) of operator A; as usual θ is the Heaviside function. Expressions for next coefficients are much more complicated but they could be also calculated explicitly. These coefficients could be defined formally but without microhyperbolicity condition they could be rather irregular. We call NW ∗ Weyl approximation and Weyl approximation is our final W goal. We denote by RW ∗ the difference between N∗ and N∗ and call it Weyl error . T T So, on the second step we estimate (NT∗ − NW ∗ ). Since Rx and R usually are estimated by O(h1−d ) (but under assumptions of the global nature one can improve them marginally) it would be sufficient to pick up in NW x and NW usually m = 0 but sometimes m = 1. However estimates of RTx,ϕ,L and RTϕ,L could be (much) better than O(h1−d ) and we need (much) more terms.

4.1. INTRODUCTION

289

Remark 4.1.1. If we consider operators acting on sections then all the symbols and thus κnx are sections as well; then Γ and κn are not defined. However, Γ tr is defined and we should use instead  κn (τ ) = tr(κnx (τ )) dx in this case as well.

4.1.2

Kind of Assumptions to be Used

The only global condition we assume is that the operator A is self-adjoint. All other conditions will be local or even microlocal: we derive asymptotics for Nx,∗ or N∗ with x belonging and symbols of qj supported in the ball B(0, 12 ) under the assumption that the ball B(0, 1) is supp(Q) contained in the domain and all conditions are imposed in this ball. This is exactly true only if we want to derive asymptotics of Nx or Circle of Light N with an error estimate O(h1−d ); then we pick up T as a small conDark Territory stant. Figure 4.1: We impose condition in However if we want to derive the “Circle of Light”; the only global (marginally) better remainder esticondition is self-adjointness. mates we need to pick up large T and impose some conditions at the points which we can touch by some generalized Hamiltonian trajectory of the length T starting in B(0, 12 ). Finally, there are two microhyperbolicity conditions: microhyperbolicity and ξ-microhyperbolicity: Definition 4.1.2. We say that (i) Symbol a(x, ξ) is microhyperbolic at energy level τ in Ω if at each point z ∈ Ω symbol (τ − a(x, ξ)) is microhyperbolic at z in some direction  = (z) (|(z)| ≤ 1) in the sense of Definition 2.1.1; (ii) Symbol a(x, ξ) is ξ-microhyperbolic at energy level τ in Ω if in the framework of (i) one can select ((z) = (0, ξ (z)) at each point.

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CHAPTER 4. GENERAL THEORY IN THE INTERIOR

Normally microhyperbolicity is used to derive asymptotics with the spatial averaging (i.e. of N and Nϕ,L while ξ-microhyperbolicity is used to derive asymptotics without spatial averaging (i.e. of Nx and Nx,ϕ,L but in Section 4.6 we will always need ξ-microhyperbolicity.

4.1.3

Plan of the Chapter

In Section 4.2 we introduce the basic notions, obtain the first rough estimates and under microhyperbolicity or ξ-microhyperbolicity conditions estimate   Γ Ft→h−1 τ χ¯T (t)Q1x u(x, y , t) tQ2y (4.1.11) and (4.1.12)

  Γx Ft→h−1 τ χ¯T (t)Q1x u(x, y , t) tQ2y

by Ch1−d as T  1; we also derive certain estimates for larger T under non-periodicity or no-loop condition. Then using Tauberian technique we derive from (4.1.11) or (4.1.12) estimates respectively RT ≤ Ch1−d and RTx ≤ Ch1−d and certain estimates for RTϕ,L and Rx,ϕ,L . Then in Section 4.3 by means of the successive approximations we 1 construct u(t, x, y ) in a bounded domain in phase space for |t| ≤ h 2 +δ with an arbitrarily small exponent δ > 0 and then we construct (Γu)(t) and (Γx u)(t, x, x) and related functions. Next, in Section 4.4 we prove the main general theorems. Namely, the construction of Section 4.3 immediately yields the complete asymptotic 1 decomposition of Nx,ϕ,L as ϕ ∈ C0∞ (R) and L ≥ h 2 −δ . Further, under microhyperbolicity or ξ-microhyperbolicity condition respectively we prove that coefficients κn or κnx are smooth as τ ∈ [−, ]. Moreover, the same condition allows us in the Tauberian approximation (4.1.5) for N∗ and Nx,∗ to replace small constant T by any lesser T ≥ h1−δ and thus we can apply construction of Section 4.3 and pass from Tauberian to Weyl approximations. In Section 4.5 we using the long time propagation theorems increase T from small constant T0 either to arbitrarily large constant or even to larger T ≤ h−δ and derive even sharper remainder estimates; actually we apply these estimates not everywhere but with exception of sets of small measure (not exceeding ε). Combining with estimates there we arrive to remainder estimates C0 (T −1 + ε)h1−d + C  h1−d+δ

4.2. PRELIMINARY ANALYSIS AND TAUBERIAN METHOD

291

for errors RW or RW x and improved estimates with averaging. Further, in Section 4.6 we partially generalize the results of the previous sections to operators with “rough” with respect to x symbols. As an application we derive spectral asymptotics for number of negative eigenvalues for operators with non-smooth coefficients on compact closed manifolds; here we combine results for “rough” operators with variational ideas. This is a first taste of things to come much later. Finally, in Section 4.7 we consider operators with non-smooth ( Cl ) coefficients directly by the perturbation method. In Appendix 4.A.1 we consider the boundary values of almost-analytic functions (which are useful for the calculation of successive approximations via the Fourier transform) and in Appendix 4.A.2 we study approximations of non-smooth functions by “rough” functions. Remark 4.1.3. (i) During the whole chapter we consider pseudodifferential operators cut-offs Qj of different classes; we include ψj into them: Qj := Qj ψj and thus tQj := ψj tQj . (ii) Further, we assume by default that sup ||qj0 (x, ξ)|| ≤ 1

(4.1.13)

∀j

(x,ξ)

which simplifies some statements; obviously there is no loss of the generality.

4.2 4.2.1

Preliminary Analysis and Tauberian Method General Estimates

In this chapter we consider the differential operator (4.2.1)

A = Opw (A(x, ξ, h)) =

|α|+|β|≤m

(α + β)! (hD)α aα+β (x, h)(hD)β α!β!

with the full symbol (4.2.2)

A(x, ξ, h) =

|α|≤m

aα ξ α

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and h ∈ (0, 1], given on the manifold X some part of which is identified with the ball B(0, 1) where the coefficients aα (x, h) are linear (not necessarily bounded) symmetric operators in some auxiliary Hilbert space H. Alternatively, we consider pseudodifferential operators A ∈ Ψh (Rd , H, H) d ˜ (m) and A ∈ Ψ h (R , H, H) and more general operators; at this moment we need only to assume that (4.2.3) A with the given domain D(A) is a self-adjoint operator in L2 (X , H) (this condition will be formulated more precisely later). Then there are Hermitean spectral projectors and an unitary propagator. Let E (τ ) be the spectral projector of A corresponding to the interval (−∞, τ ); then E (τ1 , τ2 ) = E (τ1 ) − E (τ2 ) for τ1 < τ2 is the spectral projector corresponding to the interval [τ1 , τ2 ). Our goal is to derive asymptotics as h → +0 for various functions associated with e(x, y , τ1 , τ2 ), the Schwartz kernel of E (τ1 , τ2 ). Our main tool is the analysis of u(x, y , t), the Schwartz kernel of the −1 propagator U(t) := e ih tA . Obviously,  −1 (4.2.4) u(x, y , t) = e ith τ dτ e(x, y , τ1 , τ ) with an arbitrary τ1 ∈ [−∞, +∞]. Equality (4.2.1) is understood in the sense of distributions and in the general case we cannot set x = y here. However, Proposition 1.1.11 and Theorem 1.1.12 yield that we can set x = y if we first apply operators Q1x and tQ2y on the left and right respectively provided the symbols of Q1 and Q2 are compactly supported. Namely, since the operator norms of U(t) and E (τ1 , τ2 ) do not exceed 1 Theorem 1.1.12 yields the following Proposition 4.2.1. Let A be an operator self-adjoint in L2 (X , H) and let functions ψ1 , ψ2 ∈ L∞ be supported in B(0, 1) and |ψj | ≤ 1. Moreover, let q1 , q2 ∈ Sh,ρ,1,K (T ∗ Rd , H, H) be supported in {|ξ| ≤ ρ} and Qi = Opw (qi ) (i = 1, 2), with ρ ≥ h. Then (4.2.5) ||Q1x ψ1 (x)e(x, y , τ1 , τ2 )ψ2 (y ) tQ2y || ≤ C 

 ρ d h ∀x, y ∈ Rd

∀τ1 , τ2 ∈ R

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293

and    ρ d (4.2.6) ||Ft→h−1 τ χT ,T  (t) Q1x ψ1 (x)u(t, x, y )ψ2 (y ) tQ2y || ≤ C  T h ∀x, y ∈ Rd ∀T ≥ h, T  ∈ R where here and below C  = C |||q1 ||| · |||q2 |||, |||.||| means the Sh,ρ,1,K -norm; further  K = K (d), C = C (d) here, χ ∈ C0 ([−1, 1]) and χT ,T  (t) = χ( t−T ). T In the case of dim H = ∞ we need to obtain a more refined assertion. Let us assume that B1 is a (self-adjoint) operator in H such that B1† = B1 ,

(4.2.7)

B1 ≥ I

where ≥ and ≤ denote the standard order of symmetric operators; furthermore, assume tat the estimate (4.2.8)l

B1l uL2 (B(0,1),H) ≤ c(Al uL2 (X ,H) + uL2 (X ,H) )

∀u ∈ D(Al )

holds with some l > 0. Proposition 4.2.2. Let the assumptions of Proposition 4.2.1 and conditions (4.2.7) and (4.2.8)l be fulfilled. (i) Then (4.2.9)l ||Q1x ψ1 (x)B1j e(x, y , τ1 , τ2 )B1k ψ2 (y ) tQ2y || ≤  ρ d (|τ1 | + |τ2 | + 1)j+k C h ∀x, y ∈ Rd ∀τ1 , τ2 ∈ R ∀j, k ∈ [0, l] where here and below K and C also depend on l and C also depends on c in condition (4.2.8)l . (ii) Moreover, if χ ∈ C0k+j ([−1, 1]) and T ≥ h, T  ∈ R then (4.2.10)l ||Ft→h−1 τ χT ,T  (t)Q1x ψ1 (x)B1j u(t, x, y )B1k ψ2 (y ) tQ2y || ≤  ρ d (|τ | + 1)j+k C  T |||χ||| h ∀x, y ∈ Rd ∀τ ∈ R ∀j, k ∈ [0, l] where |||χ||| means the Ck+j -norm of χ. In particular, in the framework of Proposition 4.2.1 this estimate holds for j = k = 0.

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294

Proof. Let us note that the interpolation theorems of J. L. Lions, and E. Magenes [1] imply that (4.2.8) holds for l replaced by arbitrary j ∈ [0, l] and hence B1j (A ± i)−j  ≤ c,

(4.2.11)

(A ∓ i)−j B1j  ≤ c

where we derived the second inequality just taking an adjoint operator. Therefore B1j E (τ1 , τ2 )B1k  ≤ c 2 (A + i)j E (τ1 , τ2 )(A + i)k  ≤ C (|τ1 | + |τ2 | + 1)j+k ; the resulting inequality combined with Theorem 1.1.12 implies (4.2.9)l . Similarly Ft→h−1 τ χT ,T  (t)B1j U(t)B1k  ≤ C Ft→h−1 τ (A + i)j χT ,T  (t)U(t)(A + i)k  = C Ft→h−1 τ ((−hDt + i + τ )j+k χT ,T  )U(t) ≤ CT (|τ | + 1)j+k |||χ||| where the inequality T ≥ h was also used. Applying Theorem 1.1.12 again we obtain (4.2.10).

4.2.2

Semiclassical Spectral Gaps. I

Now we are interested in the spectral gaps (of the simplest nature) appearing because some interval in R is free of Spec a(x, ξ) for all (x, ξ) ∈ supp(q) 1) . To do so let us now investigate the equations (hDt − Ax )u = 0,

(4.2.12) (4.2.12)

†

u(hDt − tAy ) = 0.

from the elliptic point of view. Since it is relatively simple, we do it in the general case of L(H, H) valued symbols. First Standard Set of Assumptions Let us assume that |α|

(4.2.13)1 ||Dxβ aα (x, h)B −(1− m ) || ≤ c ∀x ∈ B(0, 1) ∀β : |β| ≤ K 1)

∀α : |α| ≤ m

More complicated smaller gaps of the different nature will be examined much later.

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295

and (4.2.14)1

||a0 (x, h)w || ≥ 0 ||Bw || − c||w ||

∀x ∈ B(0, 1) ∀w ∈ D(B)

where (4.2.15) B is an operator in H satisfying (4.2.7) 2) and the meaning of the subscript “1” will be clear later (see (4.2.13)l , (4.2.14)l below). Moreover, let us assume that (4.2.16) A is a self-adjoint operator in L2 (X , H) and D(A) ⊃ C0m (B(0, 1), D(B)) and (4.2.17) a0 (x, h) with D(a0 ) = D(B) is a self-adjoint operator in H. Then (4.2.18) aα (x, h) with D(aα ) = D(B) are symmetric operators in H for all x ∈ B(0, 1), h and α : |α| ≤ m and (4.2.19) A(x, ξ, h) with the domain D(B) is a self-adjoint operator in H for all x ∈ B(0, 1), ξ ∈ Rd , h and, moreover, the conditions |α|

|α|

(4.2.13)l ||Dxβ B −(1− m )(1−l) aα (x, h)B −(1− m )l || ≤ c ∀x ∈ B(0, 1) ∀β : |β| ≤ K

∀α : |α| ≤ m,

(4.2.14)l B l−1 a0 (x, h)w  ≥ 0 ||B l w || − c||B l−1 w || ∀x ∈ B(0, 1) ∀w ∈ D(B) are fulfilled for all l ∈ [0, 1]. 2) But not necessarily coinciding with B1 ; in what follows B1 ≤ B. In Remark 4.2.22 we explain why we need B1 different from B.

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296

In fact, (4.2.18) follows from (4.2.16), while (4.2.13)l and (4.2.14)l follow from (4.2.13)1 and (4.2.14)1 by means of transition to the adjoint operator, (44.2.18) and interpolation. Finally, (4.2.19) follows from (4.2.13)1 with K = 0, (4.2.14)1 and (4.2.17). Finally, let us assume that (4.2.20)1 Condition (4.2.8)1 is fulfilled with B1 = B. Conditions (4.2.13)1 , (4.2.14)1 , (4.2.15)–(4.2.17) and (4.2.20)1 form the first standard set of assumptions. Second Standard Set of Assumptions However, we can also deal with the second standard set of assumptions obtained by replacing in the previous set conditions (4.2.13)1 , (4.2.14)1 , (4.2.16) and (4.2.20)1 respectively by (4.2.13) 1 , 2

(4.2.14)±

1 2

± a0 (x, h)w , w  ≥ 0 ||B w ||2 − c||w ||2

∀w ∈ D(B),

(4.2.21) ±A is a self-adjoint semibounded from below operator in L2 (X , H) 1 1 and D(A 2 ) ⊃ C0m (B(0, 1), D(B 2 )). and (4.2.20) 1 Condition (4.2.8) 1 is fulfilled with B1 = B 2

2

and preserving (4.2.15). Obviously, then (4.2.17) and (4.2.18) hold and, moreover, (4.2.19) is replaced by (4.2.19)± ±A(x, ξ, h) is a self-adjoint semibounded from below operator 1 1 in H with D(A(x, ξ, h) 2 ) = D(B 2 ) for all x ∈ B(0, 1), ξ ∈ Rd , h; in particular, it holds for a0 (x, h). Estimates to Propagator (i) If the first standard set of assumptions is fulfilled then multiplying the operator (hDt − A) by B −1 on the left or on the right we see that

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297

the parametrices G1 and G0 of the operators P1 = B −1 (hDt − A) and P0 = (hDt − A)B −1 can be constructed on supp(q) × Rt × [τ1 , τ2 ] provided ||(τ − A(x, ξ, h))B −1 w || ≥ 0 νw 

(4.2.22) and

(4.2.23) ||A(β) (x, ξ, h)(τ − A(x, ξ, h))−1 || ≤ cρ−|α| γ −|β| (α)

∀(x, ξ) ∈ supp(q) ∀τ ∈ [τ1 , τ2 ] ∀w ∈ H ∀α, β : |α| + |β| ≤ K where (4.2.24)

γ ≤ 1,

ργ ≥ h1−δ ω δ ,

ω ≥ 1,

τ 2 − τ1 ≥ ν ≥

 h n ω

with an arbitrarily small exponent δ > 0 and arbitrary n (surely, K depends on n). (ii) Similarly, if the second standard set of assumptions is fulfilled then 1 multiplying the operator (hDt − A) by B − 2 on the left and right we see that 1 1 the parametrix of the operator P 1 = B − 2 (hDt − A)B − 2 can be constructed 2 on supp(q) × Rt × [τ1 , τ2 ] provided (4.2.22)±

1

± (A(x, ξ, h) − τ )w , w  ≥ 0 ν||B 2 w ||2

∀w ∈ D(B)

and (4.2.23) ||(A(x, ξ, h) − τ )− 2 A(β) (x, ξ, h)(A(x, ξ, h) − τ )− 2 || ≤ cρ−|α| γ −|β| 1

(α)

1

∀(x, ξ) ∈ supp(q) ∀τ ∈ [τ1 , τ2 ] ∀w ∈ H ∀α, β : |α| + |β| ≤ K where condition (4.2.24) is assumed to be fulfilled in this case also. Let q ∈ Sh,ρ,γ,K (Rd , H, H) and Q = Opw (q). Then in all three cases (l = 0, 1, 12 ) the operator norm of Q(Gl Pl − I ) does not exceed C  |||Q|||(hω −1 )s where here and in what follows s is arbitrarily large and K , C depend on d, m, δ, n, s and C  also depends on c. Until further notice l = 1 or l = 12 . Then Theorem 1.1.12 immediately implies

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Proposition 4.2.3. (i) Let conditions (4.2.13)1 , (4.2.14)1 , (4.2.15), (4.2.16), (4.2.17), (4.2.20)1 and (4.2.22)–(4.2.24) be fulfilled with either q = q1 or q = q2 and let us assume that (4.2.25)

|ξ| ≤ cω

∀(x, ξ) ∈ supp(q1 ) ∪ supp(q2 )

where q1 , q2 ∈ Sh,ρ,γ,K (Rd , H, H) and Q = Opw (q) and Qi = Opw (qi ). Let ψ1 , ψ2 ∈ C0K (B(0, 1 − )),  > 0 be an arbitrarily small constant, (4.2.26)

T ≥ h1−δ ω δ

and χ ∈ C0K ([−1, 1]). Then (4.2.27)l ||Ft→h−1 τ χT ,T  (t)Q1x B j ψ1 (x)u(t, x, y )B k ψ2 (y ) tQ2y || ≤  h s ω ∀j, k ∈ [0, l]

C  T (|τ | + 1)j+k ∀x, y ∈ Rd

∀τ ∈ [τ1 , τ2 ]

with l = 1 where here C  = C |||q1 ||| · |||q2 ||| · |||ψ1 ||| · |||ψ2 ||| · |||χ|||, all the norms are calculated in the spaces indicated above, and C also depends on . (ii) Let conditions (4.2.13) 1 , (4.2.14)± , (4.2.15), (4.2.21), (4.2.20) 1 and 2 2 (4.2.22)± , (4.2.23) , (4.2.24), (4.2.25) be fulfilled with either q = q1 or q = q2 as well as the conditions of (ii) on Q1 , Q2 , ψ1 , ψ2 , χ. Then estimate (4.2.27)l holds with l = 12 . (iii) In particular, conditions (4.2.22)–(4.2.23) are automatically fulfilled in ¯ with γ = h 12 −δ provided (4.2.22) is fulfilled at the γ-neighborhood of (¯ x , ξ) 1 this point with ν = h 2 −δ . On the other hand, one can use the sharp Ga ˚rding inequality (Theorem 1.1.29) and its sharpest form for scalar operators (Theorem 1.1.31) 1 1 combined with the fact that B − 2 [A, ψ]B − 2 is a skew-self-adjoint operator modulo O(h2 )) instead of the parametrix construction and easily obtain Proposition 4.2.4. Let conditions (4.2.13) 1 , (4.2.14)± , (4.2.15), (4.2.19)± 2 (4.2.21), be fulfilled for τ = τ¯ and let condition (4.2.22)± be fulfilled in the -neighborhood f supp(q2 ) where  > 0 is an arbitrarily small constant. Then

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299

(i) Estimate (4.2.27)l holds with l = 12 provided ν = Ch with a large enough constant C and τ ∈ [τ1 , τ2 ] with [τ1 , τ2 ] = [¯ τ , ∞) respectively. (ii) Further, if a(x, ξ) = λ(x, ξ) is a scalar-valued symbol this estimate remain true provided ν = Ch2−δ with a large enough constant C .   AI 0 where (iii) Furthermore, reducing A to the block-diagonal matrix 0 AII Spec AII (x, ξ) does not intersect [τ1 − , τ2 + ] we obtain the same assertion as in (ii) provided (4.2.28)

Spec a(x, ξ) ∩ [τ1 − , τ2 + ] = {λ(x, ξ)}.

¯ either Finally, if B = I and if in the -neighborhood of (¯ x , ξ) (4.2.29)+

Spec A(x, ξ, h) ∩ [¯ τ − , τ¯ + ν] = ∅

or (4.2.29)−

Spec A(x, ξ, h) ∩ [¯ τ − ν, τ¯ + ] = ∅ 

 A+ 0 then reducing the operator to the block structure with positively 0 A− defined ±(A± − τ¯) and applying the same arguments as above we obtain Proposition 4.2.5. Let condition (4.2.13)1 , (4.2.14)1 , (4.2.15), (4.2.16) and (4.2.17), (4.2.20)1 be fulfilled with B = I . Furthermore, let condition (4.2.29)± be fulfilled with ν = Ch in the general case and with ν = h2−δ in the case of condition (4.2.28). Then for τ residing in the indicated interval estimate (4.2.27)l holds with l = 12 . Estimates to Projector In order to pass from estimates of u(·, ·, ·) to estimates of e(·, ·, ·, ·) we apply a simplified Tauberian technique.3) . 3)

In what follows using these words we refer to the proof of Proposition 4.2.6 below.

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300

Proposition 4.2.6. (i) In the frameworks of Proposition 4.2.3(i) or (ii) the estimate (4.2.30) ||Q1x ψ1 (x)B j e(x, y , τ  , τ  )B k ψ2 (y ) tQ2y || ≤  h s  (|τ | + |τ  | + 1)j+k (|τ  − τ  | + 1) C ω ∀x, y ∈ Rd ∀τ  , τ  ∈ [τ1 , τ2 ] ∀j, k ∈ [0, l] holds with l = 1 and l = 12 respectively with the same K and C as before and with C  = C |||q1 ||| · |||q2 ||| · |||ψ1 ||| · |||ψ2 |||. (ii) Moreover, the estimate holds in the frameworks of Propositions 4.2.4 and 4.2.5. Proof. We apply Tauberian arguments. Namely, equality (4.2.4) implies the equality (4.2.31) Ft→h−1 τ χT (t)Q1x ψ1 (x)B j u(t, x, y )B k ψ2 (y ) tQ2y =  (τ − λ)T T χ( ˆ ) dλ Nx,y (λ) h with (4.2.32)

Nx,y (λ) = Q1x ψ1 (x)B j e(x, y , λ1 , λ)B k ψ2 (y ) tQ2y .

Let us pick j = k, Q2 = Q1∗ , ψ2 = ψ1† (where † means complex conjugation) and x = y ; then we get (4.2.33) Γx Ft→h−1 τ χT (t)Q1x ψ1 (x)B j u(t, x, y )B k ψ2 (y ) tQ2y =  (τ − λ)T T χ( ˆ ) dλ Nx (λ) h with the L(H, H)-valued function Nx (λ) = Γx Nx,y (λ). Note that in this special case Nx (λ) is monotone4) with respect to λ because in this case Nx (λ) is a restriction to the diagonal of the Schwartz kernel of the monotone operator-valued function RE (λ1 , λ)R ∗ with R = Q1 ψ1 B j and A ≤ B =⇒ KA (x, x) ≤ KB (x, x) where in the second case we 4)

Recall that Hermitian operators on Hilbert space are partially ordered.

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301

talk about natural partial order in L(H, H) and that that operator, trace and Hilbert-Schmidt are monotone with respect to this partial order on the space of self-adjoint operators. Let us pick a H¨ ormander function χ 5) . Lemma 4.2.7. (i) Let f be monotone increasing function, χ H¨ormander function and  (τ − λ) ) dλ f (λ) ≤ g (τ ) (4.2.34) χ( ˆ  Then 



∗





(4.2.35) |f (τ ) − f (τ )| ≤ g (τ , τ , τ ) := C0 −1



τ 

g (τ ) dτ + g (τ ) τ

∀τ  , τ  , τ : τ  −  ≤ τ ≤ τ  + , τ  ≤ τ  where C0 depends only on the choice of H¨ ormander function. (ii) Above assertion remains true for symmetric operator-valued functions as well. Proof. Note first that g (τ ) ≥ 0. Observe that the right-hand expression of (4.2.34) will not increase if we take the integral only on the interval [τ − , τ + ]  λ and replace after this χ( ˆ (τ −λ) ) by 0 = minτ ∈[−1,1] χ(τ ˆ ) > 0;  then we obtain in the right-hand expression 0 f (τ − , τ + ). Therefore (4.2.36)

|f (τ  ) − f (τ  )| ≤ C0 g (τ )

∀τ  , τ  : |τ  − τ | ≤ , |τ  − τ | ≤ .

Then (4.2.35) is proven as |τ  − τ  | ≤ 2 and moreover, as |τ  − τ  | = 2 we can skip the second term in the right-hand expression. This immediately yields (4.2.35) without the second term in the right-hand expression as |τ  − τ  | ≥ 2. Combining with (4.2.36) we arrive to (4.2.35) without this assumption. This Lemma 4.2.7 with  = hT −1 and f (τ ) = Nx (τ ) combined with estimate (4.2.27)l yield that Nx (τ −

 h s h h , τ + ) ≤ C (|τ | + 1)2j IH T T ω

∀τ ∈ [τ1 , τ2 ]

5) That means that χ ∈ C0∞ ([−1, 1]) is even and satisfies χ(0) = 1, χ(τ ˆ ) > 0 ∀τ ∈ R where the hat means Fourier transform. Such a function exists (see, e.g., Shubin [3]).

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where IH is the identity operator. This inequality generalizes immediately to (4.2.37)

Nx (τ  , τ  ) ≤ C 

 h s  |τ  − τ  |T  + 1 (|τ  | + |τ  | + 1)2j IH . ω h

where here and in what follows N∗ (τ  , τ  ) = N∗ (τ  ) − N∗ (τ  ). Taking an appropriate constant T we obtain (4.2.30) in the special case Q2 = Q1∗ , ψ2 = ψ1† and k = j provided the conditions of the proposition are fulfilled with q = q1 . But in this case the left-hand expression in (4.2.30) equals RR ∗ where for fixed x we define operator R from L2 (X , H) to H by   (4.2.38) Rv = Q1 ψ1 B j E (τ  , τ  )v (x) and hence R ∗ acts in the opposite direction. Therefore the already proven version of (4.2.30) implies that (4.2.39)

R = R ∗  ≤ C 

 h s  1 (|τ | + |τ  | + 1)j (|τ  − τ  | + 1) 2 . ω

On the other hand, Proposition 4.2.2 and the same arguments as before yield the estimate (4.2.40)

S = S ∗  ≤ C 

 h − d2  1 (|τ | + |τ  | + 1)k (|τ  − τ  | + 1) 2 ω

where S is defined in the same way as R was defined but with Q1 , ψ1 , x replaced by Q2 , ψ2 , y . Since the left-hand expression in (4.2.30) equals the norm of RS ∗ we obtain (4.2.30) in complete form; recall that exponent s is chosen arbitrarily. Let us consider explicitly the case in which the second standard set of assumptions holds. Then (4.2.14)± implies that if (4.2.41)±

± τ ≤ −C0 ξm

with C0 = C0 (d, m, c) 6) then conditions (4.2.22)± and (4.2.23) are fulfilled 1 with ν = 1, ρ = |τ | m , γ = 1 and moreover (4.2.42) 6)

||(τ − A(x, ξ, h))−1 || ≤ C |τ |−1 .

We use the notation C0 instead of C in the case when in its definition c can be calculated for K replaced by K0 = 1, 2, 3 where the definition of K0 in every case we leave to the reader.

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In fact, (4.2.13) 1 implies that 2

|(A(x, ξ, h) − a0 (x, h))w , w | ≤ C0 |ξ|m ||w ||2 + C0 |ξ| · ||B

m−1 2m

w ||2

and then (4.2.14)± and (4.2.41)± yield that ±(A(x, ξ, h) − τ )w , w  ≥ C0−1 (|τ | · ||w ||2 + ||B 2 w ||2 ) 1

and this inequality yields (4.2.22)± , (4.2.42) and (4.2.23) . Therefore if qi ∈ Sh,ρ,γ,K (T ∗ Rd , H, H), Qi = Opw (qi ) and (4.2.24), (4.2.25) hold and if ±τ ≤ −C0 ω m , then (4.2.30) holds for τ  = τ , τ  = 2τ ; hence the estimate (4.2.43) ||Q1x ψ1 (x)B j e(x, y , τ , 2τ )B k ψ2 (y ) tQ2y || ≤ C  hs |τ |−s ∀x, y ∈ Rd

∀j, k ∈ [0, l]

holds with Q1 = ζ(¯ ω −1 hD) where ω ¯ = C0−1 |τ | m , ζ ∈ C0K (B(0, 1)). It easily follows that this estimate holds for Q1 = Opw (q1 ) provided 1 q1 ∈ Sh,ρ,γ,K is supported in {|ξ| ≤ C0−1 |τ | m }, ±τ ≤ −C0 and condition (4.2.24) is fulfilled; here l = 12 . Moreover, if conditions (4.2.13)1 , (4.2.14)1 and (4.2.20)1 are fulfilled then one can take l = 1. Replacing τ by 2p τ and summing with respect to p = 0, 1, 2, ... we obtain 1

Proposition 4.2.8. Let conditions (4.2.13) 1 , (4.2.14)± , (4.2.15), (4.2.21) 2 and (4.2.20) 1 be fulfilled and let qi ∈ Sh,ρ,1,K (T ∗ Rd , H, H) satisfy (4.2.25) 2 and ρ and γ = 1 satisfy (4.2.24). Let ψ1,2 be as in Proposition 4.2.3. Then (4.2.44)± ||Q1x ψ1 (x)B j e(x, y , ∓∞, τ )B k ψ2 (y ) tQ2y || ≤ C  hs |τ |−s ∀x, y ∈ Rd

∀τ : ±τ ≤ −C0 ω m

∀j, k ∈ [0, l]

with l = 12 ; moreover, if conditions (4.2.13)1 , (4.2.14)1 and (4.2.20)1 are fulfilled then one can take l = 1. Estimates without Cut-offs The following two assertions show how one can set x = y without multiplication by cutting operators.

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Proposition 4.2.9. Let conditions (4.2.13)1 and (4.2.14)1 be fulfilled and let (4.2.45) |ξ|m ||w || + ||Bw || ≤ c(||A(x, ξ, h)w || + ||w ||) ∀x ∈ B(0, 1)

∀w ∈ D(B).

∀ξ ∈ Rd

(i) Then (4.2.20)1 holds. w ∗ d ˜0 Further let us assume that q1 , q2 ∈ S h,ρ,γ,K (T R , H, H), Qi = Op (qi ), (2.1.25) holds and ψ1 , ψ2 are as in Proposition 4.2.3. Then

(ii) If either q1 or q2 is supported in {|ξ| ≥ ω} with ω ≥ C0 then  h s ω ∀x, y ∈ Rd ∀τ  , τ  : |τ  | ≤ C0−1 ω, |τ  | ≤ C0−1 ω

(4.2.46) ||Dxα Dyβ Q1x ψ1 (x)B j e(x, y , τ  , τ  )B k ψ2 (y ) tQ2y || ≤ C 

∀j, k ∈ [0, l] ∀α, β : |α| + |β| ≤ r ˜ where K and C  now also depend on r , r arbitrarily large, and the S-norms  of q1,2 are now used in the definition of C , and l = 1 here. (iii) In the general case (4.2.47) ||Dxα Dyβ Q1x ψ1 (x)B j e(x, y , τ  , τ  )B k ψ2 (y ) tQ2y || ≤ C  h−d−|α|−|β| (|τ  | + |τ  | + 1) m (d+|α|+|β|)+j+k 1

∀x, y ∈ Rd

∀τ  , τ 

∀j, k ∈ [0, l]

∀α, β : |α| + |β| ≤ r .

Proof. (i) We will prove our claim that (4.2.20)1 is fulfilled later (see Subsection 4.2.3). (ii) Proposition 4.2.6 implies estimate (4.2.46) under the additional assumption that q1 and q2 are supported in {|ξ| ≤ 4ω} and either q1 or q2 is supported in {|ξ| ≥ ω} where as usual Qj = Op(qj ). Replacing ω by 2ω and summing over a partition of unity by means of pseudodifferential operators with symbols supported in {2p ω ≤ |ξ| ≤ 2p+2 ω}, p = 0, 1, 2, ... we obtain (ii); more precisely, we can take two partitions (one to the left of Q1 and another to the right of Q2 , and p means the largest index).

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(iii) Assertion (iii) follows from (i) and Proposition 4.2.3.

Proposition 4.2.10. Let conditions (4.2.13) 1 , (4.2.14)± , (4.2.15), (4.2.21) 2 be fulfilled and let (4.2.48)±

1

± A(x, ξ, h)w , w  ≥ 0 (|ξ|m ||w ||2 + ||B 2 w ||2 ) − c||w ||2 ∀x ∈ B(0, 1)

∀ξ ∈ Rd

∀w ∈ D(B).

(i) Then (4.2.20) 1 holds. 2

w ∗ d ˜0 Further, let us assume that q1 , q2 ∈ S h,ρ,γ,K (T R , H, H), Qi = Op (qi ), (4.2.24) holds and ψ1,2 are as in Proposition 4.2.3. Then

(ii) The estimate (4.2.49)± ||Dxα Dyβ Q1x ψ1 (x)B j e(x, y , ∓∞, τ )B k ψ2 (y ) tQ2y || ≤ C  hs |τ |−s ∀x, y ∈ Rd

∀ ± τ ≤ −C0

∀j, k ∈ [0, l]

holds with l = 12 ; moreover, if either the symbol of Q1 or the symbol of Q2 is supported in {|ξ| ≥ ω} then this estimate remains true with the additional factor ω −s in the right-hand expression. (iii) In the general case the following estimate holds: (4.2.50)± ||Dxα Dyβ Q1x ψ1 (x)B j e(x, y , ∓∞, τ )B k ψ2 (y ) tQ2y || ≤ C  h−d−|α|−|β| (|τ | + 1) m (d+|α|+|β|)+j+k 1

∀x, y ∈ Rd

∀j, k ∈ [0, l]

with l = 12 . (iv) Moreover, if conditions (4.2.13)1 , (4.2.14)1 are fulfilled then all these estimates hold with l = 1. Proof. The proof is similar to that of Proposition 4.2.9 but also uses Proposition 4.2.8.

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Semiclassical Spectral Gaps under Reduced Smoothness Assumptions The following assertions follow from Propositions 4.2.6–4.2.10 under smoothness conditions which are too restrictive; here we give an independent proof. Proposition 4.2.11. Let conditions (4.2.13)1 , (4.2.14)1 , (4.2.15), (4.2.16) and (4.2.17) be fulfilled with K = max(1 + d2 , m) + r . Then (i) If condition (4.2.45) is fulfilled then (4.2.51) ||Dxα Dyβ B j e(x, y , τ  , τ  )B k || ≤ C  h−d−|α|−|β| (|τ  | + |τ  | + 1) m (d+|α|+|β|)+j+k 1

∀x, y ∈ B(0, 1 − )

∀τ  , τ 

∀α, β : |α| ≤ r , |β| ≤ r

∀j, k ∈ [0, l]



with l = 1 where here and below C equals C (d, m, r , c, ) multiplied by norm of all auxiliary symbols and functions. (ii) If the following condition (4.2.52) ||(A(x, ξ, h) − τ )w || ≥ ((ω + |ξ|m )||w || + ||Bw ||) ∀x ∈ B(0, 1)

∀ξ ∈ Rd

∀τ ∈ [τ1 , τ2 ]

∀w ∈ D(B)

is fulfilled with ω ≥ 1 then  h s  (|τ | + |τ  | + 1)j+k ω ∀τ  , τ  ∈ [τ1 , τ2 ] ∀α, β : |α| ≤ r , |β| ≤ r ∀j, k ∈ [0, l]

(4.2.53) ||Dxα Dyβ B j e(x, y , τ  , τ  )B k || ≤ C  ∀x, y ∈ B(0, 1 − )

with l = 1 where s is arbitrary and C depending also on s. Proposition 4.2.12. Let conditions (4.2.13) 1 , (4.2.14)± , (4.2.15), (4.2.21) 2 and (4.2.48)± be fulfilled with K = max(1 + d2 , m) + r . Then (4.2.54) ||Dxα Dyβ B j e(x, y , ∓∞, τ )B k || ≤ *  s C h (|τ | + 1)−s C  h−d−|α|−|β| (|τ | + 1) ∀x, y ∈ B(0, 1 − )

∀τ ≤ −C0 1 (d+|α|+|β|)+j+k m

∀α, β : |α| ≤ r , |β| ≤ r

∀τ ∀j, k ∈ [0, l]

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with l = 12 . Moreover, if conditions (4.2.13)1 and (4.2.14)1 are fulfilled then this estimate holds with l = 1. Proof of propositions 4.2.11 and 4.2.12. Let us note that if (4.2.45) is fulfilled, v ∈ C0m (B(0, 1), D(B)) and Av = (hD)β wβ |β|≤m−l

with k = 0, ... , m then 1 (4.2.55) (hD)α B m (l−|α|) v  ≤ |α|≤k



1

C B m (−m+k+|β|) wβ  + C v .

|β|≤m−k

In fact, one can easily prove this estimate in the constant coefficient case by means of the Fourier transform. In the general case let ψn be a partition of unity in a neighborhood of B(0, 1) with small enough supports and let An be obtained from A by freezing coefficients at the point x n ∈ B(0, 1) ∩ supp(ψn ). Then An ψn v = ψn (hD)β wβ + (An − A)ψn v + [A, ψn ]v ; |β|≤m−k

representing the last two terms in the form (hD)β bβα (hD)α v |α|≤k,|β|≤m−k

with small coefficients bβα for |α| = k, |β| = m − k and applying estimate (4.2.55) to An we obtain that M1 ≤ νM1 + Cν M2 where M1 and M2 are the left-hand and right-hand expressions respectively in (4.2.55) and one can choose ν > 0 arbitrarily small (before choosing the partition of unity). Hence (4.2.55) holds in the general case. Similarly one can prove that for k = m + 1, ... , K the estimate 1 (hD)α B m (k−|α|)v  ≤ C C (hD)β Av  + C v  (4.2.56) l−m≤|α|≤k

|β|≤m−k

holds for v ∈ C0m (B(0, 1), D(B)); one should use in this deduction estimate (4.2.55).

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Repeatedly substituting ψv for v where ψ ∈ C0∞ (B(0, 1 − )) in (4.2.55) and (4.2.56) we obtain that both these estimates remain true for v ∈ Cp (B(0, 1), D(B)) with p = max(m, k) if in the left-hand and right-hand expressions we take the norms in the balls B(0, 1 − ) and B(0, 1 − 2 ) respectively; here C also depends on  > 0. Further, iterating if necessary we obtain that for |α| ≤ k, j ∈ [0, 1] and 1 |α| +j ∈Z m (hD)α B j v B(0,1−) ≤ C (A m |α|+j v  + v ). 1

Substituting E (τ  , τ  )v for v we conclude that (4.2.57) The operator norm of the operator (hD)α B j E (τ  , τ  ) : L2 (X , H) → 1 L2 (B(0, 1 − ), H) does not exceed C (|τ  | + |τ  | + 1) m |α|+j ; interpolation and estimate (4.2.56) again yield that this estimate remains true without restriction m1 |α| + j ∈ Z. Then the embedding inequalities imply that (4.2.58) The operator norm of the operator R : L2 (X , H)  v → ((hD)α B j E (τ  , τ  )v )(x) ∈ H with x ∈ B(0, 1 − ) does not exceed Ch− 2 −|α| (|τ  | + |τ  | + 1) 2m + m |α|+j . d

d

1

Obviously, Dxα (−Dyβ )B j e(x, y , τ  , τ  )B i = RS ∗ where S is defined in the same way as R but with β, i, y instead of α, j, x. Therefore we obtain (4.2.51). In a similar way one can prove two estimates below: (4.2.59) Dxα Dyβ Ft→h−1 τ χT (t)B j U(t)B i  ≤ C  Th−|α|−|β| (|τ | + 1) m (|α|+|β|)+j+i 1

∀τ

∀α, β : |α| ≤ k, |β| ≤ k

∀j, i ∈ [0, 1]

where T ≥ h−1 , χ ∈ C0∞ ([−1, 1]) is a fixed function,  ·  means the operator L(L2 (B(0, 1 − ), H), L2 (B(0, 1 − ), H))-norm and the operator is applied only to functions supported in B(0, 1 − ); and also (4.2.60) ||Dxα Dyβ Ft→h−1 τ χT (t)B j u(t, x, y )B i || ≤ C  Th−d−|α|−|β| (|τ | + 1) m (d+|α|+|β|)+j+i 1

∀x, y ∈ B(0, 1 − ) ∀τ

∀α, β : |α| ≤ k, |β| ≤ k

∀j, i ∈ [0, 1]

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where || · || means the L(H, H)-norm. In this deduction we use (4.2.56), (4.2.12), (4.2.12)† and (4.2.59) with α = β = 0, j = i = 0 which follows from the fact that U(t) is an unitary operator. On the other hand, if condition (4.2.52) is fulfilled then the arguments used to deduce (4.2.55) and (4.2.56) yield that these estimates remain true if in them as well as in their conditions A is replaced by A − τ with τ ∈ [τ1 , τ2 ] and the last term in their right-hand expressions (namely C v ) is replaced by  ( ωh )s v . These modified estimates, (4.2.12), (4.2.12)† and (4.2.59) imply that for T ≥ h1−δ ω −δ , τ ∈ [τ1 , τ2 ] the left-hand expressions in (4.2.59) and (4.2.60) do not exceed  h s  h s 1 1 (|τ | + 1) m (|α|+|β|)+j+k−1 and C  (|τ | + 1) m (d+|α|+|β|)+j+k−1 C ω ω respectively. Then the Tauberian arguments used in the proof of Proposition 4.2.6 yield estimate (4.2.53). So Proposition 4.2.11 is proven. Let us assume that (4.2.48)± is fulfilled. Then the arguments used to derive estimates (4.2.55) and (4.2.56) imply that (i) If Av =



(hD)β wβ ,

|β|≤m−k m 2

k = 0, 1, ... , (obviously m is even in this case) then 1 (hD)α B m (k−|α|) v  ≤ (4.2.61) |α|≤k



C B − 2 + m (− 2 +k+|β|)+ wβ  + C B 2 v  1

1

m

1

|β|≤m−k

for v ∈

C0m (B(0, 1), D(B 1/2 )).

(ii) For k ≥ (4.2.62)

m 2



(hD)α B m (k−|α|) v  ≤ 1

m ≤|α|≤k 2



C B m − 2 Av  + C B 2 v  i

1

1

|β|≤k− m −i 2 1

where v ∈ C0p (B(0, 1), D(B 2 )), i = 0, 1, ... , min( m2 , k − m2 ), p = max(m, k) is arbitrary.

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(iii) Moreover, these estimates remain true if we replace A by A − τ with ±τ ≤ −C0 , add 1 i |τ | (hD)α B − 2 + m v  |α|≤k− m − 2i 2 1

to the left-hand expression and in the right-hand expression replace C B 2 v  1 by Chs |τ |−s B 2 v . 1

(iv) Finally, all these estimates remain true for v ∈ Cp (B(0, 1), D(B 2 )) if in their left-hand and right-hand expressions we take norms on B(0, 1 − ) and B(0, 1 − 2 ) respectively. Then the arguments used in the proof of Proposition 4.2.11 yield Proposition 4.2.12.

4.2.3

Main Tauberian Estimate

Now we are going to discuss the main scheme used to derive spectral asymptotics. We will use the series of different abstract statements. Because of the fundamental importance of these statements we will discuss them in all details. Main Tauberian Formula Lemma 4.2.13. Assume that f satisfies (4.2.35). Let χ ∈ C0∞ ([−1, 1]) and χ(0) = 1. Then (4.2.63)

  | f (τ  ) − f (τ  ) − f T (τ  , τ  ) | ≤ C1 R(τ  ) + C1 R(τ  )

with the Tauberian expression (4.2.64)





f (τ , τ ) :=  T

−1



 (4.2.65)

R(τ ) := g (τ ) +

τ   τ

  (τ − λ)  χˆ dλ f (λ) dτ , 

g (τ + z)(1 + |z|2 )−m dz

and C0−1 C1 depending only on χ and arbitrarily large m.

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311

Proof. Let us rewrite the left-hand expression in (4.2.34) as   (τ − λ)  d χˆ f (λ) dλ. dτ  Integrating from τ  to τ  we get that f T (τ  , τ  ) = f T (τ  ) − f T (τ  )

(4.2.66) where T

f (τ ) := 

(4.2.67)

−1



 (τ − λ)  χˆ f (λ) dλ. 

Then χ(0) = 1 implies that −1

f (τ ) − f (τ ) =  T

 χˆ

 (τ − λ)   f (λ) − f (τ ) dλ; 

then inequalities |χ(η)| ˆ ≤ C (|η| + 1)−2m−1 and (4.2.35) imply that |f T (τ ) − f (τ )| ≤    −1 −1 −2m−1 −1 (|τ − λ| + 1)  | C0 

λ τ

 g (τ  ) dτ  | + g (τ ) dλ.

Opening parenthesis, changing order of integration in the double integral and calculating the simple integrals we arrive to  C1 −1 g (τ  ) dτ  (|τ  − λ|−1 + 1)−2m + C1 g (τ ); this expression is obviously equal to C1 R(τ ). This estimate combined with (4.2.66) implies (4.2.63)–(4.2.65). We will call (4.2.63)–(4.2.65) main Tauberian formula. (with an error estimate). Asymptotics with Mollification We consider now mollifications with kernel L−1 ϕL (τ ) where ϕ ∈ C0∞ ([−1, 1]) and (4.2.68)

ε := /L ≤ 1.

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312

Let us consider     −1 T (4.2.69) L ϕL (τ − τ¯) f (τ ) − f (τ ) dτ = L−1 ϕ˜L (τ − τ¯)f (τ ) dτ with ϕ˜ := ϕ ∗ ε−1 χˆε − ϕ. Assuming that (4.2.70)

χ(0) = 1,

χ(k) (0) = 0 ∀k = 1, 2, ...

we note that ε−n ϕ˜ is an uniformly C0∞ function for all n and then repeating arguments of the proof of Lemma 4.2.13 one can prove easily Lemma 4.2.14. Let f satisfy (4.2.35). Let (4.2.68), (4.2.70) be fulfilled. Let ϕ ∈ C0∞ ([−1, 1]). Then (4.2.71)

 |

    s τ) ϕL (τ − τ¯) f (τ ) − f T (τ ) dτ | ≤ C R(¯ L

where s is arbitrarily large and C0−1 C depends on χ, ϕ, s, and m. Moreover (4.2.72)

 |

  τ ). ϕL (τ − τ¯) f (τ , τ¯) − f T (τ , τ¯) dτ | ≤ C R(¯

Formula (4.2.71) is a mollified Tauberian formula, and (4.2.72) is an alternative Tauberian formula. The advantage of the latter is that it invokes only one energy level. Asymptotics with Averaging Let us consider averaging functions with singularities7) . Let us assume that (4.2.73) ϑ ∈ C(R+ ) is a positive monotone function on R+ , such that (4.2.74)

ϑ(2z) ≤ cϑ(z)

∀z ∈ (0, 1)

and ϕ ∈ C∞ (R˙ + ), compactly supported in [0, ∞] such that its derivatives ϕ(m) satisfy (4.2.75) 7)

|ϕ(m) (z)| ≤ cm ϑ(z)z −m

∀z > 0 ∀m ≥ 0.

The term “mollifying” is reserved for smooth functions.

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Lemma 4.2.15. Let f satisfy (4.2.35) and let (4.2.68) and (4.2.70) be fulfilled. Further, assume that ϕ satisfies (4.2.73)–(4.2.75). Then  |

(4.2.76)

   τ )ϑ ϕL (τ − τ¯) f (τ ) − f T (τ ) dτ | ≤ C R(¯ L

where C0−1 C depends on χ, ϕ, ϑ, and m. Moreover as ϕi , ϑi , Li satisfy (4.2.73)–(4.2.75), (4.2.68) then  (4.2.77) |

  ϕ1L1 (τ − τ¯1 )ϕ2L2 (τ  − τ¯2 ) f (τ , τ  ) − f T (τ , τ  ) dτ dτ  | ≤ C

i=1,2

R(¯ τi )ϑi

 Li

Proof. It is sufficient to prove only estimate (4.2.76) since (4.2.77) follows from it due to the same arguments as before. Without any loss of the generality one can assume that as L = 1; otherwise we can make a rescaling.  One can find ψ ∈ C0∞ ([ 13 , 1]) such that n≥1 ψ(2−n z) = 1 on (0, 1). Then ϕ(z) =



¯ ϑ(Ln )φn (zL−1 n ) + ϑ(h)φ(z/h)

n≥0,Ln ≥h

where φn ∈ C0∞ ([ 13 , 1]) uniformly and Ln = 2−n and φ¯ ∈ L∞ is supported in [0, 2]. Then according to estimates with mollification contribution of each m term ϑ(Ln )φn (zL−1 τ )(hL−1 n ) to (4.2.76) does not exceed C R(¯ n ) ϑ(Ln ) with arbitrarily large m. One can see easily that L−m/2 ϑ(L) ≤ ch−m/2 ϑ(h) for large enough m and therefore the total contributions of these terms does not exceed the same expression as Ln = h which is exactly the right-hand expression of (4.2.76). On the other hand, estimate without mollification implies that contribution ¯ −1 ) does not exceed the same result. of the last term ϑ(h)φ(zh Remark 4.2.16. (i) The most prominent application are asymptotics of Riesz means with ϕ(z) = ϑ(z) = z+s /Γ(s + 1).

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(ii) Obviously Lemma 4.2.15 extends to functions      with |dμ(z  )| ≤ c (4.2.78) ϕ(z) = φ(z − z ; z ) df (z ), where φ(z; z  ) satisfy (4.2.75) with respect to z uniformly with respect to z  and μ is a function of the bounded variation. One may try to find an alternative description of functions of this type. Operator Valued Functions Remark 4.2.17. Obviously all these results hold when f and g are functions with the values in the space of symmetric operators. In this case each inequality of the type |F | ≤ G is replaced by F  ≺ G which is a short notation for |(Fw , w )| ≤ (Gw , w ) ∀w ∈ D(G ) where G is self-adjoint operator. We say that G dominates F .

4.2.4

Applications

Scheme of tThings Let us consider some applications to the spectral asymptotics. From now again χ ∈ C0∞ ([−1, − 12 ] ∪ [ 12 , 1]) and χ¯ ∈ C0∞ ([−1, 1]), χ¯ = 1 on [− 23 , 23 ]. Let Qk = Opw (qk ) be a fixed h-pseudodifferential operators with symbols qk ∈ Sh,K (Rd , H, H), supp(qk )  Ω and let us assume that A is microhyperbolic on Ω on energy level 0. This is a crucial assumption. Then one can apply Theorem 2.1.19 estimates (2.1.58)

||Ft→h−1 τ χT  (t)Γ(Q1x u tQ2y )|| ≤ Ch1−d

 h s T

and (2.1.59)

||Ft→h−1 τ χT (t)Γx (Q1x u tQ2y )|| ≤ Ch1−d

 h s T

∀x ∈ Rd

hold for all τ : |τ | ≤ ε0 , where for the second estimate we need ξ-microhyperbolicity. Due to self-adjoint properties of A we have M = 1 in (2.1.43)–(2.1.45) and therefore in (2.1.58), (2.1.59). Recall that in these estimates T ∈ [T¯ , T0 ] is arbitrary where T¯ = h and T0 is a small constant.

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315

Then making t-admissible partition on [T¯ , T0 ] we derive that (4.2.79) and (4.2.80)

  ||Ft→h−1 τ χ¯T0 (t) − χ¯T¯ (t) Γ(Q1x u tQ2y )|| ≤ Ch1−d   ||Ft→h−1 τ χ¯T0 (t) − χ¯T¯ (t) Γx (Q1x u tQ2y )|| ≤ Ch1−d

∀x ∈ Rd

respectively. On the other hand, there are “dumb” estimates without any microhyperbolicity condition (4.2.81)

||Ft→h−1 τ χ¯T (t)Γ(Q1x u tQ2y )|| ≤ CTh−d

and (4.2.82)

||Ft→h−1 τ χ¯T Γx (Q1x u tQ2y )|| ≤ CTh−d

∀x ∈ Rd .

Here T ≥ h is arbitrary. Plugging T = T¯ = h we get (4.2.83)

||Ft→h−1 τ χ¯T (t)Γ(Q1x u tQ2y )|| ≤ CMT h1−d

and (4.2.84)

||Ft→h−1 τ χ¯T Γx (Q1x u tQ2y )|| ≤ CMT h1−d

∀x ∈ Rd

as T = T¯ . Then by virtue of (4.2.79), (4.2.80) these estimates hold with T = T0 where actually MT = 1, T = T0 but we do not need to make these assumptions if just assume that either (4.2.83) or (4.2.84) hold. Note that all these estimates (2.1.58), (2.1.59), (4.2.83), (4.2.84) remain valid if we substitute χ¯T (t) by χT0 (t)χ¯T (t). Therefore estimate (4.2.83) or (4.2.84) hold with χT0 (t) instead of χ¯T0 (t). In particular, it holds with H¨ormander function χT (t) where from now we use T instead of T0 .  −1 Let us recall that U(t) = e ih τ t dτ E (τ ) and thus we are dealing with (4.2.85)

f (τ ) = ΓQ1x e(., ., τ ) tQ2y w , w 

and (4.2.86)

fx (τ ) = Γx Q1x e(., ., τ ) tQ2y w , w 

respectively which are monotone functions as Q1 = Q2∗ ; here w ∈ H is arbitrary with ||w || = 1.

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316

Thus we conclude that (4.2.34) holds with * as |τ | ≤ 0 CMT h1−d (4.2.87) g (τ ) = −d 2l Ch T (|τ + 1|) as |τ | ≥ 0 Further, plugging Q2 = Q1∗ and j = k we have monotone function f (τ ) and due to virtue of Lemma 4.2.7 we conclude that in this case f (τ ) satisfies (4.2.35). Now it is time to drop this restriction. Since (4.2.88) Q1 E (τ , τ  )Q2∗ = Q1 E (τ , τ  ) · E (τ , τ  )Q1∗ = (E (τ , τ  )Q1∗ )∗ E (τ , τ  )Q1∗ we conclude that Proposition 4.2.18. Under microhyperbolicity condition (4.2.89)

R(τ , τ  )HS ≤ C (g ∗ (τ , τ  , τ )) 2 1

and under ξ-microhyperbolicity condition (4.2.90)

Rx (τ , τ  )HS ≤ C (g ∗ (τ , τ  , τ )) 2 1

with g ∗ defined by (4.2.35) and g defined by (4.2.87) where R(τ , τ  ) := Rw (τ , τ  ) : L2 (X , H)  f → Q1 E (τ , τ  )f , w  ∈ L2 (X , C) and Rx (τ , τ  ) := Rw ,x (τ , τ  ) : L2 (X , H)  f → Q1 E (τ , τ  )f , w (x) ∈ C. However the same is true for operator S defined as R but with Q2 , w  instead of Q1 , w ; further, in the second case one may substitute x by y . Then the trace norm (and thus the trace) of operator RS ∗ does not exceed the right-hand expression of (4.2.35). One can see easily that the trace of (RS ∗ )(τ  , τ  ) is exactly (4.2.91) and (4.2.92)

f (τ , τ  ) = ΓQ1x e(., ., τ , τ  ) tQ2y w  , w  fx,y (τ , τ  ) = Q1x e(., ., τ ) tQ2y w  , w 

where in the second case we have f depending not only on w , w  but on x, y as well. Let us recall that w , w  ∈ H are arbitrary with ||w || = ||w  || = 1. Thus we can junk w , w  and estimate L(H, H) norms of operators rather than their “bra-kets” with w , w  . Therefore we arrive to the following

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317

Proposition 4.2.19. (i) Under microhyperbolicity assumption f satisfies (4.2.35) with g defined by (4.2.87). (ii) Under ξ-microhyperbolicity assumption fx,y satisfies (4.2.35) with g defined by (4.2.87). So, What We Get? Now we can apply Lemmas 4.2.13, 4.2.13 and 4.2.15; plugging T = T0 and calculating g ∗ and R according to (4.2.35), (4.2.65) and (4.2.87) we get immediately Proposition 4.2.20. (i) Under microhyperbolicity assumption on energy levels τ¯ and τ¯ on supp(q1 ) ∪ supp(q2 ) (4.2.93) and (4.2.94)

||N(¯ τ , τ¯ ) − NT (¯ τ , τ¯ )|| ≤ Ch1−d h ||Nϕ,L (¯ ; τ ) − NTϕ,L (¯ τ )|| ≤ Ch1−d ϑ L

(ii) Under ξ-microhyperbolicity assumption on energy levels τ¯ and τ¯ on supp(q1 ) ∪ supp(q2 ) (4.2.95) and (4.2.96)

||Nx,y (¯ τ , τ¯ ) − NTx,y (¯ τ , τ¯ )|| ≤ Ch1−d h ||Nx,y ,ϕ,L (¯ . τ ) − NTx,y ,ϕ,L (¯ τ )|| ≤ Ch1−d ϑ L

Remark 4.2.21. (i) In Statement (i) we need microhyperbolicity only on supp(q1 ) ∩ supp(q2 ) since this expression change negligibly as we replace Q1 and Q2 by Q1 ≡ Q1 and Q2 ≡ Q2 in the vicinity of supp(q1 ) ∩ supp(q2 ). (ii) From Proposition 4.2.18 it follows that 1 d (4.2.97) Nx,y (τ  , τ  )w H ≤ Ch− 2 |τ  − τ  | + h 2 as both τ ; , τ  are close to τ¯ and w ∈ H, ||w || = 1 where H = L2 (R2d x,y , H) under microhyperbolicity assumption and H = L2 (Rdy , H) under ξ microhyperbolicity assumption. Then repeating arguments of the proofs of Lemmas 4.2.13–4.2.15 we arrive to   d−1  Nx,y (¯ (4.2.98) τ , τ¯ ) − NTx,y (¯ τ , τ¯ ) w H ≤ Ch− 2

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318 and

  d−1 τ ) − NTx,y ,ϕ,L (¯ τ ) w H ≤ Ch− 2 .  Nx,y ,ϕ,L (¯

(4.2.99)

(iii) We also get the same array of formulae with Q1x and tQ2y replaced by Q1x B1j and B1k tQ2y respectively (thus Q2 is replaced by Q2 B1k ). It looks that instead of difficult to calculate expressions N∗ we got equally difficult to calculate expressions NT∗ . However it is not completely true. First of all in some special cases we can construct u(x, y , t) as an oscillatory integral and calculate corresponding Tauberian expressions. However for NT , NTϕ,L or NTx , NTx,ϕ,L under microhyperbolicity or ξmicrohyperbolicity condition respectively we can apply Theorem 2.1.19 again and replace T = T0 by any T ∈ [h1−δ , T0 ]. So, we need to be able to calculate these expressions with really small T . Thus we need to be able to construct u(x, y , t) for |t| ≤ T . We will do it in sSection 4.3 by the successive approximation method. On the other hand, there is no Theorem 2.1.19 for expressions with x =  y ; it is just plain false. Still we will be able to get some less sharp results. We would be able to derive better estimates if we increased T without significant increase MT . Everything is left for Sections 4.4, 4.5 after calculations of Section 4.3.

4.2.5

Estimates in the Hilbert Scale

Hilbert sScale In conclusion we return to the scale of Hilbert spaces Hj = D(B1j ) with norms ||u||j = ||B1j u||, j ∈ [0, l] where B1 satisfies (4.2.8)l . Moreover, we introduce the spaces Hj = H†−j for j ∈ [−l, 0]. So far we used sometimes Hilbert scale associated with B and with l = l¯ where l¯ = 1 or l¯ = 12 depending on assumptions8) . Let us assume that (4.2.100) 8)

||(a0 (x, h) + i)−1 w ||j ≤ c||w ||j−1

∀w ∈ Hj−1 ,

We use l as notation for “new” l and adopt l¯ as a value of “old” l = 1, 12 .

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319

(4.2.101) ||(a0 (x, h) + i)−1 D β aα (x, h)w ||j ≤ c||w ||j− |α| m

∀w ∈ Hj− |α| ∩ D(B l ) ∀x ∈ B(0, 1) ∀β : |β| ≤ K m

∀j ∈ [0, l]

and let us also assume that (4.2.102)l

¯

¯

¯

||B1l w || ≤ c||B l w ||

∀w ∈ D(B l ).

Remark 4.2.22. Let us explain why we need to have different operators B and B1 . Consider for example applications to asymptotics of number of eigenvalues of the elliptic operator in domain with thick cusps9) . After rescaling cusp is replaced by a cylinder X × Y where X and Y are the base and cross-section of the cusp respectively and instead of the old spatial variable we have (x, y ). We replace L2 (X ×Y ) by L2 (X , H) with H = L2 (Y ); actually there will be weight but it is not important at the moment. However we are interested in the genuine trace Tr(E (τ )) = Γ tr(e(., ., τ )) where e(x, z, τ ) is a “matrix” operator (i. e. operator in L(H, H)) and tr is a “matrix trace” (i.e. trace in L(H, H)). In this example B = (h2 ΔY + I )m/2 with the Dirichlet boundary conditions where ΔY is positive Laplacian in Y , so associated Hilbert scale would be Hj = H0jm (Y ) (we consider Dirichlet boundary condition on the border of the cusp X × ∂Y 10) . There will be a problem since we need to increase l¯ = 1 (or l¯ = 12 ) to l large enough to have operator of imbedding I : Hl/2 → H−l/2 of the trace class but increasing l in the above scale would not be possible. However it will be possible in the scale Hj = H jm (Y ). So as B1 we will take operator generating this scale. Remark 4.2.23. Note that in our estimates B1j and B1k appear inside, between Q1x and e(., ., τ ) and between e(., ., τ ) and tQy 2 respectively and thus in order to estimate trace we need to move them outside. Thus we need to assume that B1j Q1 B1−j and B1−k Q2 B1k are (uniformly) bounded operators. However usually Q1 and Q2 are scalar h-pseudodifferential operators. Further, let us assume that K = K (d, m, s) is large enough. Then ¯

s s (B(0, 1), Hk ∩ D(B 1−l )), Au ∈ Hloc (B(0, 1), Hj ) imply (4.2.103) u ∈ Hloc s  that u ∈ Hloc (B(0, 1), Hj+1 ) with k = j, s = s − m, s ∈ R 9)

See in details in Section 12.2. Cusp is thick if the standard Weyl asymptotics either fails or does not provide good remainder estimate. 10) Neumann Laplacian there is a completely different animal; see Section 12.7.

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320

This deduction remains true for k = 0, s  = s − (j + 1)m. Therefore ¯

s s (B(0, 1), D(B 1−l )) and Al u ∈ Hloc (B(0, 1), H), l ∈ Z+ (4.2.104) u ∈ Hloc s  imply u ∈ Hloc (B(0, 1), Hl ) with an admissible s .

Then these statements imply the inequality (4.2.105)

ψB1l u(s) ≤ C  (Al u + u)

where ψ ∈ C0∞ (B(0, 1 − )); here  · (s) means the Sobolev H s (Rd , H)-norm, and s = s(m, l) < 0, C  = C |||ψ||| CK , C = C (d, m, l, c, ). On the other hand, let us assume that inequality (4.2.105) is fulfilled for all ψ ∈ C0K (B(0, 1 − 2 )) and that under certain conditions we have proven either estimate (4.2.61) or estimate (4.2.62) for some q1 and q2 and for all ψ1 , ψ2 ∈ C0K (B(0, 1 − 2 )) with q1 = I , q2 = I in neighborhoods of supp(q1 ), supp(q2 ) respectively. Then one can see easily that this estimate ((4.2.61) or (4.2.62)) will also be fulfilled for q1 , q2 , B1j , B1k with j, k ∈ [0, l] instead of q1 , q2 , B j , B k ¯ respectively; however, one should replace in the modified with j, k ∈ [0, l] estimate T , , s by T (1 − ), 2, s − K0 (m, l) respectively. Estimates in the Hilbert Scale In particular, we arrive to Proposition 4.2.24. Let conditions (4.2.100)–(4.2.102)l be fulfilled. Then one can enhance Propositions 4.2.6, 4.2.8–4.2.12, 4.2.18–4.2.20 to 4.2.6∗ , 4.2.8∗ –4.2.12∗ , 4.2.18∗ –4.2.20∗ where ∗ means the following modification: ¯ by B j , B1k with j, k ∈ [0, l]. One (4.2.106) we replace B j , B k with j, k ∈ [0, l] 1 can enhance (4.2.97)–(4.2.99) in the same way. Moreover, let us assume that (4.2.107) ||(A(x, ξ, h) + i)−1 w ||j + |ξ|m · ||(A(x, ξ, h) + i)−1 w ||j−1 ≤ c||w ||j−1 ∀w ∈ Hj−1 and (4.2.108) ||(A(x, ξ, h) + i)−1 D β aα (x)w ||j ≤ c||w ||j− |α| m

∀w ∈ Hj− |α| ∩ D(B ) ∀x ∈ B(0, 1) ∀ξ ∈ Rd . l

m

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321

Then one can easily prove by the standard elliptic methods that for B(0, 1−) estimate (4.2.8)l holds instead of (4.2.105) and then it is easy to prove by the same methods Proposition 4.2.25. Under conditions (4.2.100)–(4.2.102)l and (4.2.107)– (4.2.108) one can enhance propositions 4.2.9–4.2.11 to propositions 4.2.9∗ – 4.2.11∗ where ∗ means the same modification (4.2.106). We leave trivial details to the reader. Again all final statements will come in Section 4.4. Here we are more interested in the scheme of things.

4.3

Method of Successive Approximations for Propagator

Now our purpose is to construct propagator u(x, y , t) on a rather short 1 interval [−T , T ] with T ≥ h1−δ . We will do it for T = h 2 −δ .

4.3.1

The Method of Successive Approximations in the Standard Case

Pilot-Construction Let us temporarily assume that X = Rd and that A ∈ Ψh,K (Rd , H, H) (so we consider neither operator B nor Hilbert scale Hj ). We also assume that K is large enough (a non-smooth case will be treated separately in Section 4.6). Moreover, let us assume that A is a self-adjoint operator in L2 (Rd , H) −1 and let u(t, x, y ) be the Schwartz kernel of the operator U(t) = e ih tA . Then (4.2.12) (namely, (hDt − Ax )u = 0) and the initial condition u|t=0 = δ(x − y )I imply the equations (4.3.1)

(hDt − A)u ± = ∓ihδ(x − y )δ(t)I

where u ± = θ(±t)u, θ(.) is the Heaviside function, I = IH is the identity operator in H and here and in what follows all the operators without index y act on x and t. Let (4.3.2) A¯ = a(y , hDx ) so the operator A¯ is the principal part of operator A frozen at the point x = y ;

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

322 then

A = A¯ + R

(4.3.3) where R=

(4.3.4)



(x − y )α hj Aα,p ,

1≤|α|+p≤N

(4.3.5) Operators Aα,p = Aα,p (y , hDx ) for |α|+p < N and Aα,p = Aα,p (x, y , hDx , h) for |α| + p = N belong to Ψh,K −N (Rd , H, H). Then (4.3.1) yields the equality (4.3.6)

¯ ± = ∓ihδ(x − y )δ(t)I + Ru ± (hDt − A)u

and hence u ± = ∓ihG¯± δ(x − y )δ(t)I + G¯± Ru ±

(4.3.7)

where G¯± and G ± are parametrices of the problems (4.3.8)±

¯ =f, (hDt − A)v

supp(v ) ⊂ {±(t − t0 ) ≥ 0},

(4.3.8)±

(hDt − A)v = f ,

supp(v ) ⊂ {±(t − t0 ) ≥ 0}

respectively with supp(f ) ⊂ {±(t − t0 ) ≥ 0} for some t0 ∈ R. Moreover, equation (4.3.1) yields that (4.3.9)

u ± = ∓ihG ± δ(x − y )δ(t)I .

Iterating (4.3.7) N times and then substituting (4.3.9) we obtain the equality (4.3.10) u ± = ∓ih



(G¯± R)n G¯± δ(x − y )δ(t)∓

0≤n≤N−1

ih(G¯± R)n G ± δ(x − y )δ(t). Therefore the method of the successive approximations works provided  (4.3.11) An operator norm of G¯± R does not exceed hδ with an arbitrarily small exponent δ  > 0;

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323

then one needs to take N = N(s, δ  ) to get O(h−d+s ) error. Observe that on the time interval [0, T ] the operator norms of the parametrices G¯± do not exceed CTh−1 (due to the Duhamel principle; see below) and R = O(|x − y | + h) = O(T ) for T ≥ h due to the finite speed of propagation property and normally one cannot improve this.  Therefore the restriction to T is T 2 ≤ h1+δ or (4.3.12)

1

T ≤ h 2 +δ

with δ > 0.

This is the heuristic idea but the actual implementation will be slightly different. Transformation of the Successive Approximation Formula Namely, let us consider the factors (xj − yj ) entering into all the copies of R defined by (4.3.4) and let us transfer them from the left to the right (to δ(x − y )). If some factor reaches δ(x − y ) then the corresponding term vanishes and so only those terms survive in which all factors (xj − yj ) are killed. There are two possibilities for killing (xj − yj ): (i) Commuting it with a pseudodifferential operator; then there arises a pseudodifferential operator again and an additional factor h; (ii) Commuting it with either G¯± or G ± . Let us note that for P¯ = hDt − A¯ ¯ − [A, ¯ (xj − yj )]v ¯ j − yj )v = (xj − yj )Pv P(x and supp((xj − yj )v ) is contained in {±(t − t0 ) ≥ 0} provided supp(v ) is contained here; therefore ¯ − G¯± [A, ¯ (xj − yj )]v (xj − yj )v = G¯± (xj − yj )Pv and we finally obtain the equality (4.3.13) similarly (4.3.13)

¯ (xj − yj )]G¯± ; (xj − yj )G¯± = G¯± (xj − yj ) − G¯± [A, (xj − yj )G ± = G ± (xj − yj ) − G ± [A, (xj − yj )]G ± .

Therefore as a result of commuting of (xj − yj ) with G¯± there appear a pseudodifferential operator, a factor h and an additional factor G¯± ; the same is also true for G ± .

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324

Now our goal is to prove that (4.3.14) All the terms in (4.3.10) containing more than (N − 1) factors h and (xj − yj ) are negligible in some appropriate sense; in particular, one can then drop the last term in (4.3.10) and in all the remaining terms in R (defined by (4.3.4)) we can drop all terms with |α| + p = N; all the remaining terms we call regular . Recall that we consider the case X = Rd , A ∈ Ψh,N (X , H, H); later we will justify the same procedure in other situations. Then the Duhamel equality  ¯ − t  )v (t  , .) dt  U(t (4.3.15) (G¯± v )(t, .) = ih−1 ±t  ≤±t

¯ = e ih−1 t A¯ and a similar formula for G ± imply with the unitary operator U(t) that   (4.3.16) L L2 ([−T , T ] × X , H), L2 ([−T , T ] × X , H) -norms of G¯± and G ± do not exceed 2Th−1 . Moreover, the equality (4.3.17)

¯ G¯± (δ(t)v ) = ±ih−1 θ(±t)U(t)v

and a similar equality for G ± imply that the   (4.3.18) L L2 ([−T , T ] × X , H), L2 (X , H) -norms of the operators v → G¯± (δ(t)v ) and v → G ± (δ(t)v ) do not exceed T 1/2 h−1 . Hence we conclude that   (4.3.19) L L2 ([−T , T ] × X , H), L2 ([−T , T ] × X , H) -norm11) of every term 1 of decomposition (4.3.10) does not exceed C  hr −k T p+k+ 2 where r , p and k + 1 are the original numbers of factors h, (xj − yj ), G¯± (or G ± ) respectively. Obviously k ≤ p + r and therefore the norm in question does not exceed C  hσ(p+r ) with σ = σ(δ) > 0 provided T satisfies (4.3.12) with an arbitrarily small exponent δ > 0. Therefore (4.3.14) is proven. Applying Theorem 1.1.12 we arrive to 11) One can consider this term as the Schwartz kernel of an operator and calculate its operator norm.

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325

Proposition 4.3.1. Let A ∈ Ψh,K (Rd , H, H), q1 , q2 ∈ Sh,K (T ∗ Rd , H, H) be supported in {|ξ| ≤ c}, Qi = Opw (qi ), i = 1, 2, χ ∈ C0K ([−1, 1]) and 1 T ∈ [h1−δ , h 2 +δ ] with δ > 0. Then  (4.3.20) ||Ft→h−1 τ χT (t)Q1x u ± ±  ih (G¯± R  )n δ(x − y )δ(t) tQ2y || ≤ C  hs 0≤n≤N−1

∀x, y ∈ Rd where (4.3.4)



R =

∀τ ∈ R

(x − y )α hj Aα,p (y , hDx ),

1≤|α|+p≤N−1

δ > 0 and s are arbitrary, N = N(d, s, δ) and K = K (d, s, δ, N) are large enough, C = C (d, s, δ, N, c), C  = C |||q1 ||| · |||q2 ||| · |||χ||| and all the norms are calculated in the corresponding spaces. Minor Modifications Let us make minor modifications to the standard case. Namely, let us replace condition A ∈ Ψh,K (Rd , H, H) by one one of the following conditions: (4.3.21)1

ζA ∈ Ψh,ρ,γ,K (Rd , H, H) with ρ, γ ∈ [0, 1], ζ ≥ hs , ργ ≥ h1−δ ;

(4.3.21)2

d ˜m A∈Ψ h,K (R , H, H);

(4.3.21)3 A is a differential operator with coefficients satisfying (4.2.13) with B = I . Now we consider the domain Ω ⊂ T ∗ X (or Ω ⊂ T ∗ B(0, 1 − 0 ) in the third case) and we assume that (4.3.22) In Ω the symbol of A coincides with that of A ∈ Ψh,K (Rd , H, H). Moreover, let us assume that (4.3.23)

||∇x,ξ A(x, ξ, h)|| ≤ c0

∀(x, ξ) ∈ Ω

Then Corollary 2.1.7 yields that if q1 , q2 ∈ Sh,ρ,γ,K (Rd , H, H) with ρ, γ ∈ [0, 1], ργ ≥ h1−δ are supported in Ω(0 ) and Q1 = Opw (q1 ), Q2 = Opw (q2 ), then

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326

(4.3.24) For t ∈ [−T0 , T0 ] with T0 = T0 (c0 , 0 ) > 0 u(t, x, y ) tQ2y 12) is negligible outside of Ω 1 0 where Ωε = {(x, ξ) : dist((x, ξ), Ω) ≥ ε} and 0 is 2 arbitrarily small. Hence (4.3.1) yields that (4.3.1)

(hDt − A )u ± tQ 2y ≡ ∓ihδ(t)K(x, y )

in [−T0 , T0 ] × Rd × Rd

modulo negligible functions where K(x, y ) = KQ2 (x, y ) = δ(x − y ) tQ2y is the Schwartz kernel of Q2 ; hence one can replace (4.3.10) by (4.3.10) u ± tQ2y ≡ ∓ih



(G¯± R)n G¯± K(x, y )δ(t)

0≤n≤N−1

∓ ih(G¯± R)n G ± K(x, y )δ(t) ¯ R, G¯± and G ± are constructed for A instead of A 13) . where A, Let us apply the previous procedure of the transfer of the factors (xj − yj ) to the right; now these factors can be also killed in the commutator with t Q2y but it is still in consistent with the case (i) in the beginning of the previous subsubsection. So we obtain the following estimate  (4.3.20) ||Ft→h−1 τ χT (t)Q1x u ± tQ2y ±  (G¯± R  )n G¯± δ(t)(δ(x − y ) tQ2y || ≤ C  hs . ih−1 0≤n≤N−1

Remark 4.3.2. (i) This estimate differs from (4.3.20) only by position of Q2y ; let us remember that while operator A does not depend on y , operators A¯ and R do; t

(ii) Moreover, let us replace here 0 by 13 0 and q2 by q2 equal to 1 in Ω 2 0 ; 3 let us multiply the function inside the norm by tQ2y ; then Corollary 2.1.7 yields that the obtained function will not change modulo negligible terms if  we replace the operator tQ2y by I . So we have proven In the case (4.3.21)3 one should replace Qi by Qi ψi with ψi supported in B(0, 1 − 0 ); usually we do it without special mention. 13) Later we will see that in the final answer this replacement is not necessary. 12)

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327

Proposition 4.3.3. (i) The assertion of Proposition 4.3.1 remains true for operators A of the type (4.3.21)j –(4.3.22), j = 1, 2, with K , C  depending on m and C  also depending on c0 , 0 , c = |||R||| + |||A||| where all the norms are calculated in the corresponding spaces. (ii) Moreover, for operators A of the type (4.3.21)3 –(4.3.22) this assertion remains true if we replace Q1 , Q2 by Q1 ψ1 , Q2 ψ2 respectively with ψi ∈ C0K (B(0, 1 − 0 )); i.e., the estimate  (4.3.20) ||Ft→h−1 τ χT (t)Q1x ψ1 (x) u ± ±  (G¯± R  )n δ(x − y )δ(t) ψ2 (y ) tQ2y  ≤ C  hs ih 0≤n≤N−1

∀x, y ∈ Rd

∀τ ∈ R

holds, where now C  = C |||q1 ||| · |||q2 ||| · |||ψ1 ||| · |||ψ2 |||.

4.3.2

Successive Approximations in the Hilbert Scale

Let us consider the general case of a differential operator with the coefficients which are unbounded operators with the coefficients satisfying (4.2.13)l – (4.2.14)l . We will use the successive approximations method with the operator A replaced by the operator   (4.3.25) A = a0 (x, h) + ζ(D)ψ3 (x) A − a0 (x, h) ψ3 (x)ζ(D) where a0 (x, h) is a coefficient in A without derivatives extended to Rd preserving properties (4.2.13)l –(4.2.14)l (for this coefficient) and here and in what follows ψ3 , ψ4 , ψ5 ∈ C0K are real-valued functions supported in B(0, 1 − 79 0 ), B(0, 1 − 59 0 ), B(0, 1 − 39 0 ) and are equal to 1 in B(0, 1 − 89 0 ), B(0, 1 − 29 0 ), B(0, 1 − 49 0 ) respectively (recall that ψ1 and ψ2 are supported in B(0, 1 − 0 )), ζ ∈ C∞ (Rd ) equal 1 as |ξ| ≤ c. Then due to finite speed of propagation u t Q2 satisfies propagation equation (hDt − A )u t Q2 ≡ 0 modulo negligible f . Recall that a0 (x, hD) is “the most powerful” coefficient in A as operator in H and we do not cut it of because we want   (a0 ± iIH )−1 ∈ L L2 (X , Hj ), L2 (X , Hj−1 )

328

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uniformly. Such an extension for “potential” a0 (x, h) can be easily chosen provided in the case (i) we replace B(0, 1) by any ball from its -covering (if necessary). The principal difficulty now is that the coefficients of A and thus perturba¯ A¯ = A (y , 0, hD) are unbounded operators and parametrices tion R = A − A, G ± and G¯± do not compensate this unboundedness in an explicit way14) . To tackle this unboundedness we apply the sandwich procedure. Namely, let us note that (4.3.26)

G¯± = −(A¯ + i)−1 + G¯± (hDt + i)(A¯ + i)−1

and that all three operators G¯± , (A¯ + i)−1 and (hDt + i) commute. Let us transform in this way all the terms in the right-hand expression in (4.3.10) and let us move all the copies of the factor (hDt +i) to the left; in this way we can replace G¯± in the original version of the right-hand expression in (4.3.10) either by −(A¯ + i)−1 or (A¯ + i)−2 or (A¯ + i)−1 G¯± (A¯ + i)−1 (with the additional factor (hDt + i)k with k = 0, 1, 2 respectively).  One can easily see that for k ≥ 1 the L L2 (Rd , H), L2 (Rd , H) -norms of the operators B 1−j (A¯ + i)−k B j do not exceed C where here and what follows j ∈ [0, 1] is arbitrary in the case of the first standard set of assumptions and j = 12 in the case of the second standard set of assumptions. Therefore   (4.3.27) L L2 ([−T , T ]×Rd , H), L2 ([−T , T ]×Rd , H) -norm of the operator B 1−j (A¯ + i)−1 G¯± (A¯ + i)−1 B j does not exceed CTh−1 . Similarly we can deal with G ± or Gˆ± . Hence we conclude that (4.3.28) Let us consider a term (in the right-hand expression in (4.3.10) which was negligible in the context of Subsection 4.3.1. Then after multiplication from the left by Ft→h−1 τ χT (t)Q1x ψ1 (x) with Q1 = Opw (q1 ), q1 supported in {|ξ| ≤ c} we obtain an expression with L(H, H)-norm not exceeding C  hs (|τ | + 1)μ with μ = μ(d, m, s, δ) for all x, y ∈ Rd . Moreover, this assertion remains true if we replace Q1x and tQ2y by Q1x B j and B k tQ2y respectively where j, k ∈ [0, 1], j + k ≤ 1 in the case Actually, there is unboundedness with respect to ξ and the failure of the coefficients to be L(H, H)-operators: they are only L(H, B −1 H)-operators. The unboundedness with respect to ξ will be compensated by cutting of A by ζ(D). 14)

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329

of the first standard set of assumptions and j, k ∈ [0, 12 ] in the case of the second standard set of assumptions. However, we should remember that instead of equation (4.3.1) we now have (4.3.1) where

(hDt − R)u ± B k ψ2 (y ) tQ 2y = ∓ihδ(t)δ(x − y )B k ψ2 (y ) tQ 2y + f ± f ± = (A − A )u ± B k ψ2 (y ) tQ 2y

and hence we need to examine corresponding terms which were automatically negligible in the context of Subsection 4.3.1. Let us note that if we replace u ± by its cut-offs u1± = ψ4 (x)u ± then (4.3.1) remains true with f ± given now by formula f ± = ψ4 (x)(A − A )ψ5 u ± B k ψ2 (y ) tQ2y where u ± = ∓ihGˆ± δ(t)δ(x − y )IH and Gˆ± is a parametrix of the problem (4.3.8)± (for the operator hDt − A while G ± are reserved for parametrices of hDt − A ). One can apply the sandwich procedure to the parametrices Gˆ± as well; then we obtain terms in which Gˆ± is sandwiched between ψ5 (A + i)−1 on the left and (A + i)−1 ψ4 from the right and also we obtain the terms containing no factor Gˆ± at all. Condition (4.2.20)l 15) yields that (4.3.29) For ψ ∈ C0K (B(0, 1 − ε)) operator ψ(A ± i)−1 acts from H r (X , Hj−1 ) to H r −m (X , Hj ) where H r is a Sobolev space and r is arbitrary (but K depends on r )16) . Then the same is true for the adjoint operator (A±i)−1 ψ 17) . Further, let us consider the commutator of (A ± i )−1 with an operator of the form ψQψ where Q is a quantization of a compactly supported scalar symbol. Then [(A ± i)−1 , ψQψ] = −(A ± i)−1 [A, ψQ  ψ](A ± i)−1 15)

Namely, either (4.2.20)1 or (4.2.20) 1 in the cases of the first or second standard set 2 of assumptions. 16) And j, 1 − j ∈ [−l, l]. 17) Condition (4.2.13)l is assumed to be fulfilled for given l and for 1 − l simultaneously.

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and hence (4.3.29) yields that its operator norm from L2 (X , Hj−1 ) to L2 (X , Hj ) 16) does not exceed Ch. This assertion immediately yields that if q1 , q2 ∈ Sh,K (T ∗ Rd , H, H), q2 is compactly supported with respect to (x, ξ) and its support does not intersect that of q1 then the operator norms from L2 (X , Hj−1 ) to L2 (X , Hj ) 16) of the operator ψQ1 ψ(A ± i)−1 ψQ2 ψ do not exceed C  hs and the same is true for (1 − ψ  )(A ± i)−1 ψQ2 ψ provided ψ  = 1 in the neighborhood of supp ψ; here Qi = Opw (qi ). This assertion and Corollary 2.1.7 imply that (4.3.30) Ft→h−1 τ B −j χ2T (t)f ± w  ≤ C  hs ||w || ∀y ∈ Rd

∀τ : |τ | ≤ 2c

∀w ∈ H

where χ ∈ C0K ([−1, 1]) is arbitrary and we estimate the L2 (Rd , H)-norm with respect to x. Therefore the extra terms arising in (4.3.10) in connection with f ± after multiplication by Ft→h−1 τ χT (t)Q1x ψ1 (x)B j with |τ | ≤ c have L(H, H)-norms not exceeding C  hs . So we have proven Proposition 4.3.4. (i) Let conditions (4.2.13)1 , (4.2.14)1 , (4.2.15), (4.2.16), (4.2.17) and (4.2.20)1 be fulfilled and let T , χ, ψ1 , ψ2 , Q1 , Q2 , N be as in Proposition 4.3.1; further, let Q1 and Q2 be scalar operators. Then the estimates  (4.3.31)± ||Ft→h−1 τ χT (t)Q1x ψ1 (x)B j u ± ±  (G¯± R  )n G¯± δ(t)δ(x − y ) B k ψ2 (y ) tQ2y || ≤ C  hs ih 0≤n≤N−1

∀x, y ∈ Rd

∀τ : |τ | ≤ c

¯ j +k ≤1 ∀j, k ∈ [0, l],

hold with l¯ = 1 8) . (ii) Let conditions (4.2.13) 1 , (4.2.14)± , (4.2.15), (4.2.21), (4.2.17), (4.2.20) 1 2 2 be fulfilled and let T , χ, ψ1 , ψ2 , Q1 , Q2 , N be as in Proposition 4.3.1. Then estimates (4.3.31)± hold with l¯ = 12 . Furthermore, let us assume that conditions (4.2.100)–(4.2.102)l are fulfilled. Consider first terms which do not contain Gˆ± . Then (4.3.26) yields that every copy of G¯± can either be sandwiched between two copies of (A¯ + i)−p

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331

with arbitrarily large p or replaced by (A¯ + i)−q with q = 1, ... , 2p, and the same is true for G ± . Let us perform this procedure for the leftmost and the rightmost copies of G¯± or G ± . If after this a factor G¯± or G ± disappears we go to the next factor of this type. Then two final outcomes are possible: (a) The term contains a factor (A¯ + i)−p or a factor (A + i)−p with no factor G¯± , G ± to the left of it, and the term contains a factor (A¯ + i)−p or a factor (A + i)−p or a factor (A + i)−p with no factor G¯± , G ± to the right of it; (b) The term contains no factor G¯± , G ± . In the case (a) (with large enough p) after multiplication on the left by ¯ by B1k an operator Ft→h−1 τ χT (t)Q1x ψ1 (x)B1j and replacing B k with k ∈ [0, l] with k ∈ [0, l] we conclude that the L(H, H)-norm of the term obtained does not exceed C  hs provided this term was negligible under the assumptions of Subsection 4.3.1. In the case (b) it is true only if j + k ≤ 1 but if we consider u = u + + u − then two copies of this term will be annihilated because they are equal but of the opposite sign. Further, let us consider terms containing Gˆ± (and hence originating from ± f ). Let us note that ((a0 − A)(a0 + i)−1 )n + (A + i)−1 = (a0 + i)−1 0≤n≤N−1

((a0 − A)(a0 + i)−1 )N (A + i)−1 ; therefore the arguments used in the proof of Proposition 4.3.4 and conditions (4.2.100)–(4.2.102)l¯ imply that if q2 is compactly supported with respect to (x, ξ) then the operator norm from L2 (X , Hj−1 ) to L2 (X , Hj ) 16) of the operator (A ± i)−1 ψQ2 ψ does not exceed C |||q1 ||| · |||q2 ||| · |||ψ1 ||| · |||ψ2 |||. Moreover, if the support of q2 does not intersect the support of q1 then the operator norms from L2 (X , Hj−1 ) to L2 (X , Hj ) of the operators ψQ1 ψ(A ± i)−1 ψQ2 ψ do not exceed C  hs , and the same is true for (1−ψ  )(A±i)−1 ψQ2 ψ provided ψ  = 1 in a neighborhood of supp ψ; here Qi = Op(qi ) and ν ∈ [0, l] is arbitrary. Then applying the sandwich procedure in the recently modified form and referring in case (a) to Corollary 2.1.7 we obtain that terms of type (a) satisfy the same estimate as before and terms of type (b) annihilate when we add u + and u − . So we have proven

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Proposition 4.3.5. Suppose that either conditions (4.2.13)1 , (4.2.14)1 , (4.2.15), (4.2.16), (4.2.17), (4.2.20)1 or conditions (4.2.13) 1 , (4.2.14)± , (4.2.15), (4.2.21), 2 (4.2.17), (4.2.20) 1 are fulfilled and let T , χ, ψ1 , ψ2 , Q1 , Q2 , N be as in Propo2 sition 4.3.1. Moreover, let (4.2.100)–(4.2.102)l be fulfilled (surely N, K and C now also depend on l). Then  (4.3.32) ||Ft→h−1 τ χT (t)Q1x ψ1 (x)B j u+  ς(G¯ς R  )n G¯ς δ(t)δ(x − y ) B k ψ2 (y ) tQ2y  ≤ C  hs ih ς=± 0≤n≤N−1

∀x, y ∈ Rd

4.3.3

∀τ : |τ | ≤ c

∀j, k ∈ [0, l].

Calculations

Let us go from estimates to calculations. For a sake of simplicity we include now powers of B1 into Q1 and Q2 if needed. Since A¯ = a(y , hDx ) is an operator with symbol which does not depend on x, an obvious method is a Fourier transform x → h−1 ξ with ξ ∈ Rd and the Fourier-Laplace transform t → h−1 τ with τ ∈ C∓ \ R. Then the following table of transformations is valid:

δ(x − y )ψ2 (y ) tQ2y

e ih

−1 x−y ,ξ

q2 (y , ξ, h)

δ(t)

1

G¯±

(τ − a(y , ξ))−1 as ± Im τ < 0

b(y , hDx , h)

b(y , ξ, h)

(xj − yj )

ih∂ξj

Table 4.1: Table of transformations

4.3. THE METHOD OF SUCCESSIVE APPROXIMATIONS

333

−1

Namely, δ(x −y )ψ2 (y ) tQ2y is transformed into e ih x−y ,ξ q2 (y , ξ, h) where q2 (y , ξ, h) is the pq-symbol of operator Q2 (replaced by ψ2 Q2 in the formulas and in this definition if necessary)18) , and δ(t) is transformed into 1; parametrices G¯± are transformed into operators of multiplication by (τ − a(y , ξ))−1 19) ; let us recall that a(x, ξ) = A(x, ξ, 0) is a principal symbol of A. Finally, operators b(y , hDx , h) and (xj − yj ) are transformed into b(y , ξ, h) and ih∂ξj respectively. Therefore for τ ∈ C∓ \ R (4.3.33) Ft→h−1 τ

  G¯± R  )n G¯± δ(t)δ(x − y ) tQ2y = 0≤n≤N−1

∓i



(2π)

−d−1 −d+n

h

 e ih

−1 x−y ,ξ

Fn (y , ξ, τ )[q2 (y , ξ, h)] dξ

0≤n≤N 

where (4.3.34) Fn are differential operators20) with respect to ξ of degree not exceeding n with L(H, H)-valued coefficients which are holomorphic with respect to τ ∈ C \ R and Cs with respect (x, ξ) where s is arbitrarily large. (4.3.35) q2 = symbqp (Q2 ) (and q1 = symbqp (Q1 ) later). Moreover (4.3.36) Under the assumptions of Propositions 4.3.1, 4.3.3 the L(H, H)norms of these coefficients21) do not exceed C | Im τ |−2n−1−|β| . The sandwich procedure yields that (4.3.37) Under the assumptions of Proposition 4.2.3 the L(H, H)-norms (with the same j as in this proposition) of these coefficients21) do not exceed C (|τ | + 1)2r | Im τ |−r with large enough r depending on n. Finally, the modified sandwich procedure yields that 18) 19) 20) 21)

In the previous analysis the method of quantization was not essential. With τ ∈ C∓ \ R; so the only difference here is the range of τ . Applied to q2 . And their derivatives of degree β, |β| ≤ s with respect to (x, ξ) ∈ Ω.

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

334

(4.3.38) Under the assumptions of Proposition 4.3.5 these estimates hold for Q1 , Q2 incorporating B1j and B1k with j, k ∈ [0, l]. Finally, Fn (y , ξ, τ ) depend only on the values of A(., ., h) in a neighborhood of (y , ξ) and hence the replacement of A by A was not essential from the formal point of view. Remark 4.3.6. The way in which we obtained formula (4.3.33) yields that its right-hand expression is approximately equal22) to    −1 −d−1 −d (4.3.39) ∓ i(2π) e ih x−y ,ξ symbpq(N  ) (τ − A)−1 ψ2 Q2 dξ h where (τ − A)−1 means a parametrix formally constructed in terms of the symbol calculus (or in terms of the formal calculus of pseudodifferential operators), “symb” means symbol, the subscript pq means the type of the symbol and the subscript (N  ) means that we cut of the decomposition at  hN ; this subscript is not written in what follows. In fact, for every fixed τ ∈ C \ R the construction of this parametrix by the method of successive approximations gives exactly the same formulas (without the factor ±(2πi)−1 ). Therefore 1 (τ − (i) Expression (4.3.39) is the Schwartz kernel of the operator ∓ 2πi −1 A) ψ2 Q2 ;

(ii) In particular, (4.3.40)

 −1 F0 (y , τ , ξ) = τ − a(y , ξ)

and  −1 (4.3.41) F1 (y , τ , ξ)[q2 ] = τ − a(y , ξ) ·  s   −1  i (j) a (y , ξ) + a(j) (y , ξ) + a(j) (y , ξ)(i∂ξj ) τ − a(y , ξ) q2 2 j j where as is the subprincipal symbol of A. 22)

Errors will be only in terms with n ≥ s + d.

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335

Formula (4.3.33) yields that   Q1x G¯± R  )n G¯± δ(t)δ(x − y ) tQ2y = (4.3.42) Ft→h−1 τ 0≤n≤N−1

∓i



(2π)



−d−1 −d+n

e ih

h

−1 x−y ,ξ

q1 (x, ξ, h)Fn (y , ξ, τ )[q2 (y , ξ, h)] dξ

0≤n≤N 

where q1 is the qp-symbol of Q1 (replaced by Q1 ψ1 in the formulas and in this definition if necessary). On the other hand, Appendix 4.A.1 yields that one can set τ ∈ R in formulas (4.3.33) and (4.3.42) and then one should replace the τ -analytic functions Fn (y , ξ, τ ) by distributions   (4.3.43) Fn∓ (y , ξ, τ ) = Fn (y , ξ, τ ∓ i0) ∈ S Rτ , Cs (Ω, L(Hl¯, Hl−1 ¯ )) . More precisely, the coefficients of Fn∓ belong to this space; recall that Fn and hence Fn∓ and Fn (see below) are differential operators. Thus the left-hand expression of (4.3.42) equals as τ ∈ R (in the sense of distributions) to (2π)−d−1 h−d+n × (4.3.44) ∓ i 0≤n≤N 

 e ih

−1 x−y ,ξ

q1 (x, ξ, h)Fn∓ (y , τ , ξ)[q2 (y , ξ, h)] dξ

and therefore (4.3.45) Ft→h−1 τ





  Q1x ς G¯ς R  )n G¯ς δ(t)δ(x − y ) tQ2y =

ς=± 0≤n≤N−1

(2π)

−d −d+n

h



e ih

−1 x−y ,ξ

q1 (x, ξ, h)Fn (y , ξ, τ )[q2 (y , ξ, h)] dξ

0≤n≤N 

with (4.3.46) Obviously (4.3.47) Moreover,

Fn = ResR (Fn ) =

1 (F − − Fn+ ). 2πi n

supp Fn = {(x, ξ, τ ) : τ ∈ Spec a(x, ξ)}.

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336

(4.3.48) Under the assumptions of Proposition 4.3.5 (4.3.49)

Fn = ResR (Fn ) =

  1 (Fn− − Fn+ ) ∈ S Rτ , Cs (Ω, L(Hl , H−l )) . 2πi

Therefore Propositions 4.3.1–4.3.5 yield the following 1

Proposition 4.3.7. Let T ∈ [h1−δ , h 2 +δ ] with an arbitrarily small exponent δ > 0. Then (i) Under the assumptions of Proposition 4.3.1 the following estimates hold ∀x, y ∈ Rd ∀τ ∈ R: (4.3.50) ||Ft→h−1 τ χT (t)Q1x u ± tQ2y ± i



(2π)−d−1 h−d+n+1 ×

0≤n≤N 

 T χ( ˆ



(τ − τ )T ih−1 x−y ,ξ )e q1 (x, ξ, h)Fn∓ (y , ξ, τ  )[q2 (y , ξ, h] dτ  dξ|| ≤ h C  hs

and (4.3.51) ||Ft→h−1 τ χT (t)Q1x u tQ2y −



(2π)−d h−d+n+1 ×

0≤n≤N 

 T χ( ˆ



(τ − τ )T ih−1 x−y ,ξ )e q1 (x, ξ, h)Fn (y , ξ, τ  )[q2 (y , ξ, h] dτ  dξ|| ≤ h C  hs .

(ii) Under the assumptions of Proposition 4.3.3 this estimate also remains true; in terms of condition (4.3.21)3 one should replace Q1 and Q2 by Q1 ψ1 and ψ2 Q2 respectively. (iii) Under the assumptions of Proposition 4.3.4 this estimate also holds with Q1 , Q2 and q1 , q2 replaced by Q1 ψ1 B j , ψ2 B k Q2 and q1 B j , B k q2 respectively ¯ j + k ≤ 1) and with τ ∈ [−2c, 2c]; (with j, k ∈ [0, l],

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337

(iv) Under the assumptions of Proposition 4.3.5 the following estimate holds ∀x, y ∈ Rd ∀τ ∈ [−2c, 2c] ∀j, k ∈ [0, l]: (4.3.52) ||Ft→h−1 τ χT (t)Q1x ψ1 (x)B1j uB1k ψ2 (y ) tQ2y −  (τ − τ  )T ih−1 x−y ,ξ −d −d+n+1 T χ( ˆ )e (2π) h q1 (x, ξ, h)× h 0≤n≤N  B1j Fn (y , τ  , ξ)[B1k q2 (y , ξ, h)] dτ  dξ|| ≤ C  hs ; further, in this case Fn is L(Hl , H−l )-valued distribution. Remark 4.3.8. (i) Surely, Remark 4.3.6 remains true with the obvious modifications for the further formulas. In particular, the sum of the left-hand expressions in (4.3.33) (with both signs) is approximately equal22) to    −1 −d −d e ih x−y ,ξ ResR symbqp(N  ) (τ − A)−1 ψ2 Q2 dξ (4.3.53) (2π) h   which is the Schwartz kernel of the operator ResR (τ − A)−1 ψ2 Q2 . (ii) We also have the important formula     (4.3.54) F0 (y , ξ, τ ) = δ τ − a(y , ξ) = ∂τ θ τ − a(y , ξ) (where θ(.) is the Heaviside function) and (  (4.3.55) F1 [q2 ] = ResR (τ − a)−1 as +  i  a(j) (τ − a)−1 a(j) − a(j) (τ − a)−1 a(j) (τ − a)−1 q2 + 2 j (j) )  i  (j) (τ − a)−1 a(j) (τ − a)−1 q2 − (τ − a)−1 a(j) (τ − a)−1 q2 . 2 j (iii) In particular, as a is scalar (4.3.56) F1 [q2 ] = −δ (τ − a)as q2 + (j)   i   (j) −δ (τ − a)a(j) q2 + δ (τ − a)a(j) q2 . 2 j

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

338

4.3.4

Exploiting Microhyperbolicity and Increasing T 1

Proposition 4.3.7 holds with any T ∈ [h1−δ , h 2 +δ ] and without microhyperbolicity condition. Let us consider the possibility to increase T in the corollaries of estimates (4.3.51) and (4.3.52) to T which is a small constant and may be beyond; as we know it is very important for the Tauberian method. We also discuss the property of coefficients under microhyperbolicity condition. Exploiting Microhyperbolicity Let us set x = y in (4.3.51)   (4.3.57) Ft→h−1 τ χT (t)Γx Q1x u tQ2y ≡ (2π)−d h−d+n+1 × 0≤n≤N 

 T χ( ˆ

(τ − τ  )T )q1 (x, ξ, h)Fn (y , ξ, τ  )[q2 (y , ξ, h] dξdτ  h

and integrate with respect to x   (4.3.58) Ft→h−1 τ χT (t)Γ Q1x u tQ2y ≡ (2π)−d h−d+n+1 × 0≤n≤N 

 T χ( ˆ



(τ − τ )T )q1 (x, ξ, h)Fn (y , ξ, τ  )[q2 (y , ξ, h] dxdξdτ  ; h

these formulae hold modulo matrices with L(H, H) or even L(Hl , Hl ) norms23) bounded by C  hs . Proposition 4.3.9. Let the operator A be ξ-microhyperbolic on energy level 0 on supp q1 ∪ supp q2 where q1 , q2 ∈ Sh,K (T ∗ Rd , H, H). Then τ − , τ¯ + ] with small constants T0 > 0, (i) For T ∈ [h1−δ , T0 ] and τ ∈ [¯  > 0 equality (4.3.57) holds modulo matrices with L(H, H) norms bounded by C  hs . (ii) Further, then (4.3.59)

κnx (τ )

= (2π)

−d

 q1 (x, ξ, h)Fn (x, ξ, τ )[q2 (x, ξ, h)] dξ

  τ − , τ¯ + ], L(H, H) . belong to C0s Rd × [¯ 23)

In the framework of the corresponding statement of Proposition 4.3.7.

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339

(iii) Moreover, in the framework of Proposition 4.3.523) equality (4.3.57) holds with Q1 , Q2 and q1 , q2 replaced by Q1 B1j , Q2 B k and q1 B1j , q2 B1k with j, k ∈ [0, l]. Proposition 4.3.10. Let the operator A be microhyperbolic on energy level 0 on supp(q1 ) ∪ supp(q2 ) where q1 , q2 ∈ Sh,K (T ∗ Rd , H, H). Then τ − , τ¯ + ] with small constants T0 > 0, (i) For T ∈ [h1−δ , T0 ] and τ ∈ [¯  > 0 equality (4.3.57) holds modulo matrices with L(H, H) norms bounded by C  hs . (ii) Further, then (4.3.60)

κn (τ )

= (2π)

−d

 q1 (x, ξ, h)Fn (x, ξ, τ )[q2 (x, ξ, h)] dxdξ

  belong to C0s [¯ τ − , τ¯ + ], L(H, H) . (iii) Moreover, in the framework of Proposition 4.3.523) equality (4.3.58) holds with Q1 , Q2 and q1 , q2 replaced by Q1 B1j , Q2 B k and q1 B1j , q2 B1k with j, k ∈ [0, l]. Proof of Propositions 4.3.9, 4.3.10. Let us note that under the microhyperbolicity condition the assertions on the regularity properties of κnx and κn follow from Appendix 4.A.1. 1 As we mentioned, for T ∈ [h1−δ , h 2 + δ] asymptotics (4.3.57) and (4.3.58) are due to (4.3.51), (4.3.52). Let us take χ = χ¯ where here we assume that χ¯ ∈ C0K ([−1, 1]) equals 1 on [− 34 , 34 ]. Then Corollary 2.1.17 yields that under corresponding microhyperbolicity condition in the left-hand expressions of (4.3.57) and (4.3.58) one can replace χ¯T by χ¯T  with arbitrary T  ∈ [h1−δ , T0 ]; moreover, in the framework of Proposition 4.3.5 we also use the arguments of Subsection 4.2.4. Further, Proposition 4.3.11 below implies that under corresponding microhyperbolicity condition in the right-hand expressions of (4.3.57) and (4.3.58) one can replace χ¯T by χ¯T  with arbitrary T  ∈ [h1−δ , ∞]. Therefore for χ = χ¯ all statements are proven. Since replacing a factor χ¯ by a factor χ = χχ¯ (with χ supported in [−1, 1]) is equivalent to convoluting the Fourier transform with 2πiT χ(τ ˆ Th−1 ) we conclude that the proposition has also been proven for χ.

340

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

Proposition 4.3.11. (i) Let χ = χ¯ (equal 1 on [− 34 , 34 ]). Then under ξmicrohyperbolicity condition or microhyperbolicity condition one can replace in the right-hand expression of (4.3.57) or (4.3.58) respectively χ¯T by χ¯T  with arbitrary T  ∈ [h1−δ , ∞]; then L(H, H)-norm of error does not exceed C  hs . (ii) Moreover, in the framework of Proposition 4.3.5 Statement (i) remains true with Q1 , Q2 and q1 , q2 replaced by Q1 B1j , Q2 B k and q1 B1j , q2 B1k with j, k ∈ [0, l]. Remark 4.3.12. As T = ∞ and χ = χ¯ the right-hand expressions of (4.3.57) and (4.3.58) become (4.3.61)



h−d+n+1 κnx (τ ),

0≤n≤N 

(4.3.62)



h−d+n+1 κn (τ )

0≤n≤N 

respectively. Proof of Proposition 4.3.11. Since χˆ ∈ S then in the right-hand expressions of (4.3.57) and (4.3.58) for τ ∈ [¯ τ − , τ¯ + ] we can insert an additional factor φ(τ  ) with φ ∈ C0K ([¯ τ − 2, τ¯ + 2] equal to 1 on [¯ τ − 32 , τ¯ + 32 ]. Let us note that if χ(t) is replaced by tχ(t) then χ((τ ˆ − τ  )Th−1 ) should −1   −1 be replaced by −ihT ∂τ χ((τ ˆ − τ )Th ) and integration by parts moves ∂τ  from χ((τ ˆ − τ  )Th−1 ) to φ(τ  )κnx (τ  ) or φ(τ  )κn (τ  ). Repeating this procedure we conclude from the regularity properties of κnx or κn that if we replace χT (t) by t r χT (t) then we obtain in the right-hand expressions of (4.3.57) and (4.3.58) negligible terms. Taking χ(t) = t −r (χ(tζ ¯ −1 ) − χ(t)) ¯ we conclude that we can replace T by ζT in the right-hand expression of (4.3.57) and (4.3.58) provided χ = χ, ¯ ζ ∈ [1, 2]. T Repeating this procedure "log2 ( T )# + 1 times we conclude that we can replace T by T  provided χ = χ; ¯ obviously, we can skip φ(τ  ) now. While the previous arguments used microhyperbolicity indirectly through smoothness of κnx and κn with respect to τ , the following arguments use it directly through propagation:

4.4. GENERAL BASIC THEOREMS

341

Alternative proof of Proposition 4.3.11. Note that without any loss of the generality one can assume ξ-microhyperbolicity; otherwise we can apply metaplectic transformation (exactly as in the proof of Theorem 2.1.19). Let us remember that the right-hand expressions of (4.3.57) came from parametrices G¯∓ and one can apply the same propagation results to them but with any T ≥ h1−δ (since those are parametrices of operators with symbols A(hD) in Rd scaling brings any T ≥ T0 to T = T0 ). Remark 4.3.13. (i) Therefore the microhyperbolicity condition works many times: it allows us to derive Tauberian formula with a sharp remainder estimate; further, it allows us to replace in the Tauberian formula T by h1−δ without spoiling the remainder estimate and, finally, it implies regularity of the coefficients in the decompositions (4.3.61) or (4.3.62). (ii) We will discuss if it is possible to increase T further in Section 4.5.

4.4

General Basic Theorems

In this section we obtain and improve local and microlocal semiclassical spectral asymptotics. We consider here the most important and sophisticated case of partial differential operators (possibly with unbounded coefficients). Moreover, the results obtained here remain true for both classes of pseudodifferential operators introduced in Section 4.324) in which case a cutting function ψi is not necessary. We assume that (4.4.1) Qi = Op(qi )ψi , supp(qi ) ⊂ Ω ⊂ {|ξ| ≤ c},

 1  dist(supp(qi ), Ω) ≥ 0 , ψi ∈ C0∞ B(0, ) 2

while our conditions are fulfilled in B(0, 1). We start from the general case in which no microhyperbolicity condition is assumed, then treat the microhyperbolic case. In both these cases we consider microlocal spectral asymptotics with no global condition (save self-adjointness of operator) assumed. Then the most special case when conditions on Hamiltonian trajectories are assumed. These conditions are semi-global i.e. imposed where these trajectories reach. 24)

See conditions (4.3.21)1 and (4.3.21)2 .

342

4.4.1

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

General Theory: T  h 2 −δ 1

Let us start with the most general case. In this case we can derive only asymptotics for (smooth) spectral means: Theorem 4.4.1. Let either conditions (4.2.13)1 , (4.2.14)1 , (4.2.15), (4.2.16), (4.2.17), (4.2.20)1 or (4.2.13) 1 , (4.2.14)± , (4.2.15), (4.2.21), (4.2.17), (4.2.20) 1 2 2 be fulfilled. Moreover, let conditions (4.4.1) and (4.2.100)–(4.2.102)l be fulfilled. Let qi ∈ Sh,K (T ∗ Rd , H, H), |¯ τ | ≤ c where as usual ∗ means any subscript (including an empty one). Let ϕ, ϕ1 , ϕ2 ∈ C0∞ (R) and L∗ ≥ h 2 −δ . 1

(4.4.2) Then

(i) The following estimates hold:    τ − τ¯   W (4.4.3) Rx,y ,ϕ,L (¯ dτ Q1x B1j e(x, y , τ1 , τ )B1k tQ2y − τ ) := ϕ L   −d+n Kn (x, y , τ ) dτ ≺ C  hs h 0≤n≤N−1

and

  1 := (4.4.4) Q1x B1j e(x, y , τ  , τ  )B1k tQ2y − L1 L2   τ  − τ¯   τ  − τ¯  1 2 ϕ2 dτ  dτ  ≺ C  hs h−d+n Kn (x, y , τ  , τ  ) ϕ1 L L 1 2 0≤n≤N−1 RW τ1 , τ¯2 ) x,y ,ϕ1 ,ϕ2 ,L1 ,L2 (¯

∀x, y ∈ Rd

∀j, k ∈ [0, l]

where s and δ > 0 are arbitrary, N = N(d, m, s, δ), K = K (d, m, s, N, δ), C  = C |||q1 |||·|||q2 |||·|||ψ1 |||·|||ψ2 |||·|||ϕ||| or C  = C |||q1 |||·|||q2 |||·|||ψ1 |||·|||ψ2 |||·|||ϕ1 |||·|||ϕ2 |||,  −1  (4.4.5) Kn (x, y , τ ) = e ih x−y ,ξ q1 (x, ξ, h)B1j Fn (y , τ , ξ)B1k [q2 (y , ξ, h)] dξ, and (4.4.6)





Kn (., ., τ , τ ) =



τ  τ

Kn (., ., τ ) dτ ;

4.4. GENERAL BASIC THEOREMS RW ... with various subscripts means Weyl remainder estimate

343 25)

.

(ii) Without conditions (4.2.100)–(4.2.102)l estimates (4.4.3) and (4.4.4) ¯ remain true with B1j , B1k replaced by B j , B k , j, k ∈ [0, l]; (iii) Moreover, estimates (4.4.3) and (4.4.4) remain true for |¯ τ∗ | ≤ c(L∗ +1). Furthermore in the case B = I one can take arbitrary τ¯∗ ∈ R. 1

Proof. (a) Let us pick T ∈ [h1−δ , h 2 +δ ] such that (4.4.7)

TL ≥ h1−δ

(which can be done due to condition (4.4.2); we decrease δ > 0 if necessary). Applying (4.2.31) and (4.3.52) we obtain that estimate (4.4.3) holds with T χ((τ ˆ  − τ )Th−1 ) instead of ϕ((τ − τ¯)L−1 ) and with τ  ∈ [−c, c]. Let us multiply this estimate by h−1 ϕ((τ − τ¯)L−1 ) and integrate over τ ∈ R 26) . Then (4.4.3) holds with the function ϕ(λ) replaced by    h −1 ϕ(λ )χˆ (λ − λ )ε−1 dλ , . ϕε (λ) = ε ε= LT Let us pick χ = χ. ¯ Recall that χ¯ = 1 on [− 12 , 12 ]. Then one can easily prove that   ϕε (λ) = ϕ(λ) + O εs (1 + |λ|)−s . Therefore it is sufficient to estimate the norm of   |τ − τ¯|T −s s (4.4.8) ε 1+ dτ Nx,y (τ ) h with Nx,y defined by modified (4.2.32): (4.4.9)

Nx,y (τ ) := Q1x B1j e(x, y , τ1 , τ )B1k tQ2y .

Dividing this integral into a sum of integrals over intervals [n, n + 1] with n ∈ Z and applying Proposition 4.2.1 we obtain that the norm of the nth integral does not exceed Chs (|n| + 1)−s (recall that ε ≤ hδ ) and hence estimate (4.4.3) is proven. Let us recall that Kn are distributions with respect to τ ; so Kn are distributions with respect to (τ  , τ ). 26) This means exactly that we apply operator ϕ((hDt − τ¯)L−1 ) to Q1x B1j uB1k tQ2y and to its approximation obtained in the previous section and set t = 0 afterwards. 25)

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

344

(b) Let us prove (4.4.4). Integrating by parts in (4.4.3) and replacing ∂τ ϕ(τ ) by ϕ(τ ) we obtain that under the additional restriction  ∞ (4.4.10) ϕ(τ )dτ = 0 −∞

the following estimate holds: (4.4.11)

  := Q1x B1j e(x, y , τ¯1 , τ )B1k tQ2y dτ −   τ − τ¯  2 dτ ≺ C  hs h−d+n Kn (x, y , τ¯1 , τ ) ϕ L 2 0≤n≤N−1 L−1 2

RW τ1 , τ¯2 ) x,y ,ϕ2 ,L2 (¯

∀x, y ∈ Rd

∀j, k ∈ [0, l]

where under condition (4.4.10) RW ¯1 . It implies x,y ,ϕ2 ,L2 does not depend on τ (4.4.4) as one of functions ϕ1 , ϕ2 satisfies (4.4.10). Let us note that the function  ∞    ϕ(τ ) = ϕ2 (τ ) ϕ1 (τ ) dτ − ϕ1 (τ ) −∞

∞ −∞

ϕ2 (τ  ) dτ 

satisfies condition (4.4.10); substituting it into (4.4.11) we obtain the difference of two double integrals; let us replace (τ , τ  ) in the first integral by (τ  , τ  ) and in the second one by (τ  , τ  ); then using the equality e(., ., τ1 , τ  ) − e(., ., τ1 , τ  ) = e(., ., τ  , τ  ) and similar equalities for Kn we obtain (4.4.4) as L = L1 = L2 and τ¯1 = τ¯2 . Consider L = L1 ≤ L2 (which is not any loss of the generality); then (4.4.4) for L2 = L implies it for L2 ≥ L with ϕ2 := ϕ2 ∗ ϕL/L2 with ϕ ∈ C0∞ (R) and ϕε (τ ) = εϕ(τ ε−1 ). However, picking ϕ such that R ϕ(τ ) dτ = 1 we note that ϕ2 − ϕ2 ∗ ϕL/L2 satisfies (4.4.10) and thus ϕ2 is decomposed into the sum of two functions for which (4.4.4) has been proven. So, we can drop condition L1 = L2 . Note that (4.4.12) e(x, y , τ1 , τ2 ) =



 (−1)

i

e(x, y , τ  , τi )ϕi (τ  ) dτ 

i=1,2



ϕi (τ  ) dτ  = 1.

as R

4.4. GENERAL BASIC THEOREMS

345

Therefore it is sufficient to consider the case L2 = 1. In this case shifting ϕ2 we can remove assumption τ¯2 = τ¯1 . As L∗ ≥ 1 shifting ϕ∗ replace assumption |¯ τ∗ | ≤ c in (4.4.33), (4.4.34) by |¯ τ∗ | ≤ cL∗ + c Let us first note that in the case B = I , L ∈ [h 2 −δ , 1] one can choose τ¯i arbitrarily (because τ : |τ | ≥ C0 falls into the semiclassical spectral gap; recall that qi are compactly supported). 1

This theorem combined with Propositions 4.2.3–4.2.5 immediately yields Theorem 4.4.2. Let the conditions of Theorem 4.4.1 be fulfilled. (i) Moreover, let either conditions of Proposition 4.2.3 be fulfilled in an 1 interval of the length ≥ h 2 −δ containing τ¯1 or let the conditions of one of Propositions 4.2.4, 4.2.5 be fulfilled with τ¯ replaced by τ¯1 ∈ R. Finally, let |¯ τ1 | ≤ cL + c. Then (4.4.13)

RW τ1 , τ¯2 ) ≺ C  hs ; x,y ,ϕ2 ,L2 (¯

here RW τ1 , τ¯2 ) does not depend (modulo O(hs )) on τ¯1 lying in the x,y ,ϕ2 ,L2 (¯ indicated interval. (ii) Further, let the assumptions of (i) be fulfilled for τ¯ = τ¯1 as well as for τ¯ = τ¯2 . Then (4.4.14) RW τ1 , τ¯2 ) := Nx,y (¯ τ1 , τ¯2 ) − h−d+n Kn (x, y , τ¯1 , τ¯2 ) ≺ C  hs ; x,y (¯ 0≤n≤N−1

here RW τ1 , τ¯2 ) does not depend (modulo O(hs )) on τ¯1 , τ¯2 residing in the x,y (¯ indicated intervals. (iii) Moreover, if conditions (4.2.14)± , (4.2.21) are fulfilled then in Statements (i) and (ii) one can take τ¯1 = ∓∞. (iv) If B = I then in in Statements (i) and (ii) one can take τ¯1 ≤ −C0 (or τ¯1 = −∞ and τ¯2 ≥ C0 (or τ¯2 = ∞) with sufficiently large C0 . (v) Assertions (ii) and (iii) of Theorem 4.4.1 remain true for the estimates of this theorem.

346

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

Proof. The corresponding proposition of Subsection 4.2.2 yields that in the function to be estimated one can replace τ1 and (or) τ2 by any other value from the neighborhood indicated above and hence by the mean value on this interval. On the other hand, for the spectral mean the corresponding estimate is obvious. Remark 4.4.3. One can prove easily that if an averaging function ϕ is not infinitely smooth but merely satisfies (4.2.73)–(4.2.75) then the statements 1 above hold with the remainder estimate C  h−d ϑ(h 2 −δ L−1 ).

4.4.2

General Microhyperbolic Theory: T  1

In the previous subsection we found the complete Weyl asymptotics of Nx,y (τ1 , τ2 ) but with a twist: either τ∗ falls into semiclassical spectral gap or 1 it is “tamed” by mollification with L∗ ≥ h 2 −δ . Now we consider the main theory when at least one of energy levels τ∗ neither falls into semiclassical spectral gap nor is “tamed”. Remark 4.4.4. Actually we can assume that only one τi is of this type; the general case is reduced to this due to (4.4.12). Asymptotics with Mollification Still let us explore first asymptotics with mollification. We know that under ξ-microhyperbolicity (microhyperbolicity) condition complete asymptotics of Ft→h−1 τ χ¯T (t)Γx (Q1x B1j uB1k tQ2y ) (Ft→h−1 τ χ¯T (t)Γ(Q1x B1j uB1k tQ2y ) respec1 tively) was holds for T  1 rather than T = h 2 +δ . Then in virtue of the same arguments as in the previous subsection all the results with applied Γx 1 or Γ hold as condition (4.4.2) (which was L∗ ≥ h 2 −δ )is replaced by (4.4.15)

L∗ ≥ h1−δ .

Theorem 4.4.5. Let assumptions of Theorem 4.4.1 be fulfilled with (4.4.2) replaced by (4.4.15). Further, let A be ξ-microhyperbolic on supp(q1 ) ∩ supp(q2 ) on the energy level τ¯∗ . Then

4.4. GENERAL BASIC THEOREMS

347

(i) The following estimates hold  (4.4.16)

τ) RW x,ϕ,L (¯

:=

   τ − τ¯  dτ Γx Q1x B1j e(x, y , τ1 , τ )B1k tQ2y − ϕ L   −d+n κnx (τ ) dτ ≺ C  hs h 0≤n≤N−1

and (4.4.17)

  1 := Γx Q1x B1j e(x, y , τ  , τ  )B1k tQ2y − L1 L2   τ  − τ¯   τ  − τ¯  h−d+n κnx (τ  , τ  ) ϕ1 ϕ2 dτ  dτ  ≺ C  hs L L 1 2 0≤n≤N−1

RW τ1 , τ¯2 ) x,ϕ1 ,ϕ2 ,L1 ,L2 (¯

∀x, y ∈ Rd

∀j, k ∈ [0, l]

with the same s, δ > 0, N, K , C  as before. (ii)–(iii) Statements (ii)–(iii) of Theorem 4.4.1 remain true with (4.4.16), (4.4.17) instead of (4.4.3), (4.4.4). Theorem 4.4.6. Let assumptions of Theorem 4.4.1 be fulfilled with (4.4.2) replaced by (4.4.15). Further, let A be microhyperbolic on supp(q1 ) ∩ supp(q2 ) on the energy level τ¯∗ . Then all the results of Theorem 4.4.5 remain true with Γx replaced by Γ and κnx , κnx replaced by κn , κn respectively. One can modify Theorem 4.4.2 in the same way as well. Remark 4.4.7. Usually in asymptotics of Nx∗ one needs to assume ξ-microhyperbolicity at supp(q1 )∪supp(q2 ) but in this section due to the finite speed of propagation we can to assume ξ-microhyperbolicity at supp(q1 ) ∩ supp(q2 ) instead. Asymptotics without Mollification The following statement immediately follows from the Tauberian estimates of Section 4.2 and calculations of Section 4.3:

348

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

Theorem 4.4.8. Let assumptions of Theorem 4.4.1 be fulfilled with (4.4.2) replaced by L∗ ≥ h.

(4.4.18)

Further, let A be ξ-microhyperbolic on supp(q1 ) ∩ supp(q2 ) on the energy level τ¯∗ . Then (i) The following estimate holds:  τ1 , τ¯2 ) := Γx Q1x B1j e(x, y , τ¯1 , τ¯2 )B1k tQ2y − (4.4.19) RW x (¯  h−d+n κnx (τ¯1 , τ¯2 ) ≺ C  h1−d 0≤n≤N−1

(ii) As ϕ∗ satisfy (4.2.73)–(4.2.75) the following estimates hold: (4.4.20)

h RW , τ ) ≺ C  h1−d ϑ x,ϕ,L (¯ L

(4.4.21)

RW τ1 , τ¯2 ) ≺ C  h1−d ϑ1 x,ϕ1 ,L1 ,ϕ2 ,L2 (¯

h h + C  h1−d ϑ2 L1 L2

with the left-hand expressions defined by (4.4.16) and (4.4.17). Theorem 4.4.9. Let assumptions of Theorem 4.4.1 be fulfilled with (4.4.2) replaced by (4.4.18). Further, let A be microhyperbolic on supp(q1 ) ∩ supp(q2 ) on the energy level τ¯∗ . Then all the results of Theorem 4.4.8 remain true W   with RW x∗ replaced by R∗ and κnx , κnx replaced by κn , κn respectively. Remark 4.4.10. (i) In Theorem 4.4.8 one can skip ξ-microhyperbolicity condition on some level τ¯i by making mollification with ϕi ∈ C0∞ (R) and L∗ ≥ h1−δ in then (4.4.19) or just assuming that ϕ∗ ∈ C0∞ (R) and L∗ ≥ h1−δ in (4.4.21). The similar statement in the framework of Theorem 4.4.9 also holds. (ii) In asymptotics without averaging only one term κ0x h−d or κ0 h−d is needed. In the asymptotics with averaging one can skip all the terms such that t n−1  ϑ(t) for all t  1 (obviously this is true as L  1; as L  1 we apply Statement (iii) below). One can skip even more terms as L  1.

4.4. GENERAL BASIC THEOREMS

349

(iii) One can prove easily that under ξ-microhyperbolicity condition (4.4.22)

τ ) = O(L−n ) (κnx ∗ ϕL )(¯

as L ≥ 1

and under microhyperbolicity condition this is true for κn . (iv) One can easily improve asymptotics derived in the previous subsections to O(hs L−s ) as L ≥ h−M . Improved Asymptotics without Mollification Here we will improve asymptotics without mollification. Actually this improvement will be probably superficial in the current settings but it will open the door for the real improvement under conditions of the semi-global nature in the next section. Let us observe that we know now (4.1.12)- or (4.1.11)-like expressions with far better precision than before and therefore as Q2 = Q1∗ and j = k we have estimates   (4.4.23) Γx Ft→h−1 τ χT (t)Q1x B1j u(x, y , t)B1k tQ2y ≺ C0 Jx (q10 B1j , τ )h1−d + C  h2−d and   (4.4.24) Γ Ft→h−1 τ χT (t)Q1x B1j u(x, y , t)B1k tQ2y ≺ C0 J(q10 B1j , τ )h1−d + C  h2−d as |τ −¯ τ | ≤  under ξ-microhyperbolicity and microhyperbolicity assumptions respectively where  0 j (4.4.25) Jx (q1 B1 , τ ) := q10 B1j dξ μτ x B1j q10  (4.4.26) J(q10 B1j , τ ) := q10 B1j d(x,ξ) μτ B1j q10 qi0 is the principal symbol of Qi ,     (4.4.27) μτ = δ τ − a(x, ξ) = ∂τ θ τ − a(x, ξ) is a non-negative L(H, H)-valued measure and μτ x is the same measure for given x.

350

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

Finally, here and below C0 = C0 (d, ϑ, ϕ) is an absolute constant; so C0 = C0 (d) in the most important case ϑ = ϕ = 1. Indeed, the leading coefficients in the asymptotics of the left-hand expressions are κ0x (τ ) and κ0 (τ ) which due to (4.3.52), (4.3.54), (4.3.60) and (4.3.59) are exactly     −d q10 B1j δ τ − a(x, ξ) B1j q20 dξ (4.4.28) κ0x (τ ) :=(2π) and (4.4.29)

κ0 (τ )

:=(2π)

−d



  q10 B1j δ τ − a(x, ξ) B1j q20 dxdξ

respectively and coincide with J0x , J0 as q20 = q10† and j = k. Estimates (4.4.23), (4.4.24) replace previously used estimates with C  h1−d right-hand expressions. Let us recall that (4.4.30) Under ξ-microhyperbolicity condition Jx (q10 B1j , τ ) is a smooth function of (x, τ ). Under microhyperbolicity condition J(q10 B1j , τ ) is a smooth function of τ . Taking H¨ormander function χ and applying Lemma 4.2.7) we derive easily from (4.4.24) that for j = k and Q2 = Q1∗ under microhyperbolicity or ξ-microhyperbolicity on the energy level 0 (4.4.31) (4.4.32)

N(τ  , τ  ) ≺ C0 g ∗ (τ  , τ  ), Nx (τ  , τ  ) ≺ C0 gx∗ (τ  , τ  )

with g ∗ , gx∗ defined by (4.2.35) where g , gx are right-hand expressions of (4.4.24), (4.4.23) respectively as |τ − τ¯| ≤ 0 ; we do not change definition (4.2.87) of g (τ ) = gx (τ ) = C  T (|τ | + 1)2l as |τ − τ¯| ≥ 0 . Now again we want to get rid of assumption j = k, Q2 = Q1∗ . For this purpose we still under this assumption rewrite (4.4.31) or (4.4.32) respectively as |(N(τ  , τ  )w , w )| ≤ C0 (g ∗ (τ  , τ  )w , w ), |(Nx (τ  , τ  )w , w )| ≤ C0 (gx∗ (τ  , τ  )w , w ) for all w ∈ H.

4.4. GENERAL BASIC THEOREMS

351

Then arguments of the proof of Proposition 4.2.19 imply that without assumption j = k, Q2 = Q1∗ (4.4.33)

|(N(τ  , τ  )w , w )| ≤ C0 (g1∗ (τ  , τ  )w , w ) 2 (g2∗ (τ  , τ  )w , w ) 2 ,

(4.4.34)

∗ ∗ |(Nx (τ  , τ  )w , w )| ≤ C0 (gx1 (τ  , τ  )w , w ) 2 (gx2 (τ  , τ  )w , w ) 2

1

1

1

respectively where subscript 1 , Then

2

1

refers to g , g ∗ defined with Q1 B1j and Q2 B2k .

|(N(τ  , τ  )w , w )| ≤ C0 (g1∗ (τ  , τ  )w , w ) + C0 (g2∗ (τ  , τ  )w , w ), ∗ ∗ |(Nx (τ  , τ  )w , w )| ≤ C0 (gx1 (τ  , τ  )w , w ) + C0 (gx2 (τ  , τ  )w , w ) respectively and we do not need w anymore. Now we can apply Lemmas 4.2.13, 4.2.15; we need to calculate R(¯ τ) according to (4.2.65). Since due to (4.4.30)   g (τ ) = g (¯ τ ) + O h1−d |τ − τ¯| + h2−d we conclude that R(¯ τ ) = g (¯ τ ) + O(T −1 h2−d + h2−d ). Therefore we arrive to estimates with the right hand expressions defined by these calculations: τ , τ¯2 ) ≺ (4.4.35) RTϕ2 ,L2 (¯

 h h1−d  C0 J(q10 B1j , τ¯) + C0 J(q20 B1k , τ¯) + C  + C  h T T

and τ , τ¯2 ) ≺ (4.4.36) RTϕ,L,ϕ2 ,L2 (¯   h  1−d  h h C0 J(q10 B1j , τ¯) + C0 J(q20 B1k , τ¯) + C  + C  h ϑ T T LT provided τ¯2 = τ¯ and ϕ2 ∈ C0∞ (R) while ϕ satisfies (4.2.73)–(4.2.75). These estimates hold under microhyperbolicity condition; the similar T estimates (with Jx , Rx∗ ) hold under ξ-microhyperbolicity condition. Let us recall that T  1 in the current settings. Also in virtue of the calculations of the Section 4.3 we can replace RT∗ by W∗T . Then we arrive to estimates (4.4.38) and (4.4.39) below but only as τ¯2 = τ¯1 and ϕ2 ∈ C0∞ (R). However in virtue of (4.4.12) we can remove these restrictions arriving to the improved versions of Theorems 4.4.8 and 4.4.9:

352

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

Theorem 4.4.11. Let assumptions of Theorem 4.4.1 be fulfilled with (4.4.2) replaced by (4.4.18). Further, let A be ξ-microhyperbolic on supp(q1 ) ∩ supp(q2 ) on the energy level τ¯∗ . Then for j, k ≤ l (4.4.37)

RW τ1 , τ¯2 ) ≺ x (¯



C0

∗=1,2

 h1−d  Jx (q10 B1j , τ¯∗ ) + Jx (q20 B1k , τ¯∗ ) + C  h , T0

 h1−d  Jx (q10 B1j , τ¯1 ) + Jx (q20 B1k , τ¯1 ) + C  h + T0   h  h1−d  Jx (q10 B1j , τ¯2 ) + Jx (q20 B1k , τ¯2 ) + C  h ϑ2 C0 T0 L2 T0

τ1 , τ¯2 ) ≺ C0 (4.4.38) RW x,ϕ2 ,L2 (¯

and (4.4.39) RW τ1 , τ¯2 ) ≺ x,ϕ1 ,L1 ,ϕ2 ,L2 (¯   h  h1−d  Jx (q10 B1j , τ¯∗ ) + Jx (q20 B1k , τ¯∗ ) + C  h ϑ∗ C0 T0 L∗ T0 ∗=1,2 with the left-hand expressions defined respectively by (4.4.19), (4.4.16) and (4.4.17). Theorem 4.4.12. Let assumptions of Theorem 4.4.1 be fulfilled with (4.4.2) replaced by (4.4.18). Further, let A be microhyperbolic on supp(q1 ) ∩ supp(q2 ) on the energy level τ¯∗ . Then all the results of Theorem 4.4.11 remain true W   with RW x∗ replaced by R∗ and κnx , κnx replaced by κn , κn respectively. Useful Formulae Remark 4.4.13. (i) Due to (4.4.28), (4.4.29)    −d κ0x (τ ) :=(2π) q10 B1j θ τ − a(x, ξ) B1j q20 dξ (4.4.40) and (4.4.41)

κ0 (τ ) :=(2π)−d



  q10 B1j θ τ − a(x, ξ) B1j q20 dxdξ.

4.4. GENERAL BASIC THEOREMS

353

(ii) Formulae for the next coefficients are rather complicated. Formulae for κ1x , κ1 are due to (4.3.52), (4.3.55), (4.3.60) and (4.3.59). In particular, in the case of scalar a (4.3.56) implies that    i (4.4.42) κ1x = (2π)−d δ(τ − a) −q10 as q20 − {a, q10 q20 } dξ+ 2  (2π)−d θ(τ − a)(q1 q2 )s dξ where (q1 q2 )s is subprincipal symbol of Q1 Q2 . Important is that as Q1 = Q2 = I (we can set it only formally now)  −d δ(τ − a)as dξ. (4.4.43) κ1x = −(2π) Similar formulae hold for κ1 . (iii) Again, as Q1 = Q2 = I but in the matrix case there is an extra term due to the second line in (4.3.55). The explicit formula and geometric interpretation of this term see in O. Chervova, R. J. Downes and D. Vassiliev [1, 2]. Problem 4.4.14. Consider (magnetic) Schr¨odinger operator H = (−ih∇ − A(x))2 + V (x) and take Q1 = −ih∂j − Aj (x), Q2 = I . Then obviously κ0x = 0 and we leave to the reader to prove that κ1x = const ∂j (τ − V (x))d/2 and calculate the constant. Remark 4.4.15. Assume that Q1 , Q2 are scalar operators. Then one can set j = k = l in (4.4.37) and multiplying by B1−l from both sides one can rewrite it as    W 1−d C0 (|q10 |2 + |q20 |2 ) dξ μx τ¯∗ + C  hB1−2l τ1 , τ¯2 ) ≺ h (4.4.44) Rx (¯ ∗=1,2

where now the left-hand expression is defined by (4.4.19) with j = k = 0. Then   τ1 , τ¯2 ) | ≤ (4.4.45) | tr RW x (¯    1−d h C0 (|q10 |2 + |q20 |2 ) dξ (tr(μx τ¯∗ )) + C  h tr(B1−2l ) . ∗=1,2

One can modify all other estimates in the same way.

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

354

4.5 4.5.1

Long Term Propagation and Sharper Remainder Estimates Preliminary Discussion

Increasing T In the previous section we under microhyperbolicity (or ξ-microhyperbolicity) condition increased T to a small constant T0 . Now we want to increase it further but in a such way that expression    Ft→h−1 τ χ¯T (t)Γ Q1x u tQ2y (4.5.1) or

   Ft→h−1 τ χ¯T (t)Γx Q1x u tQ2y

(4.5.2)

would change negligibly for τ in the vicinity of τ¯. Then (almost) the same estimate for this expression with this new T would have place as it was for T = T0 which would allow us to estimate the Tauberian error by (4.5.3) or (4.5.4)

  h  h h1−d  C0 J(q10 B1j , τ¯) + C0 J(q20 B1k , τ¯) + C  + C  h ϑ T T LT   h  h h1−d  C0 Jx (q10 B1j , τ¯) + C0 Jx (q20 B1k , τ¯) + C  + C  h ϑ T T LT

where ϑ = 1 if there is no averaging (but then we take mollification with respect to the second energy level) where now T means “new” T . However then the Tauberian expressions should change by     Ft→h−1 τ χ¯T (t) − χ¯T0 (t) Γ Q1x u tQ2y (4.5.5) or Ft→h−1 τ

(4.5.6)



   χ¯T (t) − χ¯T0 (t) Γx Q1x u tQ2y

integrated from −∞ to τ¯. We know that this operation is equivalent to the substitution (4.5.7)



χ¯T (t) − χ¯T0 (t)



  → iht −1 χ¯T (t) − χ¯T0 (t)

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355

and pinning τ = τ¯. The same no-periodicity or no-loop condition which ensures respectively the negligibility of (4.5.5) or (4.5.6) also ensures the negligibility of these side-kicks. Since we know that under microhyperbolicity or ξ-microhyperbolicity condition one can replace Tauberian expression for T = T0 by Weyl expression, it is now true for new T as well. Therefore Weyl error will be also estimated by (4.5.3) or (4.5.4). Synthesis However it is not realistic to expect that no-periodicity or no-loop condition we use to derive such improved remainder estimates is fulfilled in every point of supp(q1 ) or supp(q2 ). To overcome this problem we will use different approaches in he cases of Γ- and Γx -related expressions. In the former (Γ) case we replace our original operators Q1 and Q2 by Q1i = Ri Q1 and Q2 = Ri∗ Q2 where (4.5.8) Ri are scalar pseudodifferential operators and assume the non-periodicity condition with time Ti on supp(ri ) ∩ supp(q1 ) or supp(ri ) ∩ supp(q2 ) for i = 1, ... , m while for i = 0 we pick up T0 as a small constant. Then we estimate the error in question by  h1−d  h  0 0 j (4.5.9) C0 ϑ J(ri q1 B1 , τ¯) + J(ri0 q20 B1k , τ¯) + C  h . Ti LTi Now we select Ri such that (4.5.10)



Ri∗ Ri ≡ I , ,

0≤i≤m

and thus (4.5.11)



|ri0 |2 = 1, .

0≤i≤m

Note that due to (4.5.8)     (4.5.12) Γ Ri Q1x v (x, y ) tQ2y = Γ Q1x e(... ) t(Ri Q2 )y . Plugging Q2 := Ri∗ Q2 we arrive due to (4.5.10) to     Γ Ri Q1x v (x, y ) t(Ri∗ Q2 )y (4.5.13) Γ Q1x v (x, y ) tQ2y ≡ 0≤i≤m

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and also to κn =

(4.5.14)



κni

0≤i≤m

where κni is coefficient κn calculated for pair Q1i = Ri Q1 and Q2 = Ri∗ Q2 . Then W (4.5.15) N= Ni , N W = NW Ri i ( =⇒ ) R = 0≤i≤m

0≤i≤m

0≤i≤m

in the obvious notations. Therefore RW ∗ will be estimated by the sum of expressions (4.5.9) which is  h1−d  h  J(ri0 q1 B1j , τ¯) + J(ri0 q2 B1k , τ¯) + C  h . ϑ (4.5.16) C0 Ti LTi 0≤i ≤m Note that due to (4.5.11), (4.5.8) (4.5.17) J(ri0 q10 B1j , τ¯) = J(q10 B1j , τ¯) 0≤i≤m

which shows that inserting ri0 into J(., .) does not hurt. On the other hand, as far as Γx -expressions are concerned the bulk of our arguments will not work; instead we simply decompose Q1 and Q2 into sums of operators. Then estimate would depend on m in a more explicit way. However in applications usually m = 2 in this case. Later we will apply these improved (no double summation) versions of (4.5.9) and (4.5.10) in the following context: T1 , ... , Tm are large, T0 is a fixed small constant. Sharp and Sharper So, to increase T we need to prove that (4.5.5)- or (4.5.6)-like expressions are negligible. For this purpose we will use propagation results of Chapters 2 and 3. However, there are two kinds of results: first there are standard results of Sections 2.1, 2.2 and 3.1–3.4 when we can take T as an arbitrarily large constant and long-term results of Sections 2.4 and 3.5 when depending on conditions we can take T up to h−δ with sufficiently small exponent δ > 0.

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357

The results of the first kind allow us to acquire an arbitrarily small constant factor ε in the estimate of the contribution of the corresponding element Qi with i ≥ 1 27) to Weyl error basically going from “O”-type estimates to “o”-type estimates. However contribution of Q0 is not improved this way and so not to lose the improvement we achieved we need to take Q0 with an arbitrarily small but fixed support; “small” means here that J∗ (q00 B1j , τ¯) is small (less than ε). Actually we cannot do better in the framework of the standard theory. In this case C  depends also on T . We will do it in Subsection 4.5.2. The results of the second kind allow us to acquire a smaller factor ε in the estimate of the contribution of the corresponding element Qi with i ≥ 1 27) to Weyl error; now ε is a small but not very small parameter ε ≥ hδ . Then not to lose this improvement we need to take Q0 with a really small support; “small” means again that J∗ (q00 B1j , τ¯) ≤ ε but “really” indicates that ε is a parameter now. We can accommodate this only with Q with variable size support and in this case the theory of the previous Section 4.4 needs to be rebuilt. We will do it in Subsection 4.5.3. Finally, Subsection 4.5.4 is devoted to simplest applications, examples and discussions.

4.5.2

Standard Sharp Theory

Analysis of Wave Front Sets Let us recall that according to Theorem 1.3.13  (4.5.18) WF(Γx Q1x u tQ2y ) ⊂ (x, ζ, t, τ ) : ∃ξ, η s.t. ζ = ξ − η and (x, ξ, x, −η, t, τ ) ∈ WF(u), (x, ξ) ∈ supp(q1 ), (y , η) ∈ supp(q2 )



and (4.5.19)

 WF(ΓQ1x u tQ2y ) ⊂ (t, τ ) : ∃x, ξ s.t.

(x, ξ, x, −ξ, t, τ ) ∈ WF(u), (x, ξ) ∈ supp(q1 ) ∩ supp(q2 ) .

Meanwhile (4.5.20)

(x, ξ, y , −η, t, τ ) ∈ / WF(u) =⇒ τ ∈ Spec a(x, ξ) ∩ Spec a(y , η)

27) We ignore double summation and even the presence of the second factor in this discussion.

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and according to Theorem 2.2.10 for bounded T (4.5.21) (x, ξ, y , −η, t, τ ) ∈ / WF(u) provided either (i) There exists a complete Hamiltonian flow Ψτ ,t  (y , η) for t  /t ∈ [0, 1] which stays where symbol A(., .) is smooth and does not hit the boundary28) and (x, ξ) ∈ / Ψτ ,t (y , η) or (ii) There exists a complete Hamiltonian flow Ψτ ,−t  (y , η) for t  /t ∈ [0, 1] which stays where symbol A(., .) is smooth and does not hit the boundary28) and (y , η) ∈ / Ψτ ,t (x, ξ). where Ψτ ,−t  is a generalized Hamiltonian flow (possibly multivalued) on the energy level τ generated by symbol −a 29) . Then combining this claim with (4.5.18) we arrive to (4.5.22) (x, ζ, t, τ ) ∈ / WF(Γx Q1x u tQ2y ) provided either (i) For each (y , η) ∈ supp q2 s.t. τ ∈ a(y , η) there exists complete Hamiltonian flow Ψτ ,t  (y , η) for t  /t ∈ [0, 1] which stays where symbol A(., .i) is smooth and does not hit the boundary28) and for each (x, ξ) ∈ Ψt (y , η) either x = y or ζ = ξ − η or (ii) For each (x, ξ) ∈ supp q1 s.t. τ ∈ a(x, ξ) there exists complete Hamiltonian flow Ψτ ,−t  (x, ξ) for t  /t ∈ [0, 1] which stays where symbol A(., .) is smooth and does not hit the boundary28) and for each (y , η) ∈ Ψ−t (x, ξ) either x = y or ζ = ξ − η. Moreover, combining claim (4.5.21) with (4.5.19) we arrive to (4.5.23) (t, τ ) ∈ / WF(ΓQ1x u tQ2y ) provided either (i) For each (x, ξ) ∈ supp(q2 )∩supp(q1 ) s.t. τ ∈ a(x, ξ) there exists complete Hamiltonian flow Ψτ ,t  (x, xi ) for t  /t ∈ [0, 1] which stays where symbol A(., .i) is smooth and does not hit the boundary28) and (x, ξ) ∈ / Ψt (x, ξ) or (ii) Above condition (i) is fulfilled for t replaced by −t. 28)

If we consider differential operators in domain with the boundary. Where z ∈ Ψτ ,±t (z  ) with t > 0 simply means by definition that (z; ±t, τ ) ∈ Kt± (τ − a, (z; 0, τ )). We prefer to use terminology of the Hamiltonian flows. 29)

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359

Periodic Trajectories and Loops Let us discuss what conditions of (4.5.22) and (4.5.23) actually mean. Assume first that the complete Hamiltonian flow exists and is a singlevalued. Then condition (4.5.22)(i) is equivalent to (4.5.22)(ii) and if we are not interested in ζ this condition simply means that x = πx Ψτ ,t (x, ξ);

(4.5.24)

simply flow does not return to point x at moment t (even from another direction). This means absence of bicharacteristic loops. In the same framework condition (4.5.23)(i) is equivalent to (4.5.23)(ii) and means that (x, ξ) = Ψτ ,t (x, ξ);

(4.5.25)

simply flow does not return to point (x, ξ) at moment t. This means absence of periodic bicharacteristics.

x

(a) Bicharacteristic loop

(b) Periodic bicharacteristics

Figure 4.2: The case of a simple hamiltonian flow Note that the looping trajectory (unless it is also a periodic trajectory) does not intersect itself, only its x projection (as shown on Figure 4.2(a)) does. Also note that x-projection of the loop could look differently from the one shown on Figure 4.2(a): it can be periodic30) or be the same curve ran twice (in one and then in the opposite direction)31) . 30) 31)

F.e. as a = (ξ1 − x2 )2 + (ξ2 + x1 )2 − x3 . F.e. as a(x, ξ) = |ξ|2 − x1 .

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Another difference is the time direction which manifests itself in more general settings. While in (4.5.23) we can choose time direction as we want, it is not the case in (4.5.22) since then we need to permute supp(q1 ) and supp(q2 ) as well. Moreover, to employ Tauberian technique we need a symmetric interval [−T , T ] and to extend [−T0 , T0 ] to it we need to verify our conditions for t ∈ [−T , −T0 ] ∪ [T0 , T ]. In the case of (4.5.23) we should do it either for t ∈ [−T , −T0 ] or for t ∈ [T0 , T ] but in the case of (4.5.22) we should do it for t ∈ [−T , −T0 ] and for t ∈ [T0 , T ]. For example, let (x, ξ) and (x, η) be such that Ψτ ,−t (x, ξ) and Ψτ ,t (x, η) hit “Dark Territory” as t > 0 z Dark and t  T1 and (x, ξ) ∈ supp(q1 ), Territory (x, η) ∈ supp(q2 ); then in (4.5.22) we cannot go beyond time interval [−T1 , T1 ] (and we need to check noFigure 4.3: In (4.5.22) analysis we must loop condition). follow both directions; in (4.5.23) analyOn the other hand, if Ψτ ,−t (x, ξ) sis we can chose direction and we choose neither hits “Dark Territory” nor the dashed one. comes back as −T2 < t < 0 (for all (x, ξ) ∈ supp(q1 ) ∩ supp(q2 )) then (4.5.23) we can extend a time interval to [−T2 , T2 ]. In the current settings a “Dark Territory” could mean even a smooth boundary. However applying results of Chapter 3 we would not consider a smooth transversally met boundary as a “Dark Territory” but as a possible source of looping. For example, for a geodesic billiard flow the trajectory which hits the boundary normally goes back the same way and we need either to take T1 < 2dist(x, ∂X ) which is not an improvement over T1 < dist(x, ∂X ) or to analyze the contribution of this loop explicitly gettinga boundary layer type correction to e(x, x, .); this would have no affect on e(x, x, .)ψ(x) dx as supp(ψ) ∩ ∂X = ∅. For details see Section 8.1. For branching flow one should replace (4.5.24) and (4.5.25) by (4.5.26) (4.5.27) respectively.

x∈ / πx Ψτ ,t (x, ξ), (x, ξ) ∈ / Ψτ ,t (x, ξ)

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361

Results Remark 4.5.1. (i) Instead of Theorem 2.2.10 which does not allow bicharacteristic trajectory to hit the boundary, we can refer to Theorem 3.2.4 or geometric versions of Theorems 3.4.6 and 3.4.14. item We definitely need to use the quantitate conditions: instead “trajectory does not hit the boundary” we need to assume that   (4.5.28) dist Ψt,τ (x, ξ), ∂X ≥ γ and instead of (4.5.26), (4.5.27) we need to assume respectively that *   |t| as |t| ≤ T0 , (4.5.29) , dist x, πx Ψτ ,t (x, ξ) ≥ γ as T0 ≤ |t| ≤ T , *   |t| as |t| ≤ T0 , (4.5.30) dist (x, ξ), Ψτ ,t (x, ξ) ≥ γ as T0 ≤ |t| ≤ T , where γ is a small constant. Conditions in this form are fulfilled automatically for |t| ≤ T0 due to microhyperbolicity and ξ-microhyperbolicity respectively. Assuming that these conditions are fulfilled for all t (may be of a selected sign) with |t| ≤ T completely removes implicit T0 from the final estimates. So, our analysis leads us to the following results: Theorem 4.5.2. Let assumptions of Theorem 4.4.1 be fulfilled with (4.4.2) replaced by (4.4.18). (i) Let A be microhyperbolic on supp(q1 ) ∪ supp(q2 ) on the energy level τ¯. Consider a generalized Hamiltonian flow Ψt,¯τ issued from γ-vicinity U of supp(q1 ) ∩ supp(q2 ). Assume that as ςt > 0 32) and |t| ≤ T conditions (4.5.28), (4.5.30) are fulfilled and also (4.5.31) In γ-vicinity of πx Ψt,¯τ (U ) assumptions of Theorem 4.4.1 are fulfilled. Then formula (4.3.58) and all its derivative formulae remain valid for indicated T (rather than T = T0 ) modulo terms not exceeding C  hs where here and in what follows constant C  depends also on T and γ. 32)

Where we can select ς = ±1.

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CHAPTER 4. GENERAL THEORY IN THE INTERIOR

(ii) Let A be ξ-microhyperbolic on supp(q1 ) ∪ supp(q2 ). Consider a generalized Hamiltonian flow Ψt,¯τ issued from supp(q1 ) ∪ supp(q2 ). Assume that as |t| ≤ T conditions (4.5.28), (4.5.29) and (4.5.31) are fulfilled. Then formula (4.3.57) and all its derivative formulae remain valid for indicated T (rather than T = T0 ) modulo terms not exceeding C  hs . Remark 4.5.3. Under additional assumptions of Proposition 4.3.5 statements of Theorem 4.5.2 hold with Q1 , Q2 and q1 , q2 replaced by Q1 B1j , Q2 B1k and q1 B1j , q2 B1k respectively with j, k ∈ [0, l]. Now combining with arguments of Subsubsection 4.4.2.2 Improved Asymptotics without Mollification we arrive to the sharp33) Theorems 4.4.12 and 4.4.11. Before doing it just note Theorem 4.5.4. Let assumptions of Theorem 4.4.1 be fulfilled with (4.4.2) replaced by (4.4.18). Then (i) In the framework of Theorem 4.5.2(i) and Remark 4.5.3 for both energy levels τ¯1 and τ¯2 sharp3) estimates (4.4.37)–(4.4.39) hold (in “without x ” version). (ii) In the framework of Theorem 4.5.2(ii) and Remark 4.5.3 for both energy levels τ¯1 and τ¯2 sharp33) estimates (4.4.37)–(4.4.39) hold. Now we are ready to the synthesis as described in Subsubsection 4.5.1.2 Synthesis. Theorem 4.5.5. Let assumptions of Theorem 4.4.1 be fulfilled with (4.4.2) replaced by (4.4.18). Let A be microhyperbolic on supp(q1 ) ∪ supp(q2 ) on energy levels τ¯∗ , ∗ = 1, 2. Let ri0 ∈ Sh,γ,γ,K (T ∗ X ) 34) be scalar symbols satisfying (4.5.11) . Consider generalized Hamiltonian flow Ψt,¯τ∗ issued from γ-vicinity Ui of supp(ri0 ) ∩ supp(q1 ) ∩ supp(q2 ). Assume that as ςt > 0 with ς := ς∗i 32) and |t| ≤ T∗i conditions (4.5.28), (4.5.30) and (4.5.31) are fulfilled with U := Ui . 33) 34)

Where “sharp” means “with T instead of T0 ” and with C  depending on γ, T . With an arbitrarily small constant γ.

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363

Then τ1 , τ¯2 ) ≺ (4.5.32) RW (¯ ∗=1,2 0≤i≤m

(4.5.33) RW τ1 , τ¯2 ) ≺ ϕ2 ,L2 (¯

C0

 h1−d  0 0 j J(ri q1 B1 , τ¯∗ ) + J(ri0 q20 B1k , τ¯∗ ) + C  h , T∗i

 h1−d ( 0 0 j J(ri q1 B1 , τ¯1 ) + J(ri0 q20 B1k , τ¯1 ) + C  h + T∗i 0≤i≤m    h ) J(ri0 q10 B1j , τ¯2 ) + J(ri0 q20 B1k , τ¯2 ) + C  h ϑ2 L2 T∗i C0

and (4.5.34) RW τ1 , τ¯2 ) ≺ ϕ1 ,L1 ,ϕ2 ,L2 (¯   h  h1−d  0 0 j C0 J(ri q1 B1 , τ¯∗ ) + J(ri0 q20 B1k , τ¯∗ ) + C  h ϑ∗ T∗i L∗ T∗i ∗=1,2 0≤i≤m with the left-hand expressions defined respectively by (4.4.19), (4.4.16) and (4.4.17) (without subscript “x ”) with J(q 0 B j , τ¯) defined by (4.4.26). Here C  depends on γ, m, maxi T∗i . Theorem 4.5.6. Let assumptions of Theorem 4.4.1 be fulfilled without (4.4.2). Let A be ξ-microhyperbolic on supp(q1 ) ∪ supp(q2 ) on energy levels τ¯∗ , ∗ = 1, 2. Let ri0 ∈ Sh,γ,γ,K (T ∗ X ) 34) be scalar symbols satisfying (4.5.35) ri0 = 1. Consider generalized Hamiltonian flow Ψt,¯τ∗ issued from γ-vicinity Ui of supp(ri0 ) ∩ (supp(q1 ) ∪ supp(q2 )). Assume that as |t| ≤ T∗i conditions (4.5.28), (4.5.29) and (4.5.31) are fulfilled with U := Ui . Then τ1 , τ¯2 ) ≺ (4.5.36) RW x (¯  h1−d  Jx (ri0 q10 B1j , τ¯∗ ) + Jx (ri0 q20 B1k , τ¯∗ ) + C  h , C0 (m + 1) T∗i ∗=1,2 0≤i≤m

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τ1 , τ¯2 ) ≺ (4.5.37) RW x,ϕ2 ,L2 (¯  h1−d ( Jx (ri0 q10 B1j , τ¯1 ) + Jx (ri0 q20 B1k , τ¯1 ) + C  h + C0 (m + 1) T∗i 0≤i≤m    h ) Jx (ri0 q10 B1j , τ¯2 ) + Jx (ri0 q20 B1k , τ¯2 ) + C  h ϑ2 L2 T∗i and (4.5.38) RW τ1 , τ¯2 ) ≺ x,ϕ1 ,L1 ,ϕ2 ,L2 (¯   h  h1−d  Jx (ri0 q10 B1j , τ¯∗ )+Jx (ri0 q20 B1k , τ¯∗ )+C  h ϑ∗ C0 (m+1) T∗i L∗ T∗i ∗=1,2 0≤i≤m with left-hand expressions defined respectively by (4.4.19), (4.4.16) and (4.4.17) with Jx (q 0 , τ¯) defined by (4.4.25). Proof. Consider scalar operators Ri with principal symbols ri0 such that  ∗ i Ri = I . Then plugging Q1 := Ri1 Q1 , Q2 := Q2 Ri2 and T = min(Ti1 , Ti2 ) we see that now we need to analyze not Nxi (τ  , τ  ) but Nxi1 i2 (τ  , τ  ) and the problem is that we have now Ti1 and Ti2 . We do not indicate which of two energy levels is considered because due to (4.4.12) they are treated independently. Still as Q1 = Q2∗ , j = k and i1 = i2 = i we using Lemma 4.2.7 estimate Nxii (τ  , τ  ) as before:  h  −d Nxii (τ  , τ  ) ≺ C0 Jx (ri0 q10 B1j , τ¯) · |τ  − τ  | + h + ... Ti where dots denote     C  |τ  − τ  | + h · |τ  − τ¯| + |τ  − τ¯| + h and therefore in the general case  h  −d h + Nxi1 i2 (τ  , τ  ) ≺C0 Jx (ri01 q10 B1j , τ¯) · |τ  − τ  | + T i1  h  −d h + ... C0 Jx (ri02 q20 B1k , τ¯) · |τ  − τ  | + T i2

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which again leads to Tauberian and Weyl errors C0

h1−d  h  0 0 j h1−d  h  0 0 k J(ri1 q1 B1 , τ¯) + C0 J(ri2 q2 B1 , τ¯) + C  h2−d . ϑ ϑ T i1 LTi1 T i2 LTi2

Now summation with respect i1 , i2 delivers announced estimates.  1 Note that if r 0 = ( i |ri0 |2 ) 2 ≤ 1 (which is easy to satisfy in the framework of (4.5.35)) then Jx (ri0 q0 , τ¯) = Jx (r 0 q0 , τ¯) ≤ Jx (q0 , τ¯). (4.5.39) i

4.5.3

Sharper Remainder Estimates

To establish truly sharper remainder estimates we need to work simultaneously in two directions: to increase T up to h−δ (depending on assumptions) and to decrease γ. While it is sufficient to decrease γ to hδ we will go here 1  further, up to γ = h 2 −δ where δ  and δ are related. Analysis as |t| ≤ T0 We need to reexamine first results which are due to propagation. Proposition 4.5.7. Let A satisfy all the assumptions of Theorem 4.4.1. Let q1 , q2 ∈ Sh,ρ,γ,K (T ∗ X , H, bH) with ργ ≥ h1−δ . Then (i) If A is ξ-microhyperbolic on supp(q1 ) on the energy level τ then   Ft→h−1 τ χT (t)Γx Q1x B1j uB1k tQ2y ≡ 0 (4.5.40) as h1−δ ρ−1 ≤ T ≤ T0 (4.5.41) where as usual χ ∈ C0∞ ([−1, − 13 ] ∪ [ 13 , 1]) and T0 is a small constant, j, k ∈ [0, l]. (ii) If ρ = γ and A is microhyperbolic on supp(q1 ) on the energy level τ then instead   (4.5.42) Ft→h−1 τ χT (t)Γ Q1x B1j uB1k tQ2y ≡ 0 as h1−δ ρ−1 ≤ T ≤ T0 . (4.5.43)

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Proof. Let A be ξ-microhyperbolic on supp q¯1 and q¯1 , q¯2 ∈ Sh,1,1,K (T ∗ X , H.bH). Let ψ ∈ C0∞ (Rd ), supported in {|x − y | ≤ } with small enough  > 0. Then ξ-microhyperbolicity and the standard propagation results imply that   Ft→h−1 τ χT (t) Q¯1x B1j uψT (x − y )B1k tQ¯2y ≡ 0 as T ≥ h1−δ . Applying Q1x and tQ2y from the left and right-respectively we derive (4.5.40) under (4.5.41) immediately. Statement (i) is proven. Finally, (i) follows from (i). Successive Approximations Let us look at successive approximation method of Section 4.3 with q2 ∈ Sh,ρ,γ,K (T ∗ X , H, bH). The only difference appears as (xj − yj ) commutes with Q2 . In this case factor [xj − yj , Q2 ] of magnitude ρ−1 h appears (instead of h) and since (xj − yj ) may be accompanied by a single parametrix G ± or G¯± with the norm  Th−1 , we gain a factor Chρ−1 · Th−1 = CT ρ−1 in the estimate of the norm. Then for the method of successive approximations to 1 work we need in addition to condition T ≤ h 2 +δ also condition T ≤ ρhδ ; so we need  1  (4.5.44) T ≤ min h 2 +δ , ρhδ . Under this condition we get asymptotic expansions    (4.5.45) κnx (τ )h−d+n+1 Ft→h−1 τ χ¯T (t)Γx Q1x B1j uB1k tQ2y ∼ n≥0

and (4.5.46)

   κn (τ )h−d+n+1 Ft→h−1 τ χ¯T (t)Γ Q1x B1j uB1k tQ2y ∼ n≥0

Actually we get a similar asymptotic expansion (with Kn (x, y , τ )) even without Γx . However, we want to increase T and we can do it only if we apply Γx . In this case we get expansion for T = T0 (where T0 is again a small constant) provided upper bound (4.5.44) to T is larger than the lower bound (4.5.41); this condition is equivalent to (4.5.47)

ρ ≥ h 2 −δ 1

where we adjust an arbitrarily small exponent δ > 0 if needed. Thus we arrive to

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Proposition 4.5.8. (i) In the framework of Proposition 4.5.7(i) and (4.5.47) asymptotic expansion (4.5.45) holds with T = T0 . (ii) In the framework of Proposition 4.5.7(ii) asymptotic expansion (4.5.46) holds with T = T0 . Further, one can prove easily Proposition 4.5.9. In the framework of Proposition 4.5.8(i) and (ii) respectively (4.5.48) and (4.5.49)

||∂τp κnx || ≤ Cnp ρ−n−p γ −n ||∂τp κn || ≤ Cnp γ −2n−p

Increasing T Now the long-range propagation results of Sections 2.4 and 3.5 allow us to increase T with negligible error in (4.5.45), (4.5.46) under no-loop and 1 no-periodicity conditions (4.5.29), (4.5.30) respectively where γ ≥ h 2 −δ now is a small parameter. Those are assumptions of Theorem 2.4.17 if there is no boundary or Theorem 3.5.3 otherwise. Let us mention that due to the nature of these theorems we must assume that ρ = γ even if we consider asymptotics without spatial averaging. Tauberian eEstimates Now due to Proposition 4.5.9 estimates (4.4.23), (4.5.25) are replaced by the same estimates but with C  h2−d replaced by C  h2−d γ −2 . Then the right-hand expression of (4.5.32) is replaced by   h1−d C0 J(q10 B1j , τ ) + C  hγ −2 T which due to Proposition 4.5.9 does not exceed   h1−d . C0 J(q10 B1j , τ¯) + C  γ −1 |τ − τ¯| + C  hγ −2 T Then the right-hand expression of (4.5.33) is replaced by    h C0 J(q10 B1j , τ¯) + C  γ −1 |τ  − τ¯| + C  γ −1 |τ  − τ¯| + C  hγ −2 · |τ  − τ  | + T

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

368

as Q2 = Q1∗ , j = k which yields the similar estimate   C0 J(q10 B1j , τ¯) + J(q20 B1k , τ¯) + C  γ −1 |τ  − τ¯| + C  γ −1 |τ  − τ¯| + C  hγ −2 × 

|τ  − τ  | +

h T

in the general case. Therefore in the end of the day we arrive to the remainder estimate C0

(4.5.50)

 h1−d  h  0 j ϑ J(q1 B1 , τ¯) + J(q20 B1k , τ¯) + C  hγ −2 T LT

which is obtained from the previous one by transformation |τ  − τ¯| → hT −1 , |τ  − τ¯| → hT −1 . First we get estimate (4.5.50) for Tauberian and then for Weyl errors. Similar estimates holds without spatial averaging. So, the only difference in comparison with the earlier estimates is the replacement of the term C  h by C  hγ −2 where C  does not depend on γ, T . The synthesis stage repeats one for γ  1. Remark 4.5.10. (i) As q1 , q2 ∈ Sh,γ,γ,K we may need to take more terms in the Weyl expansion than we used to have because junior terms are larger; however on the synthesis stage we return to our original q1 , q2 ∈ Sh,K and in the Weyl expansions for them we can drop all these extra terms. (ii) Probably one can improve term C  hγ −2 to C  hγ −1 but this term seems to be rather superficial. Finally, Results Theorem 4.5.11. Let assumptions of Theorem 4.4.1 be fulfilled with (4.4.2) replaced by (4.4.18). Let A be microhyperbolic on supp(q1 ) ∪ supp(q2 ) on the energy levels τ¯∗ , ∗ = 1, 2. Let ri0 ∈ Sh,γ,γ,K (T ∗ X ) 35) be scalar symbols satisfying (4.5.11). Consider generalized Hamiltonian flow Ψt,¯τ∗ issued from γ-vicinity Ui of supp(ri0 ) ∩ supp(q1 ) ∩ supp(q2 ). Assume that as ςt > 0 with ς := ς∗i 32) and |t| ≤ T∗i conditions (4.5.28), (4.5.30) and (4.5.31) are fulfilled and moreover 35)

With h 2 −δ ≤ γ ≤ 1. 1

4.5. SHARPER ASYMPTOTICS

369

as i = 1, ... , m along Ψt,τ (U ) conditions of Theorem 2.4.17 or Theorem 3.5.3 are fulfilled with U := Ui . Then estimates (4.5.32)–(4.5.34) hold with term C  h in the factor replaced by C  hγ −2 with the left-hand expressions defined respectively by (4.4.19), (4.4.16) and (4.4.17) (without x ) with J(q 0 B j , τ¯) defined by (4.4.26). Here C  does not depend on γ, T∗i . Theorem 4.5.12. Let all assumptions of Theorem 4.4.1 be fulfilled except may be (4.4.2). Let A be ξ-microhyperbolic on supp(q1 ) ∪ supp(q2 ) on the energy levels τ¯∗ , ∗ = 1, 2. Let ri0 ∈ Sh,γ,γ,K (T ∗ X ) be scalar symbols satisfying (4.5.35). Consider generalized Hamiltonian flow Ψt,¯τ∗ issued from γ-vicinity Ui of supp(ri0 ) ∩ (supp(q1 ) ∪ supp(q2 )). Assume that as |t| ≤ T∗i conditions (4.5.28), (4.5.29) and (4.5.31) are fulfilled and moreover as i = 1, ... , m along Ψt,τ (U ) conditions of Theorem 2.4.17 or Theorem 3.5.3 are fulfilled with U := Ui . Then estimates (4.5.36)–(4.5.38) hold with term C  h in the factor replaced by C  hγ −2 with the left-hand expressions defined respectively by (4.4.19), (4.4.16) and (4.4.17) (without x ) with J(q 0 B j , τ¯) defined by (4.4.26). Here C  does not depend on γ, T∗i .

4.5.4

Discussion

Local Results We already achieved rather general results as far as O(h1−d ) remainder estimate is concerned. Let A(x, D) be an elliptic operator on the closed manifold and we are interested in its spectral asymptotics for a large spectral parameter λ  1. So, we are looking at its spectrum in (0, λ) 36) . Alternatively we can consider operator λ−1 A(x, D) and look at its spectrum at (0, 1). Further, we can rewrite the latter operator as a(x, hD, h) where h = λ−1/m , m is an order of A(x, D). Obviously the principal symbol a(x, ξ, 0) of h-differential operator a(x, hD, h) and the principal symbol Am (x, ξ) of the differential operator A(x, D) coincide. One can see easily that 0 is a 36)

Negative eigenvalues are covered in the same manner.

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CHAPTER 4. GENERAL THEORY IN THE INTERIOR

lacunar value unless we are looking at ξ = 0 37) . All the details are left for Sections 11.2 and 11.3. On the other hand, in this case a(x, hD, 0) is microhyperbolic on the energy level 1 in the direction  = (ξ , 0) = (ξ, 0). Therefore asymptotics (4.5.51)

e(x, x, 0, λ) = κ0 (x)λd/m + O(λ(d−1)/m )

as λ → +∞

holds where we used the trick described in 37) . Also we get a host of asymptotics for spectral mollifications. These results hold for matrix operators and for operators acting on the Hermitean (or even Hilbert) sections of the bundles. All details are left for Chapter 11 and for elliptic operators on manifolds with boundaries as long as x is disjoint from the boundary. Simple Improvements However the completely local nature of the above results (where the only global condition is the self-adjointness of A) is lost if we want marginally improve the remainder estimate. Let us start from o(λ(d−1)/m ). Then we need to consider Hamiltonian dynamics Ψt on Στ = {τ ∈ Spec a(x, ξ)}; the concrete value of the energy level > 0 here is of no importance due to homogeneity. As Am (x, ξ) is the scalar symbol this is the ordinary Hamiltonian dynamics of it; further, if Spec Am (x, ξ) consists of eigenvalues ρ1 (x, ξ), ... , ρr (x, ξ) of the constant multiplicities we get a bunch of dynamics generated by these eigenvalues; finally, in the general case this is the generalized Hamiltonian dynamics as described in Chapter 2. Then the result depends strongly on what we are looking for: if we want remainder estimate o(λ(d−1)/m ) in (4.5.51) upgraded to (4.5.52) e(x, x, 0, λ) = κ0 (x)λd/m + κ1 (x)λ(d−1)/m + o(λ(d−1)/m )

as λ → +∞

we need to fight bicharacteristic loops as on Figure 4.2(a), if we want this remainder estimate in To avoid this little problem we can consider spectrum of A(x, D) in ( 12 λ, λ) and thus the spectrum of a(x, hD, h) in ( 12 , 1) and then plug 2−n λ instead of λ and finally to sum what we got with respect to n = 0, ... , "log2 λ#. 37)

4.5. SHARPER ASYMPTOTICS

 (4.5.53)

e(x, x, 0, λ)ψ(x) dx =

371

   κ0 (x)λd/m + κ1 (x)λ(d−1)/m ψ(x) dx+ o(λ(d−1)/m )

as λ → +∞

we need to fight periodic bicharacteristics as on Figure 4.2(b). Recall that the bicharacteristic loops and periodic bicharacteristics are defined by (4.5.24), (4.5.25). Let Π denote the set of all periodic points (x, ξ) and ΠT denote the set of periodic points with periods t ≤ T . The latter set is closed, the former is the union of the sequence of closed sets. For a sake of simplicity we assume that dim H < ∞. Assume that (4.5.54)

μτ (Π) = 0.

Then μτ (ΠT ) = 0 for any T . However ΠT is a closed set38) . Then if we consider its ε-vicinity ΠT ,ε (4.5.55)

μτ (ΠT ,ε ) → +0

as ε → +0.

Applying Theorem 4.5.5 with U0 = ΠT ,ε we arrive to asymptotics (4.5.53) with the remainder   (4.5.56) C T −1 + μτ (ΠT ,ε ) λ(d−1)/m + CT ,ε λ(d−2)/m with arbitrary T , ε and thus (4.5.56) is actually o(λ(d−2)/m ) due to (4.5.55). Therefore (4.5.57) Asymptotics (4.5.53) holds provided assumption (4.5.54) is fulfilled. On the other hand, without assumption (4.5.54) asymptotics (4.5.53) could fail as for Laplace-Beltrami operator on Sd or on other manifolds with all periodic bicharacteristics. We will consider such operators in Section 6.2. Meanwhile the same arguments and Theorem 4.5.6 show that asymptotics (4.5.52) holds provided (4.5.58)

μτ ,x (Λ) = 0.

38) Therefore due to μτ (ΠT ) = 0 it is of measure 0 and thus Π is of the first Baire category.

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where Λ is the set of loop points (x, ξ). For example, on the surface of ellipsoid in R3 with the semi-axis (a1 , a2 , a3 ) condition (4.5.58) is fulfilled if x is not an umbilic point39) . Similarly, considering an ellipse we arrive to asymptotic (4.5.52) in all interior points which are not focal (we need to consider billiards but Chapter 3 provides all the necessary tools). The further analysis can go in two directions: sharper reminder estimates and more general operators. Sharper Remainder Estimates In this case we need to apply the long-term propagation which involves an analysis how fast DΨt grows as t → ∞ and how fast μτ (ΠT ,ε ) tends to 0 as ε → +0. There are two cases: Example 4.5.13 (Exponential case). It is the case when one can prove that (4.5.59) and (4.5.60)

|DΨt | ≤ ce c|t| μτ (ΠT ,ε ) ≤ CT −1

as ε ≤ c −1 e −cT .

Then  λ one can derive (4.5.53) with the remainder estimate  taking T =  log O λ(d−1)/m (log λ)−1 . This happens f.e. on the manifolds on of the negative sectional curvature or on 2-dimensional manifolds without conjugate points P. Be´rard [1]. Similarly, under condition (4.5.60) replaced by as ε ≤ c −1 e −cT .   one can derive (4.5.52) with the remainder estimate O λ(d−1)/m (log λ)−1 .

(4.5.61)

μτ ,x (ΛT ,ε ) ≤ CT −1

Example 4.5.14 (Power case). It is the case when one can prove that (4.5.62) and (4.5.63)

|DΨt | ≤ c(|t| + 1)c μτ (ΠT ,ε ) ≤ CT −δ

as ε ≤ c −1 T −c .

39) There are four of such points located at x2 = 0 as a1 > a2 > a3 and there are two of them at (0, 0, ±a3 ) if a1 = a2 = a3 .

4.5. SHARPER ASYMPTOTICS

373



Then taking T = λδ one can derive (4.5.53) with the remainder estimate  (d−1−δ  )/m with unspecified small exponent δ  > 0. O λ This happens for example in some Euclidean domains (ellipse, ball, domain between two confocal ellipses or between two spheres with the same center), on the surface of 3-dimensional ellipsoids which are not spheres. In this examples dynamics is completely integrable. It happens also in Euclidean polyhedra which is not completely integrable case. Similarly, under condition (4.5.63) replaced by as ε ≤ c −1 T −c .    one can derive (4.5.52) with the remainder estimate O λ(d−1−δ )/m . This happens for example in some Euclidean domains (ellipse, ball, domain between two confocal ellipses or between two spheres with the same center), on the surface of 3-dimensional ellipsoids which are not spheres, as long as we keep away from some special points: (4.5.64)

(4.5.65)

μτ ,x (ΛT ,ε ) ≤ CT −1

dist(x, L) ≥ λ−δ



where L is the set of umbilical points for the surface of the ellipsoid, L is the set of foci for elliptical domain, L is the center for a ball. All details are left for Chapter 11. Remark 4.5.15. Bringing polygons and polyhedra into consideration brings also the question what to do with billiards which hit the “bad” points which are in this case corners or edges. One can decide that they branch here40) or simply dynamics breaks. In the second case we define dynamics not on the whole Στ but on Στ \ Ξ where Ξ is the set of points for which global dynamics does not exists. Then not ΠT or ΛT are not necessarily closed sets, but but ΞT , ΠT ∪ ΞT and ΛT ∪ ΞT are where ΞT is the set of points for which dynamics breaks for time t, |t| ≤ T . The second approach leaves us with the simple dynamics and in all above arguments we replace ΠT or ΛT by their unions with ΞT . This is the case for billiards even in the smooth domains due to appearance of the pathological billiards which either touch the boundary or make an infinite number of jumps in the finite time. 40) F.e. if we consider an incoming ray close to one which hits a vertex of the polygon the direction of the outgoing ray is defined by which side was encountered first, so for the ray which hits the vertex there will be two outgoing rays. For polyhedra there will be many rays leaving vertex.

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

374

Matrix Case and Branching Bicharacteristics There we are again with the case described in Remark 4.5.15: the billiards are branching when they hit the sets Θjk = {(x, ξ) : ρj = ρk } (j = k) where ρj are eigenvalues of Am (x, ξ) and we do not distinguish eigenvalues which coincide everywhere. However now the set Ξjk (points starting from which dynamics eventually hits the point of Θjk ) could be rather large. F.e. if λj are smooth and {λj , λk } =  0 then the set Ξjk has a positive measure (unless Θjk = ∅). Does it mean that we are doomed to consider branching bicharacteristics? Not necessarily. In the example just considered we know that the dynamics of ρj changes to those of ρk = 0 the singularity loses order by 12 (see Proposition 2.2.16). Consider more general case assuming that ρj are analytic. Then any given trajectory of ρj either belongs to Θjk completely or hits Θjk only in some locally finite number of points. However again due to analyticity μτ (Θjk ) = 0 and the first case could be ignored. Along trajectories of the second type the order of the singularity also decays by δ > 0 as the dynamics of ρj = 0 changes to those of ρk . This should allow us to ignore branching. ˆ j consisting of Further, if only Am (x, ξ) is analytic then there is a set Θ points in vicinity of which ρj fails to be analytic but this should be very ˆ j such that the generalized Hamiltonian small set and the set of points Ξ ˆ j should have measure 0. dynamics originated from them hits Θ Conjecture 4.5.16. (i) As Am (x, ξ) is analytic and there is no boundary, it is sufficient to consider only points of {ρj = τ } such that the Hamiltonian dynamics of ρj originated in these points remains in the domain where ρj is analytic and ignore branching. (ii) As there is a boundary only branching at the boundary needs to be addressed. We will discuss the affect of the branching on the boundary later.

4.6

Operators with Rough Coefficients

In this section we establish results similar to those we obtained earlier, for operators with symbols satisfying (2.3.4)–(2.3.6):

4.6. OPERATORS WITH ROUGH COEFFICIENTS

375

γ Am (x, ξ)|| ≤ c(m!|α|!|β|!)σ ρ−α γ −β ε−m (4.6.1) ||∂ξα ∂xβ ∂x,ξ

∀m, α, β : |α| + |β| + 2m ≤ N ∀γ : |γ| ≤ 1 with N = −1 ,  = h/ε and (4.6.2)

min ρj γj ≥ ε = Ch| log h|σ j

σ ≥ 1.

Actually we need σ = 1. The “pedestrian way” is to consider operators satisfying γ Am (x, ξ)|| ≤ cmαβ ρ−α γ −β ε−m (4.6.3) ||∂ξα ∂xβ ∂x,ξ

∀m, α, β : |α| + |β| + 2m ≤ N ∀γ : |γ| ≤ 1 with ρ, γ satisfying (2.3.2): min ρj γj ≥ h1−δ .

(4.6.4)

j

We will be mainly interested in the case of ρj = 1 and γj = γ  1 briefly mentioning other cases too.

4.6.1

Discussion: Rationale

Actually we are interested in the operators with irregular coefficients. F.e. let us consider elliptic self-adjoint semiclassical (matrix) operator in the divergent form (4.6.5) A(x, hD) = (hD)α aαβ (x)(hD)β α,β:|α|≤m,|β|≤m

on compact closed manifold X , defined by quadratic form (then it must be of even order 2m and semibounded). We can also consider first order (matrix) operator (4.6.6)

A(x, hD) =

 1  hDj aj (x) + aj (x)hDj + a0 (x). 2 j

We assume that the coefficients belong to Cνl (X , .) where (4.6.7) ν ∈ C(0, ∞) is a positive as t ≥ 0 function, ν(t)t −l and ν(t)t −l−1 are respectively non-decreasing and non-increasing functions on (0, ∞), l ∈ Z+

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and Definition 4.6.1. Cνl (X ) is a subspace of Cl (X ) consisting of functions with l-th order derivatives continuous with continuity modulus (4.6.8)

|∂ α f (x) − ∂ α a(y )| ≤ C ν(|x − y |) · |x − y |−l ,

as |α| = l.

Remark 4.6.2. Surely this definition would hold for vector- or matrix- valued functions and one can generalize it easily to sections of vector bundles. So we are looking for the dimension NA (τ1 , τ2 ) of the spectral projector EA (τ1 , τ2 ) of operator A corresponding to interval [τ1 , τ2 ) with τ1 < τ2 . Since we cannot apply microlocal analysis to such operators we approximate them by operators with symbols satisfying (4.6.1)–(4.6.2). The most obvious approach is to bracket such operators. Namely in the case of operator A semibounded from below we can assume without any loss of the generality that τ1 = −∞ and τ2 = 0 and we are interested in N− A := NA (−∞, 0). Let us bracket A between A− and A+ : (4.6.9)

A− ≤ A ≤ A+ .

Then (4.6.10)

− − N− A+ ≤ NA ≤ NA− .

In the case of non-semibounded A we can either modify this method or just to notice that (4.6.11)

2 1   1 NA (τ1 , τ2 ) = N− A − (τ1 + τ2 ) − (τ1 − τ2 )2 . 2 4

We want A± satisfy (4.6.1)–(4.6.4). If we manage to derive remainder estimates for such operators (4.6.12)−

W− |N− A− − N A− | ≤ R − ,

(4.6.12)+

W− |N− A+ − N A + | ≤ R +

and (4.6.13)−

− W−  |NW A− − NA | ≤ R− ,

(4.6.13)+

− W−  |NW A+ − NA | ≤ R+ ,

4.6. OPERATORS WITH ROUGH COEFFICIENTS

377

− are Weyl approximations for N− where NW A A etc, then

(4.6.14)

W−   |N− A − NA | ≤ max(R− + R− , R+ + R+ ).

So, remainder estimate for operator with irregular coefficients consists of two parts: remainder estimates R± for approximate operators and approximation error R± . While R± is the subject of this section, R± is much more simple object. Let us approximate coefficients aα by aαε according to Proposition 4.A.2. As A is defined by (4.6.5) or (4.6.6) let Aε be defined in the same way but with aα replaced by aαε . Remark 4.6.3. From the heuristic point of view would like to take A± := Aε ± C ν(ε) but it is not exactly possible since A − Aε is an unbounded operator. However we can fix it. Namely, assume that A given by (4.6.5) is a semi-bounded from below elliptic operator A ≥ −C0 + 1 and τ1 = −∞. Then the following sequence     A → A + C0 → Aε ± C ν(ε) · 1 ± C1 ν(ε) →     A± := Aε ± +C ν(ε) · 1 ± C1 ν(ε) − C0 takes care of this issue. One can prove then easily that Lemma 4.6.4. Let A be ξ-microhyperbolic on energy levels 0. Then as ε ≥ h an approximation error does not exceed Ch−d ν(ε). So we are interested to take ε as small as possible. Since our symbol is analytic with respect to ξ we can take the smallest ε satisfying logarithmic uncertainty principle: (4.6.15)

ε = Ch| log h|.

So, in this case an approximation error is Ch−d ν(Ch| log h|) and it is O(h1−d ) or o(h1−d ) provided (4.6.16)

ν(t) = O(t| log t|−1 ),

(4.6.17)

ν(t) = o(t| log t|−1 )

respectively.

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We will prove in this section that under assumption (4.6.16) the remainder estimate is O(h1−d ). Moreover under appropriate condition to the dynamic system we will prove in this section that under assumption (4.6.17) remainder estimate is o(h1−d ). Further we will prove that under assumption ν(t) ≥ t| log t|−1 for all t ∈ (0, 1) the remainder estimate is O h−d ν(Ch| log h|) . Thus spectral asymptotics for operators with symbols satisfying (4.6.1)– (4.6.4) are instrumental for spectral asymptotics for operators with Cνl coefficients. Remark 4.6.5. (i) We will assume that ρ1 = ... = ρd = ρ and γ1 = ... = γd = γ. (ii) We can see that the most important case is ρ = 1,

(4.6.18)

γ = ε.

Actually one can reduce more general case ρ ≥ γ to it by rescaling. So we assume that (4.6.18) holds. (iii) In Section 4.7 we replace the bracketing method by perturbation method which allows us to improve remainder estimates in asymptotics for Riesz means N− ϕ (as ϑ is sufficiently large) and consider asymptotics of Nx (τ1 , τ2 ) and N− . x,ϕ

4.6.2

Estimates

The simplest thing to try would be to scale x → xε−1 , t → tε−1 , ξ → ξε−1 and h → hε−1 . However then scaling back we have estimate (4.6.19)

R≺C

 h 1−d −d ε  C ε−1 h1−d ε

where R denotes both Tauberian RT and Weyl RW errors. Thus we got an extra factor ε−1 which is really bad. For errors in averaged expressions we −1−s would get where an extra factor ε−s comes from   even  worse factor ε ϑ h/Lε : ϑ h/L . These bad factors are no surprise: actually we use usual Tauberian estimate but with time T  ε instead of T  1. Fortunately we have Theorem 2.3.5(vi) which implies that

4.6. OPERATORS WITH ROUGH COEFFICIENTS

379

Proposition 4.6.6. Let ε ≥ Ch| log h|. Then under ξ-microhyperbolicity condition on the energy level 0 estimate (2.1.59) remains true as T ∈ [h, T0 ] with small constant T0 and then (4.6.20)

  ||Ft→h−1 τ χ¯T (t)Γx Q1x u tQ2y || ≤ Ch1−d

∀τ : |τ | ≤ 0

Then our standard Tauberian technique implies the standard estimate h RTϕ ≺ Ch1−d ϑ L

(4.6.21)

where ϑ is a continuity modulus of ϕ.

4.6.3

Successive Approximations and Calculations

Now we need to pass from Tauberian expression  τ    T Ft→h−1 τ χ¯T (t)Γx Q1x u(x, y , t) tQ2y dτ (4.6.22) Nx = −∞

to Weyl expression. As in Sections 4.3–4.5 Q1 and Q2 are fixed operators, χ¯ is a fixed C0∞ ([−1, 1]) function, χ¯ = 1 on [− 23 , 23 ] and T is a small enough constant. Remark 4.6.7. In order to deal with operators of (4.6.1)–(4.6.4) type we assume that Qj ∈ Ψh,an . Since we know from (2.1.59) that Ft→h−1 τ χT (t)Γx Q1x u tQ2y is negligible as T ∈ [h1−δ , T0 ] we conclude that (4.6.23) In the Tauberian expression we can take any T ∈ [h1−δ , T0 ]. Remark 4.6.8. (i) Minor improvement of the propagation technique implies that actually one can take any T ∈ [Ch| log h|, T0 ]; we will not need this improvement for a very long time and we will make it much later. (ii) Improvements connected with our ability to chose T0 as an arbitrarily large constant will be discussed later.

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

380

We will apply successive approximation method. Almost as in Section 4.3 we introduce operators A¯ = A(y , hDx ) and R = A − A¯ and arrive to the same formula (4.3.10) : (4.3.10) u ± tQ2y ≡ ∓ih



(G¯± R)n G¯± K(x, y )δ(t)

0≤n≤N−1

∓ ih(G¯± R)n G ± K(x, y )δ(t). Plugging it into (4.6.23) we conclude that (4.6.24) n-th term (with n = 0, 1, ...) in the resulting formula does not exceed (4.6.25)

Ch−d T

 T 2 n h

where we can take T = h1−δ with arbitrarily small exponent δ > 0 41) . (4.6.26) In particular, as we are looking for RW it is sufficient to take 2 terms and as we are looking for RW ϕ with L ≤ 1 it is sufficient to take 2 + "s# terms provided ϑ(z) ≥ 0 z s ∀z ∈ (0, 1). Let us prove Proposition 4.6.9. Under ξ-microhyperbolicity condition n-th term does not exceed (4.6.24) with T = h, i.e. Ch1−d+n . Proof. It is sufficient to consider n-th term with χ¯T (t) replaced by χT (t) with “new” T running from h to “old” T and to estimate this term by (4.6.25) multiplied by (hT −1 )M with sufficiently large exponent M; here and below h ≤ T ≤ 0 . Scaling x → (x − y )T −1 , t → tT −1 , h → hT −1 we conclude that it is sufficient to consider T = 0 only. Let us recall that n-th term could be rewritten as a “sandwich” un with operators G¯± interlaced with hpseudodifferential operators; the latter now are of ΨεT −1 ,1,hT −1 ,an class. 41)

Actually according to Remark 4.6.7 we can take T = Ch| log h|.

4.6. OPERATORS WITH ROUGH COEFFICIENTS

381

However Theorem 2.3.2 and ξ-microhyperbolicity condition on the energy level 0 which is preserved under rescaling imply that (4.6.27)

WF(Q1x un± tQ2y ) ∩ {|τ | ≤ } ∩ { ≤ ±t ≤ 0 } ⊂ {±ξ , y − x ≥ ±0 t}

where (0, ξ ) is the direction of microhyperbolicity and therefore   (4.6.28) WF Γx (Q1x un± tQ2y ) ∩ {|τ | ≤ } ∩ { ≤ ±t ≤ 0 } = ∅ and the required estimate is proven. Thus, if we are interested just in O(h1−d ) estimate i.e. we are looking for s = 0, we are done: we need to consider just one term, i.e. we can take N = 1 in (4.3.10) ; to be able to take A0 (y , hDx ) instead of A(y , hDx ) we will need to make assumption that their difference is O(h) (see assumption (4.6.33) below with l = 1). If we are interested in estimate better than O(h1−d ) but not better than O(h2−d ) we should take N = 2 and consider a new term ∓ihG¯± R G¯± K(x, y )δ(t),

(4.6.29) plug here (4.6.30)

R¯ =

1≤|α|+k≤l

1 (x − y )α Ak(α) (y , hDx )hk α!

with l = 1 instead of R, getting again the standard term (4.6.31)

∓ ihG¯± R¯ G¯± K(x, y )δ(t),

(and plugging it to expression we need to calculate) and find estimate for the error due to (4.6.32)

∓ ihG¯± R  G¯± K(x, y )δ(t),

¯ For remainder estimate better than O(h1−d ) we will need with R  = R − R. more regularity assumptions than (4.6.1)–(2.3.6). Further, if we are interested in the remainder estimate O(hl−d ) with l > 1 we need to assume more regularity conditions: namely that (4.6.1) holds for all γ with |γ| ≤ m and modify (4.6.2) in the similar way to keep Ak(α)

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

382

with k + |α| ≤ l bounded. Then expression due to (4.6.32) will be obviously O(hl−d ). We need to consider higher order successive approximation terms in (4.3.10) , replace R by R¯ in each of them and estimate an error. However these terms will be easier and we will be able even to shorten R¯ in them. Finally, for remainder estimates between O(hl−d ) and O(hl+1−d ) we will need intermediate regularity conditions. So, let us assume that (4.6.33) ||Ai+k(β+γ) (x, ξ)|| ≤ c(i!|α|!|β|!)ε−|β|−i (α)

∀i, α, β : |α| + |β| + 2i ≤ N ∀γ, k : |γ| + k ≤ l and (α)

(α)

(4.6.34) ||Ai+k(β+γ) (x, ξ) − Ai+k(β+γ) (y , ξ)|| ≤ c(i!|α|!|β|!)ε−|β|−i ν(|x − y |)|x − y |−l ∀i, α, β : |α| + |β| + 2i ≤ N ∀γ, k : |γ| + k = m with N = −1 ,  = h/ε where ν satisfies (4.6.7). Proposition 4.6.10. Let ξ-microhyperbolicity condition, (4.6.33) and (4.6.34) be fulfilled with ν satisfying (4.6.7). Then replacing in (4.3.10) R by R¯ defined by (4.6.30) leads to O h−d ν(h) error. Proof. So, we need to consider sandwiches G¯± R1 G¯± R2 G¯± · · · Rn−1 G¯± K(x, y )δ(t) ¯ with n factors G¯± and (n − 1) factors Rk where each Rk is either R¯ or (R − R) and there is at least one latter factor. Then acting as before we can estimate the contribution of each sandwich by Chi (h/T )−M with i = n1 + ln2 and M ≤ Ci where n1 , n2 is the number of factors of each type, n1 + n2 = n and n2 ≥ 1. Then the contribution of each sandwich does not exceed Ch−d+i−δ while ν(h) ≥ hl+1 . Obviously i ≥ l + 2 unless n ≤ 3 and the contribution of the sandwich is properly estimated. Further, we can apply the same procedure as in the proof of Proposition 4.6.9: let us pick up T = h and then the contribution of sandwich does not exceed Ch−d+l ; then let us plug χT (t) instead of χ¯T (t) with “new” T running from h to “old” T and basing on Theorem 2.3.2 estimate this

4.6. OPERATORS WITH ROUGH COEFFICIENTS

383



contribution by Ch−d+l (h/T)M −M where we can take M  arbitrarily large, say M  = M + 1 and then summation with respect to partition results in Ch−d+l . Obviously i ≥ l + 1 unless n = 2 and the contribution of the sandwich is properly estimated. So, only case n = 2 needs a special consideration and only in the case when ν(h)  hl fails. Then we have “a simple sandwich” ¯ G¯± K(x, y )δ(t); G¯± (R − R) let us insert factor ψr in the middle: ¯ G¯± K(x, y )δ(t); G¯± ψr (R − R)   where ψr = ψ (x − y )r −1 and we consider three cases: (i) r = CT , ψ ∈ S0h,an is supported in {|x − y | ≤ 1}, T = h. Then since norm ¯ does not exceed C ν(r ) 42) and and as T  h we estimate the of ψr (R − R)   contribution of such modified sandwich by Ch−d ν(r ) which is O h−d ν(h) . (ii) r = CT , ψ ∈ S0h,an is supported in {|x − y | ≤ 1, |ξ| ≤ c}, T ≥ h and χ¯ is replace by χ. Then we estimate the contribution of such modified sandwich by Ch−d ν(r )

 h M−M   h M  −M  h M−M  −l−1  Ch−d ν(T )  Ch−d ν(h) T T T

and summation with respect to t-partition results in Ch−d ν(h) since M  is arbitrarily large. I remind that we apply Theorem 2.3.2 after rescaling x → (x − y )T −1 , t → tT −1 and h → hT −1 . We again used estimate of ¯ with footnote 42) . ψr (R − R) (iii) r ≥ CT , ψ ∈ S0h,an is supported in {|x − y | ≤ 1, |ξ| ≤ c}, T ≥ h. Then we estimate the contribution of such modified sandwich by  h M−M   h −M  h M   h M−M  −l−1  Ch−d ν(r )  Ch−d ν(h) T T R r where we use the same procedure as before but we apply Theorem 2.3.2 after rescaling x → (x − y )r −1 , t → tr −1 and h → hr −1 . We again used ¯ with footnote 42) . Again summation with respect to r estimate of ψr (R − R) results in Ch−d ν(h) since M  is arbitrarily large. Ch−d ν(r )

42) Actually it may be true only if we multiply this operator by Q¯ = q¯(hD) with compactly supported symbol but we can insert such factor with the negligible error.

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CHAPTER 4. GENERAL THEORY IN THE INTERIOR

  Thus we constructed “Weyl approximation” with O h−d ν(h) error for Tauberian expressions (4.6.23) and also for     (4.6.35) Nx,ϕ,L = ϕL (τ ) Ft→h−1 τ χ¯T (t)Γx Q1x u(x, y , t) tQ2y dτ (with small constant T ). Let us recall that according to Tauberian arguments NTx (0) defined by (4.6.22) and Nx,ϕ,L (0) defined by (4.6.35) approximate Nx (0) and     (4.6.36) Nx,ϕ,L (0) = ϕL (τ )dτ Γx Q1x e(., ., 0) tQ2y with O(h1−d ) and O(h1−d ϑ(h/L)) errors. So, we arrive to Corollary 4.6.11. (i) For Nx Weyl approximation is as good as the Tauberian one provided ν(t) = O(t) as t → +0 (which we assumed). (ii) For Nx,ϕ,L Weylapproximation is as good as the Tauberian one as L ≤ 1 and ν(t) = O tϑ(t) as t → +0. (iii) In both cases the difference between Weyl and Tauberian approximations is “o” in comparison  withthe remainder estimate provided ν(t) = o(t) as t → +0 or ν(t) = o tϑ(t) as t → +0 respectively. Proof. These statements are clearly true for Nx and for Nx,ϕ,L with L  1. Then for Nx,ϕ,L with L  1 we can apply arguments of Remark 4.4.10.

4.6.4

Main Theorems

Now we arrive immediately to the following Theorem 4.6.12. Let A be a self-adjoint operator with the symbol satisfying (4.6.34) where ν satisfies (4.6.7) and l ≥ 1. Assumethat ξ-microhyperbolicity  condition on the energy level 0 at each point of πx supp(q1 ) ∪ supp(q2 ) . Then  h  1−d −1 RW + ν(h)h ≺ Ch (4.6.37) ϑ ; x,ϕ,L L in particular   1−d −1 (4.6.38) 1 + ν(h)h . ≺ Ch RW x

4.6. OPERATORS WITH ROUGH COEFFICIENTS

385

W Corollary 4.6.13. The same estimates hold for RW ϕ,L , R .

Now let us slightly improve remainder estimates using long term propagation. Theorem 4.6.14. Let conditions of Theorem 4.6.12 be fulfilled. Let ri0 ∈ Sh,γ,γ,K (T ∗ X ) 34) be scalar symbols satisfying (4.5.11). Consider generalized Hamiltonian flow Ψt,0 issued from γ-vicinity Ui of supp(ri0 ) ∩ supp(q1 ) ∩ supp(q2 ). Assume that as ςt > 0 with ς := ς∗i 32) and |t| ≤ Ti conditions (4.5.28), (4.5.30) and (4.5.31) are fulfilled with U := Ui . (i) Then  h1−d  0 0 J(ri q1 , 0) + J(ri0 q20 , 0) + C  h , Ti 0≤i≤m   h  h1−d  0 0 J(ri q1 , 0) + J(ri0 q20 , 0) + C  h ϑ ≺ C0 . Ti LTi 0≤i≤m

(4.6.39)

RT (0) ≺

(4.6.40)

RTϕ,L



C0

Here C  depends on γ, m, maxi Ti . (ii) Further, the same estimates with an extra term Ch−d ν(h) in the righthand expressions hold for RW (0) and RW ϕ,L . Theorem 4.6.15. Let conditions of Theorem 4.6.12 be fulfilled. Consider generalized Hamiltonian flow Ψt,¯τ∗ issued from γ-vicinity Ui of supp(ri0 ) ∩ (supp(q1 ) ∪ supp(q2 )). Assume that as |t| ≤ Ti conditions (4.5.28), (4.5.29) and (4.5.31) are fulfilled with U := Ui . (i) Then  h1−d  Jx (ri0 q10 , 0) + Jx (ri0 q20 , 0) + C  h , Ti 0≤i≤m   h  h1−d  . Jx (ri0 q10 , 0) + Jx (ri0 q20 , 0) + C  h ϑ ≺ C1 T LT i i 0≤i≤m

(4.6.41)

RTx (0) ≺

(4.6.42)

RTx,ϕ,L



C1

Here C1 = C0 (l + 1) and C  depends on γ, m, maxi Ti . (ii) Further, the same estimates with an extra term Ch−d ν(h) in the righthand expressions hold for RW (0) and RW ϕ,L .

386

4.6.5

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

Applications: Bracketing Method

At this moment we want to demonstrate technique rather than achieve the most general results. In particular we want to use only the simplest variational arguments. Therefore we assume that operator A defined by consider operator (4.6.5) is elliptic with the positively defined senior symbol: (4.6.43) with (4.6.44)

a0 (x, ξ)w , w  ≥ 0 |ξ|2m ||w ||2 a0 (x, ξ) :=



∀ξ ∈ Rd ∀w

.aαβ (x)ξ α+β

α:|α|+|β|=2m

Then provided aαβ ∈ C (as |α| = |β| = m) (4.6.45)

(Au, u) ≥ 0 h2m uHm − C0 u2

∀u ∈ H m

First of all, applying Theorem 4.6.12 and Corollary 4.6.13 we immediately arrive to Theorem 4.6.16. Let A be an partial differential operator in the divergent form (4.6.5) on compact closed manifold with the coefficients belonging to Cνl with ν satisfying (4.6.7) and l ≥ 1. Let (4.6.43) be fulfilled. Assume that ξ-microhyperbolicity condition on energy level 0 is fulfilled at each point. Let Q1 = Q2 = I , ψ1 = ψ2 = 1. Then  h  1−d | tr(RW + ν(ε)h−1 dim H (4.6.46) ϑ ϕ,L )| ≤ Ch L with ε = h| log h|; in particular (4.6.47)

  | tr(RW )| ≤ Ch1−d 1 + ν(ε)h−1 dim H.

Proof. Indeed, we can bracket A between two operators with the rough coefficients, different from those of A by O(ν(ε)). We leave easy details to the reader. Corollary 4.6.17. In the framework of Theorem 4.6.16 with fixed dim H < ∞ 1−d the usual remainder estimate | tr(RW ϑ(h/L) holds as L ≤ 1 proϕ,L )| ≤ Ch vided continuity modulus ν satisfies (4.6.48)

ν(ε)  ϑ(ε| log ε|−1 ) · ε| log ε|−1 ;

4.6. OPERATORS WITH ROUGH COEFFICIENTS

387

in particular, the usual remainder estimate | tr(RW )| ≤ Ch1−d holds provided ν(ε)  ε| log ε|−1 .

(4.6.49)

Now let us combine Theorem 4.6.14 with the approximation arguments. We will get an approximation error O(h−d ν(ε)) but there is a problem: in this theorem non-periodicity condition is supposed to be fulfilled with the generalized Hamiltonian flow generated by aε while it is natural to consider generalized Hamiltonian flow generated by the original symbol a. The good news are that (4.6.50) As ε → +0 the generalized bicharacteristics of (τ − aε ) converge to those of (τ − a) and therefore Ψt,a (U ) ⊃ lim Ψt,aε (U).

(4.6.51)

ε→+0

Thus non-periodicity condition (4.5.30) for Ψt,a implies the same condition for Ψt,aε with γ replaced by 12 γ and Theorem 4.6.14 implies Theorem 4.6.18. Let conditions of Theorem 4.6.16 be fulfilled. Consider generalized Hamiltonian flow Ψt,0 issued from γ-vicinity Ui of supp(ri0 ) ∩ supp(q1 ) ∩ supp(q2 ). Assume that as ςt > 0 with ς := ς∗i 32) and |t| ≤ Ti conditions (4.5.28), (4.5.30) and (4.5.31) are fulfilled with U := Ui . Then (4.6.52) RW (0) ≺

C0

0≤i≤m

 h1−d  0 0 J(ri q1 , 0) + J(ri0 q20 , 0) + C  h + C0 h−d ν(ε), Ti

(4.6.53) RTϕ,L ≺   h  h1−d  0 0 J(ri q1 , 0) + J(ri0 q20 , 0) + C  h ϑ C0 + C0 h−d ν(ε) T LT i i 0≤i≤m hold with ε = Ch| log h|.

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CHAPTER 4. GENERAL THEORY IN THE INTERIOR

Corollary 4.6.19. Assume that one can select Ti as arbitrarily large constants for i ≥ 1 and simultaneously μ0 (U0 ) to be arbitrarily small then   (4.6.54) RW = o h1−d ,  1−d  h  (4.6.55) , ϑ RW ϕ,L = o h L   provided ν(ε) = o(h1−d ) and ν(ε) = o h1−d ϑ(h/L) respectively (with ε = h| log h|) i.e.   (4.6.56) ν(ε) = o ε| log ε|−1 ,   ν(ε) = o ϑ(ε| log ε|−1 ) · ε| log ε|−1 . (4.6.57) Remark 4.6.20. (i) It is not clear if sets in both sides of (4.6.51) coincide; so may be non-periodicity assumption for a is more restrictive than for aε . (ii) Even in the scalar case when equation of the generalized bicharacteristics becomes dz = −∇# a(z) dt those could branch unless the right-hand expression satisfies Lipshitz condition i.e. l ≥ 2 and ν(ε) = O(ε2 ). Finally, let us consider less regular case of l = 0. Let us apply the same approximation as before. However we can apply Theorem 4.6.12 only after rescaling x → x/T , t → t/T with (4.6.58)

T = T (ε) = εν(ε)−1

which leads to the remainder estimates 1 (4.6.59) RTx ≺ Ch1−d = Ch1−d ε−1 ν(ε). T (ε) One can see easily that this is less then the approximation error Ch−d ν(ε) and therefore Theorem 4.6.21. Let conditions of Theorem 4.6.16 be fulfilled but with l = 0. Then (4.6.60)

−d . RW x ≺ C ν(ε)h

Since approximation error dominates, mollification with respect to the spectral parameter would not improve this estimate.

4.6. OPERATORS WITH ROUGH COEFFICIENTS

4.6.6

389

Final Remarks

The following problem may be easy, or not: Problem 4.6.22. (i) Replace assumption (4.6.43) by (4.6.61)

||a0 (x, ξ)w || ≥ 0 |ξ|2 m||w ||

∀ξ ∈ Rd ∀w .

Then as aαβ ∈ C (as |α| = |β| = m) (4.6.62) (1 + h2 |D|2 )−m/2 Au ≥ 0 (1 + h2 |D|2 )m/2 u − C0 u ∀u ∈ H m . (ii) Instead of operators (4.6.5) consider (4.6.63) A(x, hD) =

(hD)α aαβ (x)(hD)β

α,β:|α|≤m+1, |β|≤m+1, |α|+|β|≤2m+1

under assumption (4.6.61) with (4.6.64)

a0 (x, ξ) :=



aαβ ξ α+β .

α:|α|+|β|=2m+1

Then as aαβ ∈ C1 as |α| + |β| = 2m + 1 ¯ ¯ (4.6.65) (1 + h2 |D|2 )−m/2 Au ≥ 0 (1 + h2 |D|2 )m/2 u − C0 u

∀u ∈ H m¯ where here and below m ¯ = m for for operators (4.6.5) and m ¯ =m+ operators (4.6.63).

1 2

for

Remark 4.6.23. (i) We did not consider in this section operators with unbounded coefficients. Most likely that this two generalizations of the standard case do not mix well together. (ii) We did not consider in this section sharper remainder estimates. While they are possible the conditions of theorems (especially in applications) become too exotic.

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

390

Remark 4.6.24. Can we replace ξ-microhyperbolicity by microhyperbolicity retaining our results for NT , NTϕ,L , N W , NW ϕ,L ? The answer is positive but with a lot of caveats. (i) First of all, in Subsection 4.6.2 we need to replace T¯ = h1−δ by T¯ = h1−δ ε−1 . Really, as we go to ξ-microhyperbolic case applying metaplectic transformation, we get operator belonging to Ψh,ε,ε,an . Then shift  T is observable as microlocal uncertainty principle T · ε ≥ h1−δ is fulfilled; improvements as we apply logarithmic uncertainty principle are superficial. It looks really bad because then we should estimate expression (4.1.11)   Ft→h−1 τ χ¯T (t)Γ Q1x u tQ2y by Ch1−d ε−1 which would lead to remainder estimates with an extra factor ε−1 but actually we can repair this after calculations. 1

(ii) Still the method of the successive approximations requires T ≤ h 2 +δ and T¯ satisfies this if and only if ε ≥ h 2 −δ 1

(4.6.66)

where we change an arbitrarily small exponent δ > 0 if needed. Under this condition and microhyperbolicity one can prove easily that  −n−i  (4.6.67) |κ(i) ν(ε), 1 . n | ≤ C max ε Then as l ≥ 1 and ε satisfies (4.6.66) we can estimate (4.1.11) by Ch1−d and therefore no factor ε−1 actually appears. Note that we must assume that l ≥ 1; otherwise we cannot differentiate with respect to x in applications and microhyperbolicity does not make sense (while ξ-microhyperbolicity does not require such differentiation). (iii) Much larger ε given by (4.6.66) increases an approximation error and to achieve the proper remainder estimates we must assume that h−d ν(ε) ≤ h1−d ϑ(h) (as L = 1) which means now     ν(ε) = O ϑ ε2+2δ ε2+2δ ; (4.6.68) in particular, as ϑ = 1 we get (4.6.69)

  ν(ε) = O ε2+2δ .

Further to recover sharp remainder estimates we need to replace in these conditions “O” by “o”.

4.7. OPERATORS WITH IRREGULAR COEFFICIENTS

4.7 4.7.1

391

Operators with Irregular Coefficients Preliminary Analysis

Our goal is to derive spectral asymptotics for operators (4.6.5) and (4.6.63) satisfying ellipticity condition (4.6.61). For brevity of statements we consider (4.7.1) operator A of the type either (4.6.5) or (4.6.63), with aαβ ∈ Cl for all α, β and with aαβ ∈ Cl+1 as |α| + |β| = 2m + 1. Let m ¯ = 12 ordA. Proposition 4.7.1. Let us consider operator A of the type (4.7.1). (i) Then the following estimate holds ¯ ¯ Au ≤ C0 (1 + h2 |D|2 )(m+s)/2 u (4.7.2) (1 + h2 |D|2 )(−m+s)/2

∀u ∈ H m¯ ∀s : |s| ≤ l. (ii) If ellipticity condition (4.6.61) is fulfilled then ¯ Au ≥ (4.7.3) (1 + h2 |D|2 )(−m+s)/2 ¯ u − C0 (1 + h2 |D|2 )s u 0 (1 + h2 |D|2 )(m+s)/2 ¯ ∀s : |s| ≤ l. ∀u ∈ H m+s

Proof. Easy proof is left to the reader. Then in virtue of embedding theorem we immediately arrive to Corollary 4.7.2. In the framework of Proposition 4.7.1(ii) with (4.7.4)

m∗ := l + m − d/2 > 0

the following estimate holds (4.7.5)

|(hDx )α (hDy )β e(x, y , τ1 , τ2 )| ≤ Ch−d

for all α : |α| < m∗ , β : |β| < m∗ , for |τ1 | ≤ c, |τ2 | ≤ c.

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

392

4.7.2

Tauberian Estimates

Now we can unleash the full power of microlocal analysis but we need to extend it to our framework. Proposition 4.7.3. For a commutator of a pseudodifferential operator with a smooth compactly supported symbol and Cl -function b(x) a usual commutator formula holds modulo O(hl |||∂b|||l−1 ) for any l > 0 where ⎧ ⎪ sup |∂ α f (x)| l ∈ Z+ , ⎪ ⎪ ⎨ α:|α|=l x (4.7.6) |||b|||l := ⎪ ⎪ sup |x − y |l−l · |∂ α f (x) − ∂ α f (y )| l ∈ / Z+ . ⎪ ⎩ x =y α:|α|=l

Proof. Easy proof is left to the reader. For operator A of type (4.7.1) let (4.7.7) |||A|||l := |||aαβ |||l + α,β



|||aαβ |||l+1 .

α,β:|α|+|β|=2m+1

Proposition 4.7.4. Let us consider operator A of the type (4.7.1) satisfying −1 ellipticity condition (4.6.61). Let u(x, y , t) be the Schwartz kernel of e ih tHA,V . Then for T  1 (i) Estimate (4.7.8)

Ft→h−1 τ χT (t)(hDx )α (hDy )β ψ1 (x)ψ2 (y )u ≤ Chs

for all α : |α| < m ¯ + l, β : |β| < m ¯ + l, s and all ψ1 , ψ2 ∈ C0∞ (B(0, 1)), such that dist(supp(ψ1 ), supp(ψ2 )) ≥ C0 T , χ ∈ C0∞ ([−1, 1]) and τ : |τ | ≤ c0 ; in this Proposition . means an operator norm from L2 to L2 . (ii) As l ≥ 1 estimate (4.7.9) Ft→h−1 τ χT (t)(hDx )α (hDy )β ϕ1 (hDx )ϕ2 (hDy )u ≤ Chs + Chl−1 |||∂A|||l−1 holds for all α : |α| < m ¯ + l, β : |β| < m ¯ + l, s and all ϕ1 , ϕ2 ∈ C0∞ , such that dist(supp(ϕ1 ), supp(ϕ2 )) ≥ C0 T , χ ∈ C0∞ ([−1, 1]) and τ : |τ | ≤ c0 .

4.7. OPERATORS WITH IRREGULAR COEFFICIENTS

393

(iii) If also l > 1 and in B(0, 2) operator A is ξ-microhyperbolic on the energy level τ : |τ | ≤ c then then for a small constant T =  estimate (4.7.9) holds for all α : |α| ≤ m ¯ + l, β : |β| ≤ m ¯ + l, s and all ψ1 , ψ2 ∈ C0∞ (B(0, 1)), such that diam(supp(ψ1 ) ∪ supp(ψ2 )) ≤ 0 T and χ ∈ C0∞ ([−1, − 12 ] ∪ [ 12 , 1]). Proof. Let v = e ith

−1 H

f with arbitrary f .

(i) Statement (i)43) is easily proven by the same arguments as in the proof of Theorem 2.1.2: we consider just usual function φ(x) and operators of multiplication like χ(φ(x)) so there are no “bad” commutators due to nonsmoothness of the coefficients. (ii) Statement (ii)44) is also proven by the same arguments; however in this case φ = φ(x, ξ) so we need to involve “bad” commutators but their contributions are bounded by   C Q1 u · hl |||A|||l + u + h1+δ Q  u in the right-hand expression while the left-hand expression is hQ1 u2 where Q, Q1 , and Q  are operators with symbols χ(φ(x, ξ)), χ1 (φ(x, ξ)), and χ1 (φ(x, ξ) − η) respectively, η > 0 is an arbitrarily small constant (so the latter symbol has a bit larger support than the former one), δ > 0 is a small 1 −1 exponent, χ1 (t) = (−χ (t)) 2 , and f.e. χ(t) = e −|t| as t < 0, χ(t) = 0 as t ≥ 0. Therefore we conclude that Qu ≤ Chl−1 |||A|||l + u + Chδ Q  u and similarly we can estimate Q  u with Q  u in the right-hand expression etc and thus we conclude that Qu ≤ Chl−1 |||A|||l + Chs u which is what we need. 43) 44)

Finite speed of propagation with respect to x. Finite speed of propagation with respect to ξ.

394

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

(iii) Statement (iii) is easily proven also by the same arguments: we consider just usual function φ(x) and operators of multiplication like χ(φ(x)) so there are no “bad” commutators due to non-smoothness of the coefficients. However we need to consider a contribution of u which is not confined to the small vicinity of (y , η) and we need Statement (ii) for this so the last term in the right-hand expression of (4.7.9) is inherited. We leave easy details to the reader. Proposition 4.7.5. In the framework of Proposition 4.7.4(iii) with however T : h ≤ T ≤  estimates   (4.7.10) |Ft→h−1 τ χT (t) (hDx )α (hDy )β u(x, y , t) x=y | ≤   T k |||∂A|||k−1 + T l |||∂A|||l−1 Ch1−d+s T −s + Chl−1 T 2−d−l 1≤k≤l

and as l + d > 2 (4.7.11)

  |Ft→h−1 τ χ¯T (t) (hDx )α (hDy )β u(x, y , t) x=y | ≤ Ch1−d

hold for all α : |α| < m ¯ + l − d/2, β : |β| < m ¯ + l − d/2, s as T =  and |τ | ≤  where as usual χ ∈ C0∞ ([−1, − 12 ] ∪ [ 12 , 1]), χ¯ ∈ C0∞ ([−1, 1]). Proof. Observe first that estimate (4.7.10) holds as T  1 in virtue of embedding theorem. Let us consider T ∈ [C0 h, ]; then we apply the standard rescaling t → tT −1 , x → xT −1 , h → hT −1 . Then we need to add factor T −d × T to the right-hand expression where the first factor appears since u(x, y , t) is a density with respect to y and the second one appears since there is an integral with respect to t in the left-hand expression. Observation how |||.|||l scales concludes the proof of (4.7.10). To prove (4.7.11) observe first that it holds for T = 2h. To prove it for h ≤ T ≤  we use t-admissible partition on [−1, 1]\[−h, h] and apply (4.7.10) for each element of the partition. The result is Ch1−d unless l = d = 1 when the result is O(| log h|). Then for RTx (τ1 , τ2 ) and its averages the standard estimates hold: RTx ≺ Ch etc. 1−d

4.7. OPERATORS WITH IRREGULAR COEFFICIENTS

4.7.3

395

Weyl Asymptotics

Now let us pass from Tauberian expression NTx (τ1 , τ2 ) to Weyl expression NW x (τ1 , τ2 ). Theorem 4.7.6. Let A be operator (4.7.1) with l ≥ 1 satisfying ellipticity condition (4.6.61) and (4.7.4). Let on the energy levels τ1 and τ2 ξ-microhyperbolicity condition is fulfilled. (i) Then (4.7.12) Γx (hDx )α (hDy )β e(x, y ; τ1 , τ2 ) − NW x (x; τ1 , τ2 ) ≺

  C h1−d + h−d ς 1/2

as |α| < m∗ = m + l − d/2, |β| < m∗ and ς = hl where NW x (x, τ ) means corresponding Weyl expression. (ii) Further, for ϕ supported in the small vicinity of τ¯i such that ∂ α ϕ is continuous with the modulus of continuity τ r −r +1  (4.7.13)

  ϕ(τ ) dτ Γx (hDx )α (hDy )β e(x, y ; τ , τi ) − NW (x; τ , τ ) ≺ i x   C h1+r −d + h−d ς (r +1)/2 + δr 1 h−d ς| log ς|

as |α| < m∗ , |β| < m∗ and ς = hl where NW x (x, τ ) means corresponding Weyl expression including the second term (if needed).   Remark 4.7.7. It is definitely not as good as C h1−d + h−d ς estimate we got in the previous section but it is a different object (pointwise vs global), we consider a larger class of operators and it really does not matter provided ν(ε) = O(h2 ) which is the case as l ≥ 2 (at least if we set ε = h). Proof of Theorem 4.7.6. (a) We will use the method of successive approximations but in two steps and the first step will be different. As unperturbed operator we consider Aε with the coefficients aαβε = aαβ ∗ φε where φε (x) = ε−d φ(ε−1 x). Then |aαβε − aαβ | ≤ ς := ν(ε) where ν(ε) = C0 εl under assumption that coefficients belong to Cl . Therefore on the energy level τ : |τ | ≤ C the error norm does not exceed C0 ς and then according to calculations of Section 4.4 Tauberian expressions

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

396

on intervals [−T , −T /2]∪[T /2, T ] for A and Aε differ by Ch−d−1 ςT and then this is true for Tauberian expressions on interval [−T , T ]: in the obvious notations (4.7.14)

NTx (τ1 , τ2 ; T ; A) − NTx (τ1 , τ2 ; T ; Aε ) ≺ Ch−d−1 ςT

while Tauberian errors do not exceed Ch1−d T −1 . Let us set ε = C0 h| log h| 45) . Then according to previous Section 4.6 (4.7.15)

1−d NTx (τ1 , τ2 ; T ; Aε ) − NW x (τ1 , τ2 ; Aε ) ≺ Ch

and due to microhyperbolicity one can prove easily that (4.7.16)

W −d ς. NW x (τ1 , τ2 ; Aε ) − Nx (τ1 , τ2 ; A) ≺ Ch

Combining, we arrive to (4.7.17)

 1−d −1  T + h−d−1 ςT . RW x (τ1 , τ2 ; A) ≺ C h

Minimizing by T ∈ [C0 h, ] we get

 1−d 1 RW + h−d ς 2 . x (τ1 , τ2 ; A) ≺ C h

(4.7.18)

(b) Similarly, let us consider NW x,ϕ , with ϕ supported in the small vicinity of τi such that ∂ α ϕ is continuous with the modulus of continuity τ r −r +1 as |α| ≤ )r * − 1 as r > 0. Then the Tauberian error is Ch1+r −d T −1−r , Tauberian expressions on intervals [−T , −T /2] ∪ [T /2, T ] for A and Aε differ by Ch−d−1 ςT × (h/T )r and therefore this is also true for Tauberian expressions on interval [−T , T ] provided r < 1 while for r = 1 we obtain Ch−d ς(1 + | log T /h|):   (4.7.14) NTx,ϕ (T ; A) − NTx,ϕ (T ; Aε ) ≺ Ch−d−1+r ς T 1−r + C δr 1 | log T /h| . Similarly, the right-hand expression in (4.7.15) becomes Ch1+r −d but one needs to include in Weyl expressions the second term here and below (unless it is 0), the right-hand expression in (4.7.15) remains and combining we get Ch−d ς: (4.7.17) RW x,ϕ (A) ≺

   C h1+r −d T −r −1 + Ch−d−1+r ς T 1−r + C δr 1 | log T /h| .

45)

Later we will set ε = C0 h.

4.7. OPERATORS WITH IRREGULAR COEFFICIENTS

397

Minimizing by T ∈ [C0 h, ] we get  1+r −d  RW + h−d ς (r +1)/2 + δr 1 h−d ς| log ς| . (4.7.18) x,ϕ (A) ≺ C h (c) We need to prove that we can set ε = C0 h. Let us apply the previous arguments to A2ε and Aε instead of Aε and A respectively, and set ς = ν(ε), ε ≥ C0 h. One can observe easily that for operator Aε rather than A all our previous results could be improved. Namely, in Proposition 4.7.3 a usual commutator  formula holds modulo O hl (h/ε)s |||∂b|||l−1 with arbitrarily large exponent s and therefore this factor (h/ε)s pops-up in some other estimates in the correct places: in Proposition 4.7.4 it appears alongside factor hl−1 in the last term of the right-hand expression in (4.7.9). Consider now (4.7.14) . Interval [−T , −T /2] ∪ [T /2, T ] contributes now Ch−d−1 (h/T )r ςT (h/ε)s + (h/T )s . Indeed, it follows from the fact that rescaling x → x/T we have not only h → h/T but also ε → min(ε/T , 1). Then summation with respect to partition with respect to tT results in Ch−d−1 (h/T )r ςT (h/ε)s + Ch−d ς which is now the new right-hand expression in (4.7.14) . Plugging ε = εn := 2n h and taking a sum with respect to n = 0, 1, ... n¯ = )log2 (ε/h)* we conclude that (4.7.19) Nx,ϕ (A) − NTx,ϕ (T , Aε ) ≺ Ch−d (h/T )r + Ch−d−1 (h/T )r ς(h)T + Ch−d ς(ε) where the first term estimates the Tauberian error, the second term estimates an error when we replace A by Ah in the Tauberian expression, and it and the third term estimate together an error when we replace Ah by Aε . Here 0 ≤ r ≤ 1 where for r = 0 we consider instead Nx and NTx and for r = 1 we need to replace the second term by Ch−d ς(h)| log h|. Next in virtue of Section 4.6 we can replace here NTx,ϕ (T , Aε ) by NW x,ϕ (T , Aε ) as long as ε ≥ C0 h| log h| and, finally, we can replace NW (T , Aε ) by x,ϕ NW (T , A). x,ϕ Minimizing with respect to T ∈ [C0 h, ] we conclude that the left-hand expression in (4.7.19) is estimated by the right-hand expression of (4.7.18) with ς = ς(h) plus Ch−d ς(ε). One can see easily that the latter term is smaller than the right-hand expression of (4.7.18) as long as ε ≤ h1−δ for r < 1 and ε ≤ C0 h| log h| for r = 1. This concludes the proof.

398

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

Theorem 4.7.8. In the framework of Theorem 4.7.6(i) and (ii) respectively (i) Estimate   (4.7.20) |Γ tr b(x)(hDx )α (hDy )β e(x, y ; τ1 , τ2 ) − NW (τ1 , τ2 )| ≤   C h1−d + h−d ς 2/3 dim H holds where NW (τ1 , τ2 ) means corresponding Weyl expression and b ∈ C0∞ is a matrix-valued function. (ii) Estimate      (4.7.21) | ϕ(τ ) dτ Γ tr b(x)(hDx )α (hDy )β e(x, y ; τ , τi ) − NW (τ , τi ) | ≤   C h1+r −d + h−d ς 2(r +1)/3 + δr 2 h−d ς| log ς| dim H holds where NW (τ , τi ) means corresponding Weyl expression including the second term. Proof. (a) Let us apply a two term approximation rather than one term. Then the error will be the third rather than the second term; then the error estimate will be Ch−d−2 ς 2 T 2 rather than Ch−d−1 ςT and we get expression    (4.7.22) C h1+r −d T −r −1 + h−d−2+r ς T 2−r + δr 2 | log T /h| instead of the right-hand expression in (4.7.17) . Minimizing by T ∈ [C0 h, ] we get the right hand expression in (4.7.21) but without dim H which appears after we take trace. Here ς = ν(ε), ε = C0 h| log h| at the moment. Consider the second term; it appears from     (4.7.23) Tr BGε± (A − Aε )Gε± = Tr (A − Aε )Gε± BGε± with B = (−hD)β b(hD)α and since in Gε± BGε± operator B has smooth coefficients and ε = C0 h| log h| we conclude that due to propagation of singularities results contribution of time intervals [−T , −T¯ ] ∪ [T¯ , T ] with T¯ = Ch| log h| is negligible. Meanwhile on time interval [−T¯ , T¯ ] we apply our standard successive approximation method with freezing symbol at point y etc resulting in     tr (a − aε )(τ − aε )−1 b(τ − aε )−1 = tr b(τ − aε )−1 (a − aε )(τ − aε )−1

4.7. OPERATORS WITH IRREGULAR COEFFICIENTS

399

calculated at point (y , ξ). To this expression one needs to apply ResR , multiply by (2πh)−d , integrate by (y , ξ) ∈ R2d and by τ ∈ [τ1 , τ2 ] 46) . Due to microhyperbolicity of a on the energy level s τ1 , τ2 and estimate |a − aε | ≤ C0 ς the result is equal to the result of the same procedure applied to   tr b(τ − aε )−1 − b(τ − a)−1 with an error, not exceeding Ch−d ς 2 . However the latter result is equal to the difference between NW (τ1 , τ2 ; A) and NW (τ1 , τ2 ; Aε ) and therefore Statement (i) is proven albeit with ς = ν(ε). (b) To improve this statement to one with ς = ν(h) we apply the same procedure as in Part (c) of the proof of Theorem 4.7.6. Then estimate of the third (error) term in the successive approximation is trivial and is left to the reader, let us consider the second term which now is due to the following modification of (4.7.23):   (4.7.24) Tr (Aεn − Aεn+1 )Gε±n+1 BGε±n+1 0≤n≤¯ n

where h ≤ εn ≤ ε = C0 h| log h|. We want to convert both factors Gε±n+1 to Gε± . To do this properly observe that Gε±m BGε±m − Gε±m+1 BGε±m+1 contains one factor (Aεm − Aεm+1 ) and three factors Gε±m+1 and Gε±m (together). Then applying arguments Part (c) of the proof of Theorem 4.7.6 we estimate the error in (4.7.24) by

 T 3 h−3 ν(εn )ν(εm )hl−1 (h/εm )s + (h/T )s 0≤n≤m≤¯ n

and we estimate the resulting error in the second term properly.   So, instead of (4.7.24) we have Tr (A − Aε )Gε± BGε± and we are be able to pass to Weyl expression as in Part (a). Remark 4.7.9. (i) As r ≥ 12 and 1 ≤ l ≤ 1 + r we have a better estimate than one delivered by the bracketing. (ii) One can generalize our arguments for more general ν, in paricular, for ν(s) = s l (log |s|)σ . 46)

If we consider Statement (i); Statement (ii) is proven the same way.

400

4.7.4

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

Sharper Asymptotics

Observe that under assumption (4.7.25)

2

ν(s) = o(s 3 )

as s → +0.

the second terms in the right-hand expressions of (4.7.20) and (4.7.21)are “o” in comparison with the first terms. Theorem 4.7.10. In the framework of Theorem 4.7.8 let (4.7.25) be fulfilled. Further, let the standard non-periodicity condition (4.7.26)

z∈ / Φ( z)

∀t = 0 ∀z

(where Φt means the generalized (multivalued) Hamiltonian flow) be fulfilled. Then estimates (4.7.20) and (4.7.21) hold with the right-hand expressions o(h1−d ) and o(h1−d+r ) respectively. Proof. Under assumption (4.7.26) for Aε one can take T arbitrarily large (so in all estimates Ch1−d+r T −1−r with h ≤ T ≤  is replaced by Ch1−d+r T −1−r + oT (h1−d+r ) for any T ≥ . Then under assumption (4.7.26) proof of Theorem 4.7.8 delivers improved estimates.

4.A 4.A.1

Appendices On the Traces of Almost Analytic Functions

Let Ω ⊂ Rd be an open domain, Γ an open convex domain with a vertex ˜ = Ω + iΓz ⊂ Cd . Let B be a Banach space. A function at 0 and let Ω 1 ˜ F (.) ∈ C (Ω, B) is called r -almost analytic if (4.A.1)

||F (z)|| ≤ M| Im z|−l+δ

˜ ∀z ∈ Ω,

and (4.A.2)

||∂¯zj F (z)|| ≤ M| Im z|r −1+δ

where ∂¯zj = 12 (∂xj + i∂yj ) and δ > 0, l ∈ N.

˜ ∀j = 1, ... , d ∀z ∈ Ω

4.A. APPENDICES

401

Proposition 4.A.1. Let conditions (4.A.1) and (4.A.2) with r = 0 be fulfilled. Then For every y ∈ Γ with |y | = 1 there exists a limit f (x) = F (x + i0y ) = lim F (x + ity )

(4.A.3)

t→+0

in the space of B-valued distributions D (Ω, B);  k Moreover f = 0≤k≤l M fk where fk ∈ C(Ω, B) and ||fk || ≤ CM with C = C (l, δ), M = y , ∂x . f (x) does not depend on the choice of y . Proof. For l = 0 this assertion is obvious. Let us note that ∂t F (x + ity ) = −MF (x + ity ) + 2y , ∂¯z F (x + ity ) and hence (4.A.4) F (x + ity ) = F (x + iy )+  t  t   ¯ 2 y , ∂z F (x + it y )dt − MF (x + it  y ) dt  . 1

1

Then (4.A.1) and (4.A.1) yield that the first and the second terms in the right hand expression belong to C(Ωx ×Jt × Γy , B), J = [0, 1]; the same t is also true for the “shortened” third term 1 F (x + it  y )dt  provided l = 1. On the other hand, for l ≥ 2 we replace F (x + it  y )dt  in (4.A.4) by the right-hand expression of (4.A.4) with t, t  replaced by t  , t  respectively. Repeating this procedure l − 1 times we obtain that (4.A.5) F (x + ity ) = Mk Fk (x) + 0≤k≤l−1

1 (−M)l (l − 1)!



t

(t − t  )l−1 F (x + it  y ) dt 

1

where Fk with k = 0, ... , l − 1 as well as  t 1 (−M)l (t − t  )l−1 F (x + it  y ) dt  Fl (x) = (l − 1)! 1

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

402

obviously belong to C(Ωx × Jt × Γy , B). Therefore the limit F (x + i0y ) exists in the distribution sense and is the sum of Mk fk with fk (x) = Fk (x + i0y ). So Statements 4.A.1 and 4.A.1, are proven. In order to prove 4.A.1 let us note that ∂s F (x + iy 0 − isω) = M F (x + iy 0 − isω) − 2ω, ∂¯z F (x + iy 0 − isω) with M = −ω, ∂x ; setting y , y 0 ∈ Γ, s, t ∈ [0, 1] and ω = y 0 − ty we obtain a formula similar to (4.A.4) with F (x + ity ), M, y , ∂¯z , t, t  , 1 replaced by F (x + iy 0 − isω), −M , −ω, ∂¯z , s, s  , 0 respectively. Then formula (4.A.5) also holds modified in the same way: F (x + ity ) =



(−M )k Fk (x, y 0 , ω, t)+

0≤k≤l−1

1 Ml (l − 1)!



1

(1 − s  )l−1 F (x + iy 0 − is  ω) ds  ;

0

moreover, in this sum we can take the limit as t → +0 (and ω → y 0 , M → −M), and this limit obviously does not depend on y . Let us consider a symbol a ∈ Ck (Ω, H, H) (where H is an auxiliary Hilbert space) such that (4.A.6) a† (x) = a(x) in Ω, and (4.A.7) (a)(x)w , w  ≥ 0 ||w ||2 − c||(a(x) − τ )w ||2

∀x ∈ Ω ∀w ∈ H

where we write now x instead of (x, ξ) and  ∈ T Ω. Let a˜(x + iy ) be the q-analytic extension of a(x): (4.A.8)

a˜(x + iy ) =

1 (∂ α a)(x)(iy )α . α! x

|α|≤q

We assert that for small enough  ≥ 0, ν ≥ 0 (4.A.9)

˜ || ≥ 1 (ε + ν)||w || ||Pw

where here and below P˜ = a˜(x + iε) − τ + iνI and P = a(x) − τ . In fact, ˜ P(x)w = P(x)w + i(ε(a)(x) + ν)w + O(ε2 ||w ||)

4.A. APPENDICES

403

and hence (4.A.6) and (4.A.7) yield that ˜ ˜ ||P(x)w || · ||w || ≥ ImP(x)w , w ≥ 0 (ε + ν)||w ||2 − C0 ε2 ||w ||2 − c||P(x)w ||2 ≥ 1 ˜ ||2 , ( (ε + ν) − C ε2 − C ν 2 )||w ||2 − c||P(x)w 2 and (4.A.9) trivially follows from this inequality. Therefore F (z) = (˜ a(z  ) − z0 )−1 satisfies (4.A.1) and (4.A.2) with l = 2, δ = 1, r = q − 2, z = (z0 , z  ) and Γ = {y0 ≤ 0, y  = t, t ≥ 0, y0 − t < 0}. Moreover, (4.A.10)



a(x) − τ ± i0

−1

 −1 = a˜(x ± i0) − τ

(we have analyzed the “+” case; the “−” case can be analyzed in the same way) and the same is true for all products of the factors (˜ a(z  ) − z0 )−1 47) (the number of which should not exceed j) and regular q-almost analytic factors, q = q(j). We will use these properties in the forthcoming sections and chapters.

4.A.2

Smoothing of Cϑm Functions

Proposition 4.A.2. Let f ∈ Cϑm (X , .) where ϑ satisfies (4.6.7). Then for every ε > 0 there exits fε such that (4.A.11)

|∂ α fε (x) − ∂ α f (x)| ≤ C ϑ(ε)ε−|α|

∀α : |α| ≤ m,

(4.A.12)

|∂ α fε (x)| ≤ C ϑ(ε)ε−|α|

∀α : |α| > m;

here C does not depend on α. Proof. Let φ be a function such that   ˆ = δα0 where φˆ denotes Fourier transform (4.A.13) φˆ ∈ C0∞ B(0, 12 ) , ∂ α φ(0) of φ. Then |∂ α φ| ≤ Cn (1 + |x|)−n for any fixed n and every α. Let fε := f ∗ φε with φε = ε−d φ(x/ε). 47)

I.e., of the class C q up to Ω ⊂ Rd .

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

404

Then for |α| ≤ m α



α

∂ fε (x) − ∂ f (x) =

  φ(y ) ∂ α f (x − εy ) − ∂ α f (x) dy .

Taking Taylor expansion ∂ α f (x − εy ) =

β:|β|≤m−|α|

1 (−εy )β ∂ α+β f (x) + Rα β!

with Rα ≤ C (ε|y |)−|α| ϑ(ε|y |) we see that the left hand expansion in (4.A.11) does not exceed  −|α| |y |−|α| ϑ(ε|y |)(1 + |y |)−n dy Cε and the integrand does not exceed ϑ(ε)(1 + |y |)1−n for both |y | ≤ 1 and |y | ≥ 1 due to assumption f ∈ Cϑm ; so we get (4.A.11). For |α| > m we have  α m−|α| ∂ fε = ε ∂ β φ(y )(∂ α−β f )(x − εy ) dy with |β| = |α| − m; replacing (∂ α−β f )(x − εy ) by (∂ α−β f )(x) we get 0 and the difference does not exceed  C ε−|α| ϑ(ε|y |)(|y | + 1)−n dy which leads to (4.A.12).

Comments The number of papers devoted to spectral asymptotics is huge. Apart of the Tauberian approach developed by T. Carleman in [1, 2] and used here, and variational approach we will use later to cover singular zones, we should mention a method of almost spectral projector due to M. Shubin and V. Tulovsky [1].

4.A. APPENDICES

405

Depending which function f (t, A) is used and what Tauberian theorem is applied we talk about resolvent method (f = (z − A)−1 ), complex power −tA method (f = Az ), heat kernel √ method (f =√e ) or hyperbolic operator itA method (f = e , f = cos(t A) or f = sin(t A) applied here. The absolute majority of the papers using the latter use explicit construction of the hyperbolic kernel and we refer to Yu. Safarov and D. Vassiliev [3]. Here we comment only on the constructions of this Chapter. The whole construction of Sections 4.3–4.4 appeared first in paper of V. Ivrii [8] in the standard rather than semi-classical situation: first propagation results and rescaling were used to prove that singularity at t = 0 of σ(t) = Γ(Qu) was of the normal type i.e. (tDt )n σ(t) had the same order of singularity in the vicinity of 0: t n Dtn σ(t) ∈ H −M [−T0 , T0 ]

∀n ∈ Z+ .

Then the successive approximation was used and each next term contained an extra factor of t 2 Dt type: σm (t) ∈ t 2m H −M−m [−T0 , T0 ]

∀m ∈ Z+ .

Then combining these two statements it was proven that actually σm (t) ∈ H M−m [−T0 , T0 ]

∀m ∈ Z+ .

Long term propagation to prove a “sharp” remainder estimate was first used by J. J. Duistermaat and V. Guillemin [1] and without Fourier integral operators again by V. Ivrii [8]. These allowed to prove Weyl conjecture: two-term asymptotics of eigenvalue counting function under assumption that measure of the set of all periodic billiards is 0. These arguments were repeated in V. Ivrii [10] to study matrix operators with characteristic roots of variable multiplicity. “Sharper” remainder estimates were first derived by P. Be´rard [1] and then by A. Volovoy [1] and later repeated by V. Ivrii [23] in the general settings. The sharp remainder estimates without smoothness conditions appeared first in L. Zielinski [1–4], followed by V. Ivrii [26] and M. Bronstein and V.Ivrii [1]. While L. Zielinski was able to develop microlocal analysis as 1 ε = h 2 and in V.Ivrii [26] it was done as ε = h1−δ with arbitrarily small δ > 0; then it was improved to ε = h| log h|1+δ (unpublished) and in M. Bronstein– V.Ivrii [1] envelope was pushed further to the limit ε = Ch| log h|; this

406

CHAPTER 4. GENERAL THEORY IN THE INTERIOR

improvement leads to smaller approximation errors as regularity is fixed and to weaker regularity condition as approximation error is fixed. Direct approach to operators with non-smooth coefficients was developed first by V. Ivrii while researching for Chapter 27 of this book.

Chapter 5 Scalar Operators in the Interior of the Domain. Rescaling Technique 5.1 5.1.1

Introduction Pilot: Schr¨ odinger Operator

In this chapter we consider scalar and similar operators and weaken microhyperbolicity conditions while trying to retain sharp remainder estimates. Note that for such operators we can rewrite ellipticity, microhyperbolicity and ξ-microhyperbolicity on energy level τ conditions respectively as (5.1.1)

|a(x, ξ) − τ¯| ≥ 0 ,

(5.1.2)

|a(x, ξ) − τ¯| + |∇x,ξ a(x, ξ)| ≥ 0 ,

(5.1.3)

|a(x, ξ) − τ¯| + |∇ξ a(x, ξ)| ≥ 0 .

To illustrate, for Schr¨odinger operator (5.1.4)

h2 |D|2 + V (x)

and Q1 = Q2 = I these conditions become (5.1.5)

V ≤ τ¯ − 0 ,

(5.1.6)

|V (x)| + |∇V (x)| ≥ 0 ,

(5.1.7)

|V (x)| ≥ 0

© Springer Nature Switzerland AG 2019 V. Ivrii, Microlocal Analysis, Sharp Spectral Asymptotics and Applications I, https://doi.org/10.1007/978-3-030-30557-4_5

407

CHAPTER 5. SCALAR OPERATORS AND RESCALING

408

respectively. These conditions are assumed to be fulfilled in B(0, 1) and then respectively (5.1.8)

|e(x, y , τ¯)| ≤ C  hs ,

e W (x, y , τ¯) = 0

with arbitrarily large s,  | RW (5.1.9) τ )ψ(x) dx| ≤ Ch1−d x (¯ (5.1.10)

|RxW (¯ τ )| ≤ Ch1−d .

as x, y ∈ B(0, 12 ) or ψ ∈ C0∞ (B(0, 12 )) To weaken these conditions we apply rescaling with variable scale parameters. First, however, consider constant parameters ρ, γ such that (5.1.11)

ργ ≥ h

and that V is  ρ2 in an appropriate sense and our playground is B(0, γ). Then we scale x → xγ −1 , divide operator by ρ2 1) and arrive to operator (5.1.4) but with h →  = hρ−1 γ −1 and potential V  = ρ−2 V (xγ) now satisfying all the assumptions of Chapter 4. In order our new operator to be elliptic, microhyperbolic or ξ-microhyperbolic on the energy level 0 we must assume that originally (5.1.12)

V ≤ −0 ρ2 ,

(5.1.13)

|V (x)| + γ|∇V (x)| ≥ 0 ρ2 ,

(5.1.14)

|V (x)| ≥ 0 ρ2

respectively and then we get estimates

(5.1.16)

|e(x, y , 0)| ≤ C  hl ρ−l γ −l−d , e W (x, y , τ¯) = 0,  1−d d−1 −1 γ −d | RW ρ γ x (0)ψ(x) dx| ≤ Ch

(5.1.17)

|RxW (0)| ≤ Ch1−d ρd−1 γ −1 .

(5.1.15)

1)

Without any loss of the generality we can assume that τ¯ = 0.

5.1. INTRODUCTION

409

Now, if we want to get rid of assumptions (5.1.6) or (5.1.7) in the original standard problem we need to introduce scaling functions (5.1.18) or (5.1.19) 1

1  γ(x) =  |V (x)| + |∇V (x)|2 2 + γ¯ ,

ρ(x) = γ(x),

γ(x) = |V (x)| + γ¯ ,

ρ(x) = γ(x) 2

1

2

where γ¯ = h 2 , γ¯ = h 3 respectively in order to fulfill (5.1.11). Then |∇γ| ≤ 12 and therefore both γ and ρ have constant magnitudes in balls B(x, γ(x)). Further, as γ ≥ 2¯ γ in B(x, γ(x)) condition (5.1.13) (or (5.1.14)) is fulfilled. On the other hand, as γ  γ¯ these conditions are of no importance since   1 then. So, we arrive to estimates   W −d | Rx (0)ψ(x) dx| ≤ Ch ρ(x)d−1 γ −1 dx (5.1.20) or (5.1.21)

−d |RW ρ(x)d−1 γ −1 . x (0)| ≤ Ch 1

Plugging ρ = γ or ρ = γ 2 we arrive to   −d | RW γ(x)d−2 dx (5.1.22) (0)ψ(x) dx| ≤ Ch x or (5.1.23)

−d |RW ρ(x)d−1 γ (d−3)/2 . x (0)| ≤ Ch

Observe that the right-hand expression in (5.1.22) is O(h1−d ) as d ≥ 2 and 1 O(h− 2 ) as d = 1 while the right-hand expression in (5.1.23) is O(h1−d ) as 4 2 d ≥ 2, O(h− 3 ) as d = 2 and O(h− 3 ) as d = 1. Therefore we can remove in some cases microhyperbolicity or ξ-microhyperbolicity condition either without any punishment at all or with some punishment.

5.1.2

Plan of the Chapter

The goal of this Chapter is to develop this approach further. In Section 5.2 we consider operators with scalar principal symbols as well as operators with eigenvalues of the principal symbols having constant multiplicities. First, we prove results similar to those for the Schr¨odinger

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CHAPTER 5. SCALAR OPERATORS AND RESCALING

operator we discussed above; the role of the dimension is played by rank Hess a in stationary points. Then for asymptotics with spatial mollification we step by step weaken these non-degeneracy conditions until the become really weak. We also explore sharp and sharper asymptotics. In Section 5.3 we apply these results to the Schr¨odinger and Dirac operators. For the Schr¨odinger operator the conditions of these theorems are fulfilled automatically for any dimension d (if we are interested in spatial means) but for the Dirac operator some additional treatment is necessary because we want asymptotics uniform with respect to the mass which can be not only finite but also very large or very small parameter; in the latter case eigenvalues of the principal symbols are not disjoint but it happens only at one point. We also derive improved asymptotics without spatial mollification for 1- and 2-dimensional Schr¨odinger operator as operator is not ξ-microhyperbolic; then asymptotics contains a correction term due to short bicharacteristic loops. In Section 5.4 we generalize some of the results of the previous two sections to non-smooth operators. While we start from ξ-microhyperbolicity condition, at some moment we weaken it and microhyperbolicity condition also plays a role and we observe that results with or without spatial mollification become different. For example, if d = 1, 2 and we consider Schr¨odinger operator then in the framework of Subsection 5.1.1 remainder estimate is   W W 1−d γ (d−3)/2 (x) dx; (5.1.24) |R | = | Rx ψ(x) dx| ≤ Ch and under microhyperbolicity condition (5.1.25)

γ(x) + |∇γ(x)| ≥ 0 .

Then as d = 2, 1 we arrive to the remainder estimates |RW | = O(h−1 ) and |RW | = O(| log h|) respectively and in the latter case additional arguments lead to the remainder estimate |RW | = O(1). Thus we replaced ξ-microhyperbolicity by microhyperbolicity. Rescaling again, we arrive to certain estimates without microhyperbolicity condition. Furthermore, in some proofs we use not only rescaling but also long-term propagation (where long-term refers to time scale after rescaling) and ability to choose time direction arbitrarily is very handy (but this is possible only for asymptotics with spatial mollification).

5.2. SPECTRAL ASYMPTOTICS FOR SCALAR OPERATORS

411

Finally, in Appendix 5.A.1 we calculate spectral kernels for Schr¨odinger operator with potential V (x) = x1 and in Appendix 5.A.2 we list some properties of the Pauli matrices.

5.2

Spectral Asymptotics for Scalar Operators

In this Section we consider operators with scalar principal symbols2) for which we weaken the restrictions on averaging parameter L of Subsection 4.4.1 and the microhyperbolicity condition we had in Subsection 4.4.2. Our main tool is rescaling applied to the results of the previous Chapter 4. This theory is applied to more general operators as well. Namely, all the results remain true for operators such that (5.2.1)

Spec A0 (x, ξ) ∩ [−0 , 0 ] = {a(x, ξ)}

in Ω

where here A0 (x, ξ) is the principal symbol of A and a(x, ξ) is the scalar function. Then in a fixed neighborhood of supp(q1 ) ∪  supp(q2 )  the operator A 0 I A can be reduced3) to the block-diagonal form A ≡ where the 0 AII principal symbol of AI equals a(x, ξ) and Spec aII ∩ [−0 , 0 ] = ∅; let Ω be this neighborhood.   uI ,I uI ,II If u = corresponds to this reduction then the standard uII ,I uII ,II ellipticity arguments yield that (5.2.2) All the components of u apart from uI ,I are negligible in Ω × ΩT × [−0 , 0 ]  (x, y , τ ); in what follows we will consider only τ ∈ [−0 , 0 ]. Thus we can assume without any loss of the generality that (5.2.3) B = I and A is an operator with the scalar principal symbol a. 2)

Later when we consider more degenerate operators we will need to include more terms in the analysis and assume that these terms are scalar. 3) By means of multiplication on the right and left by G ∗ and G respectively where G is a pseudodifferential operator with G ∗ G ≡ GG ∗ ≡ I .

CHAPTER 5. SCALAR OPERATORS AND RESCALING

412

For operators satisfying assumption (5.2.3) the ellipticity, microhyperbolicity and ξ-microhyperbolicity conditions on the energy level τ¯ are exactly (5.1.1), (5.1.2) and (5.1.3) meaning respectively that τ¯ is not a value of a(x, ξ) at all, and τ¯ is not a critical value of a, τ¯ is not a critical value as a(x, ξ) is considered to be function of ξ only. One can easily see that  (5.2.4) dμτ = IH dxdξ : daΣτ in the sense of exterior forms.

5.2.1

From Condition “No Critical Point” to “rank Hess a ≥ 2 at Critical Points”. I

Standard Asymptotics Let us introduce the scaling functions  1 (5.2.5) γ(x, ξ) = 1 |a(x, ξ) − τ | + |∇(x,ξ) a|2 2 + γ¯ ,

ρ(x, ξ) = γ(x, ξ)

1

with γ¯ = h 2 ; this choice of γ¯ is due to the fact that at this step γ will be scale with respect x and ξ. One can easily check that γ is an admissible scaling function (5.2.6)

|∇(x,ξ) γ(x, ξ)| ≤

1 2

provided 1 > 0 is a small enough constant. Let us consider a γ-admissible partition of unity by pseudodifferential operators Ql 4) Let Ql = Op(ql ) where ql is the symbol of the same type as ql . Then one can easily see that  ∗ d (5.2.7) ρ−2 l Ql AI ∈ Ψh,γl ,γl ,K (T R , H, H) uniformly with respect to l.

Then one needs to consider the separate elements of the partition and to treat two cases: That means that Ql = Op(ql ) where ql ∈ Sh,γ(z l ),γ(z l ),K (T ∗ Rd ) are supported in the balls B(z l , 12 γl ) with γl = γ(z l ) and the multiplicity of the covering by balls B(z l , γl ) does not exceed K0 = K0 (d). Such a partition exists (see H¨ ormander [1], for example). 4)

5.2. SPECTRAL ASYMPTOTICS FOR SCALAR OPERATORS

413

(i) Regular case when γ := γl ≥ c0 γ¯ ; then for this element all the conditions including the microhyperbolicity condition will be fulfilled after the rescaling (5.2.8) x → xnew = (x − x l )γ −1 , ξ → ξnew = (ξ − ξ l )γ −1 , h → hnew = hγ −1 ρ−1 , A → Anew = ρ−2 Al and the gauge transformation Al → e ih

−1 x,ξ

l

AI e −ih

−1 x,ξ

l

.

So, there are (apart from inessential shifts with respect to (x, ξ)) rescalings with respect to (x, ξ) and the rescaling with respect to the spectral parameter τ as well τ → τnew = τ −1

(5.2.9)

with  = ρ2 here. We can assume that τ¯ = 0 to keep this energy level 1 unaffected by rescaling. We selected γ¯ = h 2 to keep hnew ≤ 1 (because ρ = γ at this moment). Therefore one can apply all the results of Subsection 4.4.2. Thus, let us apply all the results of Subsubsection 4.4.2.3 Improved Asymptotics without Mollification and then return back to the original variables. At this moment we consider only estimates of RW (τ ) and RW ϕ,L (τ ). We need to plug there Lnew = L−1 . Moreover we can apply the results of Section 4.5 as well noticing that on this step no rescaling on t is necessary −1 because hnew Anew = h−1 A modulo an inessential shift and thus Tnew = T here. Note that −1 hnew L−1 and hnew (Lnew T )−1 = h(LT )−1 new = hL

(5.2.10) and therefore (5.2.11)

1−d hnew Tnew

 (U ∩Στ )

dxnew dξnew : danew

h1−d = T

 dxdξ : da U∩Στ

(surely in the left-hand expression not (xnew , ξnew ) but (x, ξ) belongs to U ∩Στ ) and  hnew   h 1−d+δ  h  1−d+δ (5.2.12) hnew = 2 . ϑ ϑ Lnew T γ L We obtain that

CHAPTER 5. SCALAR OPERATORS AND RESCALING

414

(5.2.13) For this element of the partition all the estimates of Subsubsection 4.4.2.3 Improved Asymptotics without Mollification and Section 4.5 remain true with an additional term  h  1−d+δ (5.2.14) Ch ϑ γ −2−2δ dxdξ L U in the right-hand expression where U = B(z l , γl ) now. (ii) Irregular case when γ := γl ≤ c0 γ¯ ; then for this element all the conditions apart from the microhyperbolicity condition will be fulfilled after the same change of variables as before. However, the microhyperbolicity condition provides no advantage because hnew  1 now. Therefore one can apply Proposition 4.2.1 and Theorem 4.4.12. Returning to the original variables and operator we see that the contribution of this element to the remainder does not exceed    h (5.2.15) C0 h1−d dxdξ : da + C  hδ γ −2−2δ dxdξ ϑ ) L U  ∩Στ U where U  = B(z l , γl ) now and    dxdξ : da = O γ −2 dxdξ ; (5.2.16) U  ∩Στ

moreover one can replace “= O” by “” if in B(z l , 12 γl ) ellipticity a  γ 2 fails. Note that (5.2.17) Contribution5) of each irregular element does not exceed C0 ϑ(h/L) and (5.2.18) Contribution5) of each regular elliptic element does not exceed Ck h−d+k γ −2d−2k ϑ(h/L) with an arbitrarily large exponent k. 5)

Both to the remainder and to the main term.

5.2. SPECTRAL ASYMPTOTICS FOR SCALAR OPERATORS

415

Now let us sum the contributions of all the elements of this partition: (5.2.19)

≤ C0 h   δ +C h

|RW ϕ,L (τ )|

1−d

  h  ϑ dxdξ : da L U ∩Στ ∩{γ≥c0 γ ¯}   −2−2δ  k γ dxdξ + C h γ −2−2k dxdξ

U ∩{|a−τ |≤γ 2 }

U

where U is a fixed vicinity of supp(q). Here the last term in the right-hand expression is the contribution of all elliptic and of all irregular elements. We need to consider only one level asymptotics and RW (τ ) is also covered since we can always set the second energy level to −C since B = I . First of all we obtain immediately Theorem 5.2.1. RW (τ ) = O(hs ) with arbitrarily large s provided L ≥ h1−δ and ϕ ∈ C0∞ (R). To cover more interesting case we need to impose some non-degeneracy condition; namely let us assume that (5.2.20)r |a(x, ξ) − τ | + |∇x,ξ a(x, ξ)| ≤ 0 =⇒ Hess a(x, ξ) has r eigenvalues (counting multiplicities) f1 , ... , fr such that |f1 | ≥ 0 , ... , |fr | ≥ 0 6) . Then in the vicinity of every point z = (x, ξ) with (5.2.21)

|a(z) − τ | + |∇z a(z)| ≤ 0

one can choose a coordinate system7) z = (z  ; z  ) = (z1 , ... , zr ; zr +1 , ... , z2d ) such that ςj zj2 + V (z  ) with ςj = ±1. (5.2.22) a(z) = 1≤j≤r

Then in a smaller vicinity (5.2.23) γ(z)  |z  | + γ  (z  ), 6) 7)

 1 γ  (z  ) := 2 |V (z  ) − τ | + |∇V (z  )|2 2 + γ¯ .

The other eigenvalues are arbitrary. This system is not assumed to be symplectic.

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CHAPTER 5. SCALAR OPERATORS AND RESCALING

One can see easily that then  (5.2.24) Στ ∩U ∩{ε≤γ(x,ξ)≤2ε}

then for r ≥ 3



(5.2.25) Στ ∩Ω

and

dxdξ : da ≤ C εr −2 ;

dxdξ : da < ∞,



dxdξ : da = O(ε(r −2) )

(5.2.26) Στ ∩Ω∩{γ(x,ξ) 0. Meanwhile for r = 2 we get a logarithmic divergence in (5.2.25). On the other hand, assume instead that (5.2.20)+ 2 |a(x, ξ) − τ | + |∇x,ξ a(x, ξ)| ≤ 0 =⇒ Hess a(x, ξ) has 2 eigenvalues (counting multiplicities) f1 , f2 of the same sign such that |f1 | ≥ 0 , |f2 | ≥ 0 6) . Then (5.2.25), (5.2.26) hold. Really, in this case for each z  on Στ  magnitude logarithmic divergence. So, in  of |z | is defined which prevents 8) this case Στ dxdξ : da is uniformly bounded .   1−d Finally, under condition (5.2.20)2 the additional terms are O h ϑ(h/L) .   Thus we arrive to the remainder estimate O h1−d ϑ(h/L) under condition (5.2.20)+ 2. Now look what happens to the coefficients. They are given by the “regularized” formulae in which Q2 is replaced by Q2 (I − ζ(x, hD)) where (5.2.27)

supp(ζ) ⊂ {γ(x, ξ) ≤ c0 γ¯ },

ζ = 1 as γ(x, ξ) ≥ γ¯ , |D α ζ| ≤ cα γ −|α|

∀α;

so these regularized coefficients are given by (5.2.28) κn reg (τ , h) =    (2π)−d q1 (x, ξ, h)Fn (x, τ , ξ)[q2 (x, ξ, h)] 1 − ζ(x, ξ) dxdξ 8) However, as example a = z12 + z22 shows, this integral is not necessarily uniformly convergent.

5.2. SPECTRAL ASYMPTOTICS FOR SCALAR OPERATORS

417

instead of (4.3.60); exact choice of ζ is not important. Meanwhile integral in (4.3.60) may lead to distribution with respect to τ because there is no microhyperbolicity anymore. So, we arrive immediately to Theorem 5.2.2. (i) Under condition (5.2.20)3    h   W 1−d (5.2.29) |Rϕ,L,reg (¯ τ )| ≤ C0 h ϑ dxdξ : da + C  hδ L Στ¯ ∩U where  (5.2.30)

τ) RW ϕ,L,reg (¯

:=

ϕ(

 τ − τ¯   ) dτ ΓQ1x Be(., ., τ  , τ tQ2y )− L  h−d+n κn,reg (τ , h) dτ 0≤n≤N  −1

with regularized coefficients κn,reg . (ii) Under condition (5.2.20)+ 2 (5.2.31)

h . |RW τ )| ≤ Ch1−d ϑ ϕ,L,reg (¯ L

(iii) Under condition (5.2.20)2 (5.2.32)

 h  · | log h| + 1 . |RW τ )| ≤ Ch1−d ϑ ϕ,L,reg (¯ L

(iv) Under condition (5.2.20)1 (5.2.33)

|RW τ )| ≤ Ch 2 −d ϑ ϕ,L,reg (¯ 1

h . L

Proof. We proved Statements (i) and (ii) already. Proof of Statements (iii) and (iv) follows the same scheme but as we integrate with respect to z  we note that this integral becomes    | log |V (z  ) − τ¯| + h | dz  (5.2.34)

418 and (5.2.35)

CHAPTER 5. SCALAR OPERATORS AND RESCALING 



|V (z  ) − τ¯| + h

− 12

dz 

as r = 2, a = z12 − z22 + V (z  ) and as r = 1, a = ±z12 + V (z  ) respectively. In 1 the worst case V (z  ) = 0 these expressions become | log h| and h− 2 factors. We will use improved remainder estimates later. Remark 5.2.3. (i) We do not need to include terms such that ϑ(h) ≥ C −1 hn−1 for all h ∈ (0, h0 ); (ii) Under condition (5.2.20)+ r we do not need to regularize coefficients in the −n terms such that ϑ(L)L is a monotone non-decreasing function on (0, h0 ). Sharp and Sharper Asymptotics Let us leave analysis of Statements (iii) and (iv) of Theorem 5.2.2 for coming subsections and consider improvements of (i) and (ii) under condition to long-term dynamics. If we assume that a is scalar globally we have just Hamiltonian flow (not the generalized Hamiltonian flow) and we do not need to consider sharp asymptotics separately; however we do not make such assumption. We will use ν instead of γ of Section 4.5 because γ is in use. Then we can apply Theorems 4.5.5 and 4.5.12 in the following settings. Let Ui denote supp(ri0 ) and Ω denote supp(q1 ) ∩ supp(q2 ) (from these theorems). Assume that (5.2.36)

Ui ⊂ Ω+ ε := {z ∈ Ω : γ(z) ≥ ε}

∀i = 1, ... , m

with ν either a small constant (for sharp asymptotics) or a small parameter ν ∈ [hδ , ] (for sharper asymptotics). Then in the zone Ω+ ε we can use Theorems 4.5.5 and 4.5.12 with condition (4.5.30) replaced by *   γ|t| as |t| ≤ T0 , (5.2.37) dist (x, ξ), Ψτ ,t (x, ξ) ≥ γν as T0 ≤ |t| ≤ T . In particular, we arrive to Proposition 5.2.4. Let condition (5.2.20)+ 2 be fulfilled. Further, assume that μτ (Π ∩ Ω+ ) = 0 where Π is the set of all periodic trajectories and + Ω+ := {z ∈ Ω : γ(z) > 0}. Then for  any ε > 0 contribution of Ωε to the 1−d remainder estimate is o h ϑ(h/L) uniformly with respect to h ≥ L.

5.2. SPECTRAL ASYMPTOTICS FOR SCALAR OPERATORS

419

We leave to the reader to formulate theorems. However it would be completely futile unless we can properly estimate contribution to the remainder of near-critical zone Ωε := {z ∈ Ω : γ(x, ξ) < ε}.

(5.2.38) Near-Critical Zone

Observe first that due to (5.2.26) as r ≥ 3 contribution of Ωε to the remainder does no exceed C εh1−d and thus is small. In particular, Proposition 5.2.4 immediately implies the final Theorem 5.2.9(i). However under condition (5.2.20)+ 2 situation is more complicated. Due to (5.2.23) (5.2.39)

μτ (Ωε )  mes(Ωε ),

Ωε := {z  ∈ Ω : γ  (z  ) < ε}

where we assigned μτ (Ω0 ) = 0 because μτ is not defined on Ω0 = {z ∈: γ(z) = 0}, and Ωε = B(0, ε) × Ω . Proof of Theorem 5.2.2(ii) immediately implies Proposition 5.2.5. Let condition (5.2.20)+ 2 be fulfilled. Then contribution of Ωε to the remainder estimate does not exceed Ch1−d ϑ(h/L) mes(Ωε ). In particular, Proposition 5.2.4 immediately implies the final Theorem 5.2.9(ii) under assumption mes({V = τ¯}) = 0. However we can do better than this. Let us introduce the set (5.2.40)

Στ = {(x, ξ) : τ is an eigenvalue of a(x, ξ) + has (x, ξ) }

and the L(H, H)-valued9) measure   (5.2.41) dμτ = ∂τ θ τ − a(x, ξ) − as (x, ξ)h supported on Στ . T Proposition 5.2.6. Let condition (5.2.20)+ 2 be fulfilled. Then Rϕ,L (τ ) is dominated by   dμτ + (5.2.42) Ch1−d  Στ ∩Ω∩{γ≥ε}   h  1−d |{z1 , z2 }| dμτ + C  h1−d+δ ϑ Ch L Στ ∩Ω∩{γ≤ε} 9)

We do not assume here that the subprincipal symbol as is scalar.

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CHAPTER 5. SCALAR OPERATORS AND RESCALING

where ε = hδ with arbitrarily small δ > 0. Proof. It is sufficient to prove (5.2.42) as (5.2.43)

supp(q1 ) ∩ supp(q2 ) ⊂ Ωε .

Without any loss of the generality we can assume that two disjoint from 0 eigenvalues of Hess a are positive. Introducing appropriate local coordinates z = (z  ; z  ) = (z1 , z2 ; z  ) in R2d we obtain that a − τ = (z12 + z22 + V (z  )) 1 and then (5.2.23) holds for γ defined by (5.2.5); let us take γ¯ = h 2 −δ and consider a γ-admissible partition of unity. Then the arguments of the proof of Theorem 5.2.2 provide us with an appropriate estimate in the zone {γ ≥ c0 ε} and we need to consider only the zone {|z  | ≤ ε} × Ω . Let us cover Ω with balls of radius γ1 = hδ1 with δ1 < δ and consider balls of different types: (i) Let | Hess V | ≥ c0 γ1 at some point z  ∈ Ω. Then this remains true in B(z  , γ1 ) (with c0 replaced by 12 c0 ) and (5.2.44)

     mes Ωε ∩ B(z  , γ1 ) ≤ mes B(z  , γ1 ) hδ ,

δ  > 0.

Then the total contribution of these balls to the remainder does not exceed   C  h1−d+δ ϑ h/L); thus we need to consider only balls with | Hess V | ≤ γ1 . Moreover, the same arguments combined with  the induction with respect to m yield that one need only treat balls with 2≤|α|≤m |D α V | ≤ c0 γ1 with arbitrarily large m. Now as V is almost constant in the balls in question we need to consider {z1 , z2 } which is either rather small (less than ω) or has a constant magnitude ω in balls in question; then in both cases classical dynamics could be periodic but its period is  ω −1 . We exploit this in a bit different way. (ii) Let us consider a ball with |{z1 , z2 }| ≤ cω where ω ≥ hδ0 . This inequality holds at all points of this ball simultaneously. Then in the appropriate symplectic coordinates (5.2.45)

(a − τ¯) = ξ12 + (ξ2 + b1 (x, ξ))2 + b2 (x, ξ)

where |∇b1 | ≤ cω and | Hess b| ≤ cω 2 in this ball.

5.2. SPECTRAL ASYMPTOTICS FOR SCALAR OPERATORS

421

Let us redefine the scaling functions γ and ρ in the ball B(z  , γ1 ). Namely, pick  1   (5.2.46) ρ :=  |ξ1 | + |ξ2 + b1 | + |b2 − τ | + ω −2 |∇b2 |2 2 + ρ¯ , γ := ρω −1 , Then |∇ρ| ≤

1 2

ρ¯ = h 2 ω 2 . 1

1

(due to the condition on the Hessian) and ργ ≥ h.

Moreover, one can check that (5.2.47) distx (z  , z  ) ≤ γ(z  ), distξ (z, z  ) ≤ ρ(z  ) =⇒ ρ(z  )  ρ(z  ), γ(z  )  ρ(z  ). Let us rescale xnew = xγ −1 , ξnew = ξρ−1 . Then we should replace h by hnew = hρ−1 γ −1 and L by Lnew = Lρ−2 because we divide A by ρ2 now. Obviously, (5.2.48)

hnew h = ω. Lnew L

One can easily check that after this rescaling the derivatives (up to order K ) of the symbol of the operator are bounded. Surely, the symbol itself is not necessary bounded because of the term a hρ−2 but in this case one can introduce operator B = as (¯ z )hρ−2 where z¯ is a fixed point in the original ball of radius γ1 . s

For a scalar symbol as one can avoid this difficulty simply by replacing a with a + has in the definitions of γ and ρ. Moreover, the microhyperbolicity condition is fulfilled for ρ ≥ c0 ρ¯ 10) . Then Theorem 4.5.12 provides all the necessary estimates in this zone and its total contribution to Rϕ,L surely does not exceed the right-hand expression of estimate (5.2.41) with ω instead of |{z1 , z2 }|. Further, Theorem 5.2.2 11) applies in the domain ρ ≤ ρ¯ , the total contribution to RW ϕ,L of which does not exceed h h C  ρ¯2−d γ¯ −d ϑ ≤ C  h1−d ωϑ . L L According to our procedures, one should check this condition for the symbol a + as h after rescaling, but the derivatives of as h are negligible, so one may check the condition for the scalar symbol a instead. 11) With obvious modifications for matrix as . 10)

CHAPTER 5. SCALAR OPERATORS AND RESCALING

422

Therefore the contribution of balls with ω ≤ hδ0 is estimated properly. (iii) Finally, let us consider balls of radius γ2 = h 2 −δ2 where | Hess V | ≤ c0 γ1 and |{z1 , z2 }|  ω ≥ hδ0 . In these balls the variation of b2 is less than h1+δ3 and we simply include h−1 b2 in as . Without any  loss of the  generality one B 0 + can assume that in the ball, h−1 b2 + as − τ = where B+ ≥ νI , 0 B− B− < (1 − )νI and  ω ≤ ν ≤  ω with small enough positive constants ,  ,  . Then one can easily see that  dμτ , π− γ22−2d 1

 where π− =

 0 0 . 0 I

Στ ∩B(z,γ2 )

Let us consider A+ = Z12 + Z22 + B+ . Then Ga ˚rding inequality yields that (A+ u, u) ≥ Z1 u2 + Z2 u2 − C0 h1+2δ0 u2 − C Qu2 ≥ 1 (Z1 u2 + Z2 u2 + |([Z1 , Z2 ]u, u)|) − C0 h1+2δ0 u2 − C Qu2 ≥ 2 (Z1 u2 + Z2 u2 + ωhu2 ) − C Qu2 where Q is negligible in the ball in question. Therefore for τ  ≤ τ the operator A+ − τ  is invertible in this ball and the norm of the inverse does not exceed Ch−1 ω −1 . 

Therefore Ft→h−1 τ χT (t)(Q1 u ++ ) is negligible for τ ≤ ωh, T ≥ h−1−δ ω −1 with an arbitrarily small exponent δ  > 0 and simplified Tauberian arguments yield that Q1 E (τ  , τ ) is negligible for τ  , τ  ≤ τ + ωh. This and estimates from below of the right-hand expressions of (5.2.42) yield that (5.2.42) holds. From the proof above immediately follows W Theorem 5.2.7. Let condition (5.2.20)+ 2 be fulfilled. Then Rϕ,L,reg is dominated by (5.2.42) where now regularization also means that as h is included in the principal symbol while calculating coefficients.

Remark 5.2.8. (i) Obviously μτ (U) may be not continuous function; in particular, it has no-zero jump at τ¯ as a = z12 + z22 + τ¯; however in the

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423

case (iii) which is the only case when this really matters the bottom of the spectrum is actually pushed up and it helps; otherwise we would need to replace in the right-hand expression τ by τ + Ch . (ii) Theorem 5.2.7 clearly provides improvement for example for 2-dimensional Schr¨odinger operator without magnetic field but it provides improvement neither for 2-dimensional magnetic Schr¨odinger operator h2 D12 + h2 (D2 − x1 )2

(5.2.49)

nor for 1-dimensional Schr¨odinger operator h2 D12 + h2 x12 ;

(5.2.50)

these operators are actually equivalent. In particular, Proposition 5.2.4 immediately implies the final Theorem 5.2.9(ii) in the full generality. Sharp and Sharper Asymptotics (End) So, we can apply Theorem 5.2.7 in the near-critical zone. In particular we get Theorem 5.2.9. Assume that μτ¯ (Π ∩ Ω+ ) = 0. Then (i) Under condition (5.2.20)3   h  . RW τ ) = o h1−d ϑ ϕ,L,reg (¯ L

(5.2.51)

(ii) Let condition (5.2.20)+ 2 be fulfilled. Then as as (5.2.52)

 0}) = 0 mes({V = τ¯, {z1 , z2 } =

estimate (5.2.51) holds.

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424

5.2.2

From Condition “No Critical Point” to “rank Hess a ≥ 2 at Critical Points”. II

Now we begin treating the more complicated case in which condition (5.2.20)2 is fulfilled but condition (5.2.20)+ 2 fails. We want to obtain remainder estimate O(h1−d ϑ(h/L)) in this case. Without any loss of the generality one can assume that (5.2.20)− 2 |a(x, ξ) − τ | + |∇x,ξ a(x, ξ)| ≤ 0 =⇒ Hess a(x, ξ) has two eigenvalues (counting multiplicities) f1 , f2 such that f1 ≥ 0 , −f2 ≥ 0 6) . Then in the appropriate coordinates locally a − τ = z1 z2 + V (z  )

(5.2.53)

with z  = (z3 , ... , z2d ). The goal of this Subsection is Theorem 5.2.10. Let condition and (5.2.20)− 2 be fulfilled. Let diam Ω ≤ c. 1−d Then RW ϑ(h/L). ϕ,L,reg is dominated by Ch Proof. Let us observe first that for fixed z  hyperbole Στ ,z  = {(z1 , z2 ) : (z1 , z2 , z  ) ∈ Στ } is parametrized by z1 ± z2 as ±V (z  ) < 0 and    dμτ ,z   | log |V (0)| + h |; (5.2.54) 1 Στ ,z  ∩{|z  |≥h 2 }

after integration with respect to z  this may lead to the factor | log h| in the final estimate and our purpose is to fight it. (a) First we consider zone where V is “variable enough”. Let us introduce the scaling functions (5.2.55)

1   l+1 l+1 |D α V | l+1−|α| + ζ¯l , ζl (z  ) = 

|α|≤l

with l ≥ 1. Then (5.2.56)

|∇ζl | ≤

1 2

2 ζ¯l = h l+3

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425

as  > 0 is small enough constant and (5.2.57)

γ(z)  |z1 | + |z2 | + ζ1 (z  )

and it trivially follows from the arguments of the previous Subsection 5.2.1 12) that 1

(5.2.58) The contribution of zone Ω ∩ {γ(z) ≤ C0h 2 } to the remainder  the 1 1−d  does not exceed Ch mes {z ∈ Ω : ζ1 (z  ) ≤ C1 h 2 } ϑ(h/L) and (5.2.59) The contribution of the zone Ω∩{γ(z) ≤ C0 ζ1 (z  )} to the remainder estimate does not exceed Ch1−d ϑ(h/L) where Ω = {(z  , z  ) : |z  | ≤ , z  ∈ Ω } and domain Ω is fixed. Moreover, the same arguments yield that (5.2.60) The contribution of the zone Ω ∩ {γ(z) ≤ (z  )} to the remainder estimate does not exceed   h  (z  ) 1−d (5.2.61) Ch ϑ + 1 dz  log  L Ω ζ1 (z ) provided (z  ) satisfies conditions (z  ) ≥ ζ1 (z  ) and |∇| ≤ 1. Denote by Jl the integral in (5.2.61) with (5.2.62)

l :=



ζkpk ,

pk = 2k−1 .

1≤k≤l

Let us prove by induction with respect to l ≥ 1 that (5.2.63) Jl does not exceed Cl . 12)

One needs to consider balls in R2d of radius ζ1 (z) for each z.

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For l = 1 this is obvious. Let us assume that it is true for some l; to make an induction in (5.2.63) we should prove that   l (z  )   dz (5.2.64) Jl := log l−1 (z  ) is bounded. Let us consider a ζ-admissible covering of Ω (with ζ = ζl ) and let us  consider the contribution of  each element to Jl . Surely, one needs to consider    only elements B z , ζ(z ) with l ≥ C2 l−1 with arbitrarily large constant C2 . The latter inequality means that ζk ≤ ζ pl /pk i. e. (5.2.65)

|D α V | ≤ 2 ζ 2

l−k (k−|α|+1)

∀k : |α| ≤ k ≤ l − 1

with arbitrarily small constant 2 > 0. On the other hand, (5.2.66)

|D α V | ≤ cζ l−|α|+1

∀α : |α| ≤ l

and for some α = α ¯ with |α| ≤ l the opposite inequality holds (5.2.67)

|D α V | ≥ 0 ζ l−|α|+1

as α = α ¯.

Picking in (5.2.65) k = l − 1 we get inequality |D α V | ≤ 2 ζ 2(l−|α|)

∀α : |α| ≤ l − 1

which is stronger than (5.2.66) and incompatible with (5.2.67) since 0 is fixed and 2 is arbitrarily small; thus (5.2.67) holds with some α = α ¯ : |¯ α| = l. Meanwhile (5.2.64) can only increase if we replace l−1 by the lesser pl−1 value ζl−1 where ζl−1 in turn is greater than (5.2.68) η := |D α V | α:|α|=l−1

while l l−1  1 + ζ pl −1 l−1 . Thus contribution of B(z  , ζ(z  )) to (5.2.64) does not exceed   ζ l   dz . log (5.2.69) C0 η B(z  ,ζ(z  )) Rescaling y = z  /ζ and introducing function W (y ) = V (ζy )ζ −l we see that  2d−2 (5.2.69) does not exceed C ζ | log φ(y )| dy with φ(y ) = α:|α|=l−1 |Dyα W | and

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427

(5.2.70) For some β with |β| = l, |Dyβ W | ≥ 0 .  Therefore it is sufficient to prove that | log φ(y )| dy ≤ C but this is obvious because in the appropriate coordinate system φ(y ) ≥ |y1 | (due to (5.2.70)). Thus, inequality Jl ≤ C is proven, the induction step is complete and (5.2.63) is also proven. As a corollary we obtain that (5.2.71) The contribution of the zone

{z = (z  , z  ) : z  ∈ Ω, γ(z) ≤ l (z  ) =

ζkpk ,

pk = 2k−1 }

1≤k≤l

to the remainder does not exceed Ch1−d ϑ(h/L). Thus, we have estimated properly the contribution of the zone {l ≥ γ}; so far the non-covered zone is {ζk ≤ γ 1/pk ∀k ≤ l} or equivalently {|D α V | ≤ γ (k−|α|+1)/2

k−1

∀α, k : |α| ≤ k ≤ l}.

Picking the optimal value k = |α| we get this zone (5.2.72)

{(z1 , z2 , z  ) : |D α V | ≤ cγ 2

1−|α|

∀α : |α| ≤ l, γ  |z1 | + |z2 |}

with arbitrarily large l. (b) Now we want to estimate the contribution of the larger (than (5.2.72)) zone defined by (5.2.73)

|D α V | ≤ cγ δ

∀α : |α| = 2,

γ  |z1 | + |z2 |

with arbitrarily small δ > 0 (as l = 2 we need to do it for δ = 12 ). First of all we consider the part of this zone with (5.2.74)

|∇V | ≥ γ 1+δ1

with δ1 < δ. Let us consider a point z¯ satisfying both inequalities (5.2.73) and (5.2.74). Without any loss of the generality one can assume that all the

CHAPTER 5. SCALAR OPERATORS AND RESCALING

428

components of ∇V (¯ z  ) vanish excluding σ = ∂z3 V . Then in the ν-vicinity  U of (0, z¯ ) (5.2.75)

c −1 ≤ |∇V | ≤ cσ,

V (z) = V (¯ z  ) + σ(z3 − z¯3 ) + O(ν 2 γ δ )

provided ν ≤ min(σγ −δ , γ δ ). Let V1 = V ∩ {z  : |V (z  )| ≤ cγ 2 }. One can see easily that due to (5.2.75) mes(V1 )ν 2−2d ≤ γ 2 σ −1 ν −1 ≤ γ δ2

(5.2.76)

with δ2 > 0 provided νσ ≥ γ 2+δ2 , ν ≥ γ 1−δ2 . Obviously conditions to ν are not contradictory for small enough δ2 > 0, δ1 > 0. Therefore (5.2.77) For any ε the contribution of the zone 1  {z : ε ≤ γ(z) ≤ 2ε, |∇2 V | ≤ γ δ , |∇V | ≥ γ 1+δ } 2 to the remainder does not exceed Ch1−d εδ2 ϑ(h/L) and therefore the total contribution of the zone in question does not exceed Ch1−d ϑ(h/L). So, we should consider only zone where inequalities (5.2.73) and |∇V | ≤ γ 1+δ

(5.2.78) hold.

In this zone we apply Theorem 4.5.12 after rescaling: so we use the long term propagation of singularities. (c) In this part of the proof we establish that (5.2.79) Contribution of γ-admissible element on which ω := |{z1 , z2 }| ≤ γ δ1

(5.2.80) to RW ϕ,L,reg does not exceed (5.2.81)

  h Ch1−d γ 2d−2 γ δ2 + hδ2 γ −2δ2 ϑ . L

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429

To prove (5.2.79) let us consider V which is 0 γ-vicinity with respect to z  and γ 1−δ3 -vicinity with respect to z  . Due to (5.2.77) we can assume that both (5.2.78) and (5.2.80) hold. Then in V the propagation speed with respect to z  does not exceed cωγ + cγ 1+δ ≤ cγ 1+δ1 while the propagation speed with respect to z  does not exceed cγ. Therefore for time T = γ −δ2 Hamiltonian trajectories remain in V. Moreover, for infinitesimal variations we obtain the inequality | ddt dz| ≤ C γ |dz| and therefore |dz(t)| ≤ C |dz(0)|. Hence, |DΨt | ≤ C . δ

Finally, let us check the non-periodicity condition for |t| ≤ T . One should consider only initial points with |¯ z1 | ≥ γ (or with the same inequality for z¯2 ). Obviously there exists a linear function ψ such that {ψ, z1 } = {ψ, V } = 0 and {ψ, z2 } = 1 at z¯ in which case ddt ψ = z1 + O(γ 1+δ ) and therefore ψ(z(t)) = ψ(z(0)) + z1 (0)t + O(γ 1+δ1 ) and the non-periodicity condition is fulfilled as well. Thus (5.2.79) is proven. It immediately implies that the total contribution of zone defined by (5.2.80) and (5.2.78) to the remainder does not exceed Ch1−d ϑ(h/L) and we arrive to (5.2.82) The total contribution of zone defined by (5.2.80) to the remainder does not exceed Ch1−d ϑ(h/L). (d) So, we should now consider only points satisfying (5.2.72), (5.2.73) and ω = |{z1 , z2 }| ≥ γ δ2

(5.2.83)

where one can choose δ2 > 0 arbitrarily. Consider ω(0, z  )+h 2 as a scaling function and a corresponding partition. Now we need to prove that 1

(5.2.84) As zˆ = (0, zˆ ) contribution to RW z , η) ϕ,L of η admissible element B(ˆ does not exceed Ch1−d η 2d−2 ϑ(h/L) with η := ω(ˆ z ). Let us again consider the Hamiltonian flow (which will be hyperbolic here); to be definite we assume that {z1 , z2 } > 0. We consider point z¯ and trace the trajectory z(t) with z(0) = z¯ as long as it remains in V which is η

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CHAPTER 5. SCALAR OPERATORS AND RESCALING

vicinity of zˆ with respect to z  and 1 η 2 vicinity with respect to z  ; due to (5.2.83) z¯ ∈ V. Recall that one can choose the time direction arbitrarily. Therefore without any loss of the generality one can assume that |¯ z2 |  γ  z¯1 because |z1 |  |z2 | implies that γ  ζ1 of Part (a). Note that due to (5.2.83) γ ≤ ω M with arbitrarily large M and that |∇V | ≤ γ 1+δ ≤ ωγ 1+δ4 . So, consider Hamiltonian equation; then under extra assumptions just imposed, for time (5.2.85)

T := ω −1 | log γ| ≥ C ω −1 | log ω|

this trajectory remains in V; and for positive time T ∗ ≤ C ω −1 | log γ| it leaves V for sure through |z2 |  ω 2 . Let us try to look at this trajectory in the positive time direction; then considering equation to variations we conclude that | ddt dz| ≤ cω|dz| and therefore |dz(t)| ≤ Ce cω|t| |dz(0)|. Hence, |DΨt | ≤ Ce cω|t| . Finally, since |z2 (t) − z2 (0)| ≥ 1 ωte 1 ωt we conclude that the nonperiodicity condition is fulfilled as well. Thus we can take T defined by (5.2.85). Then the total contribution of the element in question to the remainder does not exceed Ch1−d Jω 2d−1 ϑ(h/L) with  (5.2.86) J = γ −1 | log γ|−1 dγ; this integral has loglog divergence and we can avoid it considering only  γ ≤ hδ with arbitrarily small δ  > 0; to be on the safe side we also assume 1  that γ ≥ h 2 −δ . So we arrive to (5.2.87) Contribution of subzone described in (5.2.84) with extra condition 1   {h 2 −δ ≤ γ ≤ hδ } does not exceed Ch1−d ω 2d−1 ϑ(h/L). 

Thus we need to reconsider two subzones: outer subzone {hδ ≤ γ ≤ ω M } 1 1  and inner core {h 2 ≤ γ ≤ h 2 −δ }. (e) To tackle the former subzone let us consider trajectories issued from z¯ in the negative time direction (while before we considered positive time direction) as long as it remains in V but capping T by | log h|. Then it

5.2. SPECTRAL ASYMPTOTICS FOR SCALAR OPERATORS 

431



remains in the zone {|z2 | ≥ h2δ } and |dΨt | ≤ h−3δ and non-periodicity condition is fulfilled as well. Then the contribution of subzone in question does not exceed     1−d (5.2.88) Ch T− (x, ξ)−1 + | log h|−1 dxdξ : da+ Στ ∩U

 h  C  h1−d+δ ω 2d−2 ϑ L

where U is an element in question and we do not cap T− (z) anymore so T− (z) = ∞ if the trajectory in question does not leave V for time ≥ | log h|.  Obviously | log h|−1 dxdξ : da| ≤ C and we are left with  (5.2.89) T− (x, ξ)−1 dxdξ : da; Στ

and trajectories counted can leave V only through |z1 |  ω 2 . Since the measure dxdξ : da is invariant we conclude that integral (5.2.89) is of the same magnitude as if it was restricted to { 12 ω 2 ≤ γ ≤ ω 2 } and T− was replaced by the time T  spent by the trajectory in this zone; this removes logarithmic divergency. So we arrive to (5.2.90) Contribution of subzone described in (5.2.84) with extra condition  {hδ ≤ γ ≤ ω M } does not exceed Ch1−d ω 2d−1 ϑ(h/L). z1

z2 Figure 5.1: Solid trajectories leave V for a capped time; dashed trajectories do not and are not considered in (5.2.89); the shadow indicate zone γ  ω 2 . (f) In order to treat the remaining zone let us apply the routine rescaling 1  xnew = (x − x¯)¯ γ −1 and hnew =  := h¯ γ −2 with γ¯ = h 2 −δ . We get an

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CHAPTER 5. SCALAR OPERATORS AND RESCALING

operator of the same type but with complete symbol a = z1 z2 + V¯ + ... where {z1 , z2 } = ω + ... where the dots denote symbols all the derivatives of which do not exceed M with arbitrarily large M and are therefore negligible, V¯ and ω are constant and V¯ is not necessarily a scalar; here ω ≥ δ1 (otherwise we can apply rescaling and arrive into case already considered in Part (c) 1 1  and only the contribution of the zone h 2 γ ≤  2 −δ needs to be treated (with γ = γnew = γold /¯ γ as well and δ1 and δ2 are new but still arbitrarily small). Now δ0 = 0. Without any loss of the generality one can assume that z1 = hD1 and either z2 = ωx1 or z2 = ωx1 + x2 (which can be arranged otherwise by an appropriate symplectic change of variables). Moreover, changing coordinates  x1new = x1 + ω −1 x2 , xnew = x  if necessary and multiplying the operator by −1 ω we get exactly the operator x1 D1 + i2 + V¯new (modulo negligible) with V¯new = ω −1 V¯ . ¯ with γ¯ = |¯ All that remains to do is to consider points (¯ x , ξ) x1 | ≥ |ξ¯1 | (points with the opposite inequality are treated similarly with the negative time direction) and to prove that one can take T = | log |. But for this very specific operator the proof is direct: one can consider operators of −1 −1 multiplication; then e −it A φe it A ≡ φt with φt (x) = φ(e −t x) and for the indicated t ≥  we obtain that φt φ = 0 provided supp φ is small enough. This proves that the contribution of the element of the partition in  question to the remainder estimate is O h1−d ωγ 2d−2 | log h|−1 ϑ(h/L) and the  total contribution of such elements is O h1−d ω 2d−1 ϑ(h/L) as well because factor | log h|−1 compensate for logarithmic divergence of the integral. Thus we arrive to (5.2.91) Contribution to the remainder of the subzone described in (5.2.84) 1  with extra condition {h 2 ≤ γ ≤ h1/2−δ } does not exceed Ch1−d ω 2d−1 ϑ(h/L). Now combining (5.2.87), (5.2.90) and (5.2.91) we obtain (5.2.84) which implies the proper estimate Ch1−d ϑ(h/L) of the remaining zone. Finally, Theorem 5.2.10 has been proven. 

Remark 5.2.11. (i) Thus in the inner zone {γ ≤ hδ } we considered trajectories moving from degeneration while in the outer zone we considered trajectories moving to degeneration.

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433

(ii) We did not exploit that actually instead of factor ϑ(h/L) we had factor ϑ(h/LT ). (iii) Now regularization means that as h must be included in the principal part; otherwise generally remainder estimate would contain logarithmic factor. (iv) Assuming that V = 0 identically, one can use the canonical form of such operators. Namely, it was proven (for similar proofs see, e. g., Section 13.3) that in the framework of the current analysis with {z1 , z2 }  1 (5.2.92)

F ∗ AF ≡



1 amn (x  , hD  )(x1 hD1 − ih)m hn 2 m,n:m+n≥1

for an appropriate Fourier integral operator F such that in the vicinity of the point in question F ∗ F ≡ I . (v) The most reasonable example is 1-dimensional Schr¨odinger operator (5.2.93)

h2 D12 − h2 x12 .

Actually we proved even more precise Theorem 5.2.12. Let condition (5.2.20)− 2 be fulfilled. Then contribution of Ωε to the remainder does not exceed    h |{z1 , z2 }| dz  + C  hδ ϑ (5.2.94) Ch1−d η + L {ζm 0. Proof. Consider ζm -admissible covering. Arguments of the proof of Theorem 5.2.10  imply that the total contribution of zone {ζm ≥ ε1 } does not exceed ηh1−d + C  h1−d+δ ϑ(h/L); therefore only zone {ζm ≤ ε1 } needs to be reexamined. Here ε1 is an arbitrarily small constant. Further, reexaming this proof one can notice that then only zone where {ω > ε1 } needs to be reconsidered and contribution of the latter zone does not exceed (5.2.94). This opens the door to more precise remainder estimates. In particular, we have

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Theorem 5.2.13. Assume that (5.2.95) μτ¯ (Π ∩ Ω+ ) = 0 and mes({V = τ¯, {z1 , z2 } = (5.2.96)  0}) = 0. Then estimate (5.2.51)   h  1−d RW (¯ τ ) = o h ϑ ϕ,L,reg L holds. + − So, despite very essential between   2 and (5.2.20)2  −d differences  −d (5.2.20) cases, both estimates O h ϑ(h/L) and o h ϑ(h/L) are proven in the both of them; the latter under assumptions (5.2.95) and (5.2.96). Our analysis under assumption (5.2.20)2 is complete.

5.2.3

From Condition “rank Hess a ≥ 2 at Critical Points” to “rank Hess a ≥ 1 at Critical Points”

Now we will consider operators with stronger degenerations and we consider only scalar operators. We will indicate later what order of matrix symbol can be added. So, let us assume that (5.2.97) A is a scalar operator13) . In this subsection we assume that condition (5.2.20)1 is fulfilled. Then locally in the appropriate coordinate system   z  = (z2 , ... , z2d ) (5.2.98) a = ± z12 + V (z  ) , where here and below a means the complete symbol of A 13) . However we need to pass to symplectic coordinates with z1 = ξ1 . Without any loss of the generality we can assume that |∂ξ1 z1 | ≥ 0 (locally) and then by Malgrange preparation theorem (Malgrange [1]) we rewrite (5.2.98) as  2  a = b(x, ξ) ξ1 − α(x, ξ  ) + V (x, ξ  ) . 13) In fact, in this subsection one needs to assume that this condition is fulfilled only modulo O(h2 ).

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435

Then after the symplectic transformation ξ1new = ξ1 − α(x, ξ  ), (x, ξ  )new = ψ(x, ξ  ) we obtain the same formula with α = 0. Thus, without any loss of the generality one can assume that   b(x, ξ) ≥ 0 (5.2.99) a(x, ξ) = b(x, ξ) ξ12 + V (x, ξ  ) , where we changed the sign if needed. In what follows z  = (x, ξ  ). Theorem 5.2.14. Let conditions (5.2.98) and (5.2.20)1 be fulfilled. Then (i) If (5.2.100)m



|D α V | ≥ 

|α|≤m

  1−d ϑ h/L . with some m ≥ 2 then RW ϕ,L is dominated by Ch (ii) If (5.2.101) ϑ(τ )τ −s is a monotone increasing function with some s > 0   1−d ϑ h/L as well. then RW ϕ,L is dominated by Ch (iii) In the general case estimate (5.2.102)

1−d−δ RW ϑ ϕ,L ≤ Ch

h L

holds for arbitrarily small δ > 0. In the smoothness conditions K depends on m, ϑ and δ > 0. Proof, Part I. First of all, this theorem is proven under condition (5.2.100)m for m = 2 (see Theorems 5.2.2 and 5.2.10). Let us assume that Statements (i), (ii) are proven for given m = m ¯ − 1 ≥ 2 and let us prove it for m = m. ¯ Let us introduce the function γ = ζm−1 by formula (5.2.55): (5.2.103)

1   l+1 l γ :=  |D α V | m−|α| + γ¯m ,

|α|≤m−1

2

γ¯m = h m+2

436 then |∇γ| ≤

CHAPTER 5. SCALAR OPERATORS AND RESCALING 1 2

(so γ is a scaling function) and

(5.2.104) with (5.2.105)

|D α V | ≤ C ρ2 γ −|α|

∀α : |α| ≤ K m

ρ=γ2.

Let us consider the point (¯ x , ξ¯ ). Let U  be its γ-vicinity (with respect to (x, ξ  )) with γ = γ(¯ x , ξ¯ ) ≥ h2/(m+2) . (a) Let us consider zone {|ξ1 | ≥ C0 ρ} with ρ defined by (5.2.105) and let 1 U = {(x, ξ) : (x, ξ  ) ∈ U  , |ξ1 − ξ¯1 | ≤ |ξ¯1 |} 2 with |ξ¯1 | ≥ CO ρ. Let us rescale xnew = (x − x¯)γ −1 , multiply the operator by ρ−2 and introduce hnew = hρ−1 γ −1 with redefined temporarily ρ := |ξ¯1 |. Let us note that the ellipticity condition is fulfilled and  ≤ 1,

(5.2.106)

hγ −2 ≤ h1− m 2

due to our choice of γ¯ . Therefore contribution of U to the remainder does not exceed (5.2.107) R(γ, ρ, h) := Cϑ

 hρ  h r 2d−2  hρ  h r −d+1  γ d−1 = Ch1−d ϑ γ Lγ ργ ρ Lγ ργ

with arbitrarily large r ; here factor (γ/ρ)d−1 appears since U  is covered by this number of balls of radius ρ with ξ  and we plugged Lnew = Lρ−2 and hnew /Lnew = hρ/Lγ into ϑ(.). Observe that R(γ, ρ, h)ρr /2 is decaying function of ρ provided r is large enough. Therefore after summation on a |ξ1 |-admissible partition of unity in {(x, ξ) : (x, ξ  ) ∈ U  , |ξ1 | ≥ cρ} we obtain that the contribution of this zone to the remainder estimate does not exceed CR as well with ρ restored to its (5.2.105) definition. Note that (5.2.108)

 h  ρ s  hρ  ≤ϑ ϑ Lγ L γ

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437

with s > 0 under assumption (5.2.101) and s = 0 otherwise. Using (5.2.108) and (5.2.105) we arrive to (5.2.109)

 h  h r 1 (m−2)s+2d−2 R ≤ Ch1−d ϑ . γ2 L γ 12 (m+2)

Summation with respect to (2d − 1)-dimensional γ-admissible partition results in   h   h r 1 (m−2)s−1  (5.2.110) Ch1−d ϑ γ2 dz . 1 L γ 2 (m+2) Note that under condition (5.2.100)m this integral is converging as r > 0. (b) Now let us consider zone U = {(x, ξ) : (x, ξ  ) ∈ U  , |ξ1 | ≤ cρ|}. Let us rescale and multiply the operator in the same way as above. Let us note that if γ ≥ c γ¯ then condition (5.2.100)m−1 is fulfilled14) and the contribution of U to the remainder does not exceed (5.2.107) with r = 0; this we estimate by (5.2.109) with r = 0 and summation with respect to partition results in (5.2.110) with r = 0, and this integral is bounded under condition (5.2.100)m as s > 0. (c) proof-5-2-14-c If γ ≤ c γ¯ then hnew  1 and one does not need to check condition (5.2.100)m−1 and therefore the same arguments hold. (d) So, under assumption s > 0 we can make the step of induction and RW ϕ,L is properly estimated under condition (5.2.100)m . Further, as (m − 2)s > 2 integral (5.2.109) is bounded even without condition (5.2.100)m and therefore Statement (ii) of theorem is proven. On the other hand, if s = 0 with each step of induction factor | log h| appears and we arrive to estimate (5.2.102) but with an extra factor | log h|m−3 . Further, on the step without condition (5.2.100)m with large enough m factor γ −1 appears with γ = h2/(m+2) . Therefore statement (iii) of theorem is proven as well. 14) More precisely, this condition is fulfilled if γ  ≥ γ where γ  is introduced by the same formula (5.2.103) with differentiation only on x. However before rescaling one can apply the symplectic transformation (xj , ξj ) → (ξj , −xj ) with some j = 2, ... , d if necessary and fulfill this condition.

CHAPTER 5. SCALAR OPERATORS AND RESCALING

438

Proof, Part II. Now, our goal is to get rid of logarithmic factor for s = 0 under condition (5.2.100)m . We will follow the proof of Theorem 5.2.10 with some modifications; to make comparison more clear we restart numeration of the parts of the proof. (a) First of all, without any loss of the generality one can assume that |∂zm2 V | ≥ 0

(5.2.111)m

(condition (5.2.100)m yields this condition after -partition and appropriate choice of coordinate system). Then Malgrange preparation theorem yields that   (5.2.112) V = β(z  ) z2m + W1 (z  )z2m−1 + ... + Wm (z  ) with |β(z  )| ≥ , z  = (z3 , ... , z2d ). Without any loss of the generality one can assume that W1 = 0; really we can achieve it by z2 → z2 + m1 W1 . Then  on the induction step γm−1 = 1 |z2 | + γm−1 (z  ) where  m 1  γm−1 = |D α Wk | k−|α| m (5.2.113)m 2≤k≤m |α|≤(k−1)  with |∇γm |≤

1 2

and one can see easily that due to the induction assumption

 (5.2.114) Contribution of zone {|z2 | ≤ C0 γm } to R2ϑ does not exceed 1−d Ch ϑ(h/L).  Thus we need to consider only zone {|z2 | ≥ C0 γm } which was the source of the logarithmic factor.

Now let us introduce (5.2.115)m,l

ζm−1,j = 





mj

|D α Wk | kj−|α|m

 1j

2≤k≤m |α|≤j(k−1)/m

with j ≥ m; it coincides with (5.2.113)m as j = m. Here j ∈ R, j > 0. Note that (5.2.116) Conditions (5.2.117)m,j

ζm−1,j ≥ 

are essentially different for given m only for a monotone sequence ji ∈ Q (which denotes the set of rational numbers), ji → ∞, ji+1 − ji ≤ 1 ∀i.

5.2. SPECTRAL ASYMPTOTICS FOR SCALAR OPERATORS

439

Introducing j in the same way as in the proof of Theorem 5.2.10, Part (a) and repeating relevant arguments we arrive to the proper estimate of zone {|z2 | ≤ C ρm,l } with redefined (5.2.118)

1/m  ≥ ch2/(m+2) γ := |z2 |m + |z1 |2

and in particular of zone defined by |D α Wk | ≥ γ δ |α|+k=m

with some δ > 0. Easy details are left to the reader. (b) So, we need to consider zone where opposite inequality (5.2.119)



|D α Wk | ≤ γ δ

|α|+k=m

holds; this inequality is a new incarnation of (5.2.73) and repeating arguments of the proof of Theorem 5.2.10, Part (b) we reduce it to zone defined by (5.2.120)

|D α Wk | ≤ C γ m+δ−k−|α|

∀α, k : |α| + k ≤ m;

this is a new incarnation of (5.2.78); we use induction with respect to j to prove that if (5.2.120) is fulfilled for all α, k with |α| + k > m − j and some δ > 0 and if (5.2.120) fails for some α, k with |α| + k = m − j and any δ > 0 then (5.2.76) holds with V1 = V ∩ {z  : |V (z  )| ≤ cγ (m+2)/2 }. So, δ = δj is decreasing but we take the smallest one in the end. (c) So, we need to consider zone where (5.2.120) and (5.2.80) hold. Recall that (5.2.121)

  a = ± z12 + β(z  )z2m + W (z  )

where (5.2.122)

|D α W | ≤ cγ m+δ−|α| .

As β > 0 and m is even proof is no different from one under condition m (5.2.20)+ 2 . As m is odd this is true for subzone where βz2 > 0.

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CHAPTER 5. SCALAR OPERATORS AND RESCALING

So we need to consider the remaining cases: either even m and β < 0 or odd m, β < 0 and z2 > 0 15) . Consider (5.2.123)

m/2

w1 = z1 − (−β)1/2 z2

m/2

, w1 = z2 + (−β)1/2 z2

which will play role of z1 and z2 . Then (5.2.124)

|w1 | + |w2 |  γ m/2 ,

(5.2.125)

{w1 , w2 } = 2ω(−β)1/2 z2

(m−2)/2

m/2

+ O(z2

),

ω = {z1 , z2 }

and we need to rescale time t = tnew = told γ (2−m)/2 . Then analysis in zone {|ω| ≤ γ δ2 } is exactly as it was in the proof of Theorem 5.2.10(c); as before ω = {z1 , z2 }. (d) Finally analysis in zone where |ω| ≥ γ δ2 repeats arguments of the proof of Theorem 5.2.10 (d)–(f) but with a twist: we always consider trajectory launched in the direction of decreasing γ(z(t)). Indeed, because propagation slows down near w = 0 faster than before we can cap time by T = γ )δ2 with δ2 > 0 rather than by T = | log γ/¯ γ|  (γ/¯ −1 as we did then; now Στ T dxdξ is bounded (instead of possible loglog divergence) and the same arguments of that proof Part (e) work in the whole zone {¯ γ ≤ γ ≤ }. In the case V = const (treated as a parameter) one can divide the operator by ρ2 = max(|V |, h2 ) and introduce hnew = hρ−1 (no rescaling with  respect to x is necessary). Then the remainder estimate O h1−d ϑ(h/L) holds. Therefore we arrive to Conjecture 5.2.15.   For operators with analytic symbols the remainder estimate O h1−d ϑ(h/L) holds under condition (5.2.20)1 even without condition (5.2.101).

5.2.4

From Condition “rank Hess a ≥ 1 at Critical Points” to a Weak non-Degeneracy Condition

Now even condition “rank Hess a ≥ 1” seems to be too restrictive for us and we are going to get rid of it. 15) One can reduce the case of odd m, β > 0 and z2 < 0 to this one replacing z2 , β by −z2 , −β.

5.2. SPECTRAL ASYMPTOTICS FOR SCALAR OPERATORS

441

Theorem 5.2.16. Let A be a scalar operator. Then (i) Under assumption

(5.2.126)n

|D α a| ≥ 0

|α|≤n 1−d−δ RW with arbitrarily small δ > 0. ϕ,L is dominated by C ϑ(h/L)h 1−d . (ii) Under condition (5.2.101) with s > 0 RW ϕ,L is dominated by C ϑ(h/L)h

Proof. To prove both Statements (i) and (ii) we apply induction on n. Note that for n = 2 theorem is already proven. (a) So suppose that under condition (5.2.126)n−1 both Statements (i) and (ii) are already proven and let us consider some point z¯ at which condition (5.2.126)n holds but (5.2.126)n−1 fails. Then in the appropriate symplectic coordinates |∂ξn1 a(¯ z ) ≥ 0 and then according to Malgrange preparatory theorem in the vicinity of z¯   (5.2.127) a(x, ξ) = b(x, ξ) ξ1n−j Vj (x, ξ  ) with |b(x, ξ)| ≥ 0 0≤j≤n

and with (5.2.128)

V0 = 1,

V1 = 0.

Now instead of a single function V we have several functions Vj and we want to prove remainder estimate (5.2.126) under an extra condition (5.2.129)n,l |D α Vj | ≥ 0 2≤j≤n, |α|≤lj/n

with l ≥ n; later we will drop it for large ν. First let us treat the case l = n. Let us introduce the functions  n1  n 1 α j−|α| := γn =  |D Vj | + γ¯ with γ¯ = h 2 (5.2.130)n γ 2≤j≤n, |α|≤j−1

and ρ = γ. Then |∇γ| ≤ and (5.2.131)

1 2

for small enough  > 0 (so γ is a scaling function)

|D β Vj | ≤ cρj γ −|β|

∀j, β.

CHAPTER 5. SCALAR OPERATORS AND RESCALING

442

Let us rescale as usual xnew = (x − x¯)γ −1 , multiply the operator by ρ−n and introduce hnew = hρ−1 γ −1 . Then for γ ≥ c0 γ¯ after this procedure the smoothness conditions and condition(5.2.126)n−1 are fulfilled and hence the contribution of the zone {z : z  ∈ B z¯ , γn (¯ z  ) , |ξ1 | ≤ cρ} to the remainder does not exceed  h −d+1−δ  hρn−1  ϑ (5.2.132) C ργ Lγ where δ = 0 under condition (5.2.101) with s > 0 and δ = 0 otherwise; later ρ will be different from γ. We used the fact that Lnew = Lρ−n due to multiplication of operator and therefore (5.2.133)

hnew hρn−1 = Lnew Lγ

Therefore the total contribution of the zone {z : |ξ1 | ≤ cρ, ργ ≥ h} to the remainder does not exceed   hρn−1   dz . (5.2.134) Ch1−d−δ γ −1+δ ϑ Lγ Meanwhile for ργ  h, |ξ1 | ≤ cρ (i. e. for ρ  ρ¯ and γ  γ¯ ) we obtain hnew = 1 and the contribution of the corresponding element of the partition to the remainder estimate does not exceed C ϑ(hρn−1 /Lγ) and the total contribution of zone {ργ  h, |z1 | ≤ cρ} to the remainder does not exceed (5.2.134) with integral restricted to his zone. Finally, in the zone {|ξ1 | ≥ cρ} we apply an |ξ1 |-admissible partition and standard elliptic arguments and then the contribution of each element does not exceed C (h/ξ12 )r ϑ(hξ1n−2 /L) with arbitrarily large r and therefore the total contribution does not exceed (5.2.134). (b) Let us notice that under condition (5.2.129)n,n in the appropriate coordinate system γ ≥  |z2 | and then (5.2.134) with ρ = γ does not exceed Ch1−d−δ ϑ(h/L). This is trivial as δ > 0; as δ = 0 and (5.2.101) is fulfilled we use estimate  h   ρn−1 s  hρn−1  ≤ϑ · (5.2.135) ϑ Lγ L γ with s > 0. Therefore we proved that

5.2. SPECTRAL ASYMPTOTICS FOR SCALAR OPERATORS

443

(5.2.136) Assuming that under condition (5.2.126)n−1 a proper remainder estimate is proven, we prove it also under conditions conditions (5.2.126)n and (5.2.129)n,n . It is not a complete induction; in the next step we apply induction with respect to l as well. Moreover, let us notice that (5.2.137) Under condition (5.2.101) we obtain a proper remainder estimate even without condition (5.2.129)n,n for (n − 2)s ≥ 1. So, under condition (5.2.101) the estimate will be proven for any n if it was proven for n ≤ 2 + s −1 . (c) Let us assume that Statements (i) and (ii) are proven under condition (5.2.126)n−1 and also under conditions (5.2.126)n and (5.2.129)n,l . Now our goal is to prove them under conditions (5.2.126)n and (5.2.129)n,l  with some l  > 0 and to get rid of the latter condition for l  large enough. Obviously statement similar to (5.2.116) holds. Let us introduce the scaling function  (5.2.138) γ := γn,l (z  ) = 



α

|D Vj |

nl jl−|α|n

 nl1

+ γ¯nl ,

n

γ¯nl = h n+l

2≤j≤n, |α|≤jl/n

with small enough  > 0. Further, let ρ = γ l/n . One can easily see that |∇γ| ≤ 12 and that (5.2.131) holds. Moreover, for some j and α : |α| < jl/n the opposite inequality holds (with 1 > 0 instead of c) provided γ ≥ c γ¯ . Therefore, after the routine rescaling procedure with xnew = (x − x¯)γ −1 , hnew = hρ−1 γ −1 and with multiplication operator ρ−n we obtain the routine smoothness conditions and either condition (5.2.129)n,l or hnew  1; we consider now the zone {|ξ1 | ≤ cρ} and if necessary apply a linear symplectic transformation for (x  , ξ  ). In both cases the contributions of the corresponding balls was described above and the contributions of the corresponding zones are given by (5.2.134).

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CHAPTER 5. SCALAR OPERATORS AND RESCALING

In the zone {|ξ1 | ≥ cρ} elliptic arguments are applicable and taking into account that γ 2 ≥ h1−σ for l > n we easily obtain that the contribution of this zone does not exceed the (5.2.134) as well. So, the total remainder estimate is given by (5.2.134). Recall that this deduction holds under the assumption that for given l ≥ 1 condition (5.2.129)n,l yields the proper remainder estimate. (d) Let us assume now that condition (5.2.129)n,l  is fulfilled. Then there exist α and j with |α| ≤ l  j/n such that |D α Vj | ≥ . Then in the appropriate coordinates |D β Vj | ≥ |z2 | for some β : |β| = |α| − 1 and picking l = (|α| − 1)j/n we obtain that γ ≥ |z2 |. Then integral in (5.2.134) is bounded. Therefore remainder estimate (5.2.126) under condition (5.2.129)n,l  holds provided it holds under condition (5.2.129)n,l . We can restrict ourselves to numbers l = m/(n − 1)! with integers m ≥ n and apply induction on m. Moreover, under condition (5.2.101) with s > 0 and l > l(n, s) none of the final estimates linked with z2 and condition (5.2.129)n,l are necessary. These estimates can also be avoided in the case s = 0 as well (for m ≥ mn,δ ) by doubling δ. Therefore the induction step from n − 1 to n is complete. So, Statements (i) and (ii) are proven under condition (5.2.126)n with arbitrarily large n. (e) Finally we need to get rid of condition (5.2.126)n in Statement (ii). Let us introduce the scaling functions 1  n 1 (5.2.139) γn =  |D α a| n−|α| n + γ¯n , γ¯n = h 2 . α:|α|≤n−1

Then |∇γ| ≤

1 2

and |D α a| ≤ cγ n−|α| 1

∀α : |α| < n

and either γ ≤ ch 2 or the opposite estimate holds for some α with |α| < n.

5.2. SPECTRAL ASYMPTOTICS FOR SCALAR OPERATORS

445

Then applying the rescaling procedure xnew = (x − x¯)γ −1 , hnew = hγ −2 and multiplying the operator by γ −n we get into the situation of (5.2.126)n−1 or h  1 respectively and applying an estimate Ch1−d ϑ(h/L) we obtain that the contribution of γ-element to the remainder does not exceed Ch1−d γ 2d−2 ϑ

 hγ n−2  h h ≤ Ch1−d γ 2d−2+(n−2)s ϑ ≤ Ch1−d γ 2d ϑ L L L

as (n − 2)s ≥ 2. In this case summation with respect to partition yields the required remainder estimate. Remark 5.2.17. (i) Without condition (5.2.101) with s > 0 we obtain the unpleasant factor h−δ only when we get rid of condition (5.2.129)n,l . All other steps of our analysis give only additional factors | log h|M . However, we cannot go from conditions (5.2.126)n−1 , (5.2.129)n−1,l to (5.2.126)n , (5.2.129)n,n without dropping condition (5.2.129)n−1,l first unless we assume that the condition (5.2.140)r |D α Vj | ≥  2≤j≤n−r , |α|≤j

holds with r = n − 2. Indeed, this jump invokes γn defined by (5.2.130)n and we need to check that for some k ≤ n − 1 and β k−1−|β|(n−1)/l |D α Vj | j−|α| ; |D β Vk |  γn(k−1−|β|(n−1)/l)  2≤j≤n,|α|≤j−1

one can rewrite this condition as |D α Vj |1/(j−1−|α|(n−1)/l)  2≤j≤n−1, α≤j



|D α Vj |1/(j−|α|) ;

2≤j≤n, |α|≤j−1

and this is not guaranteed unless |D α Vj |  1 for some α : |α| = 1 and j = 2; let us recall that V1 = 0. So in order to provide the remainder estimate O(h1−d | log h|M ) one should assume that condition (5.2.140)r is fulfilled for r = n−2; under this condition one can prove the estimate easily.

CHAPTER 5. SCALAR OPERATORS AND RESCALING

446

(ii) On the other hand, if we assume that V2 , ... , Vn−r −1 vanish identically while condition (5.2.140)r holds, we observe that the problem is reduced (after a few steps of the induction described above) to the problem with a = bξ1r with r = n − r − 1 and non-vanishing symbol b. Then the same partition arguments yield the remainder estimate O(h1−d ) because the zone {|ξ1 | ≥ h} is elliptic. Therefore a proper remainder estimate remains true but we do not know whether it is uniform with respect to the spectral parameter. (iii) In certain cases (under some additional conditions) propagation of singularities arguments provide the remainder estimate O(h1−d ) but I never investigated it.

The remark above leads us to W is dominated Conjecture 5.2.18. For operators with analytic symbols Rϕ,L  1−d  by O h ϑ(h/L) .

5.2.5

From Microlocal to Local Spectral Asymptotics

Let us go from microlocal to local spectral asymptotics. Then the results of the previous subsections combined with the results of Section 4.2 imply

Theorem 5.2.19. (i) Let condition (4.2.45) be fulfilled. Then the assertions of all the above statements remain true for Q1 = Q2 = I provided their conditions are fulfilled for Ω = T ∗ B(0, 1) 16) . (ii) Moreover if condition (4.2.48)± is fulfilled then in every estimate without mollification with respect to τ1 one can take τ1 = ∓∞ and in every estimate with mollification one can drop the mollification and take τ1 = ∓∞. In fact, it is necessary to check them only for Ω = B(0, 1) × B(0, C0 ) with a large enough C0 . 16)

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

5.3

447

Local and Pointwise Spectral Asymptotics for the Schr¨ odinger and Dirac Operators

In this section we treat the Schr¨odinger operator (5.3.1) A= Pj g jk Pk + V , Pj = hDj − Vj j,k

and Dirac operator (5.3.2)

A=

1 j,l

2

σl (ω jl Pj + Pj ω jl ) + σ0 m + V · I

where the components g jk of the inverse metric tensor, coefficients ω jl , the components of the vector-potential Vj and the scalar potential V are real-valued functions, σl are Pauli matrices (see Subsection 5.3.2 and Appendix 5.A.2), m = const ≥ 0 is a mass parameter, and h ∈ (0, 1). Constants in conditions and in estimates depend neither on h nor m. Remark 5.3.1. As d ≥ 2 one can consider also the Schr¨odinger-Pauli operator  1 2 (5.3.3) A= σl (ω jl Pj + Pj ω jl ) + V 2 j,l which differs from (5.3.1) only by matrix lower order terms, but in the current framework (without strong magnetic field as in Volumes III and IV) it does not make any difference. We assume that (5.3.4) A is self-adjoint operator in L2 (X , H), H = CD , D = 1 for the Schr¨odinger operator.

5.3.1

Schr¨ odinger Operator

Main Assumptions Let us start with the Schr¨odinger operator assuming that (5.3.5)

B(0, 1) ⊂ X ,

448 (5.3.6)1−3

CHAPTER 5. SCALAR OPERATORS AND RESCALING |D α g jk | ≤ c,

and (5.3.7)

0 ≤



|D α Vj | ≤ c,

g jk (x)ξj ξk |ξ|−2 ≤ c

|D α V | ≤ c

∀α : |α| ≤ K ,

∀x ∈ B(0, 1) ∀ξ ∈ Rd \ 0.

j,k −1

Remark 5.3.2. A (unitary) gauge transformation u → e ih ϕ(x) u with a realvalued function ϕ can be applied which for the Schr¨odinger and Dirac operators is equivalent to the transformation Vj → Vj + ∂j ϕ. Hence one can weaken condition (5.3.6)2 , replacing it with (5.3.6)2

|D α Fjk | ≤ c

∀α : |α| ≤ K

where (5.3.8)

Fjk = ∂k Vj − ∂j Vk

are components of the tensor magnetic intensity. Here the principal symbol of A equals g jk (ξj − Vj )(ξk − Vk ) + V (5.3.9) a(x, ξ) = j,k

and the subprincipal symbol vanishes and hence the coefficients in the spectral asymptotics are (5.3.10)

d/2 √

κ0 (x, τ ) = (2π)−d d (τ − V )+

g

and (5.3.11)

κ1 (x, τ ) = 0

where τ1 = −∞, τ2 = τ , d is the volume of the unit ball in Rd and g = det(g jk )−1 . We prefer to use notation κn (x, τ ) rather than κxn (τ ) 17) . Asymptotics with Spatial Mollification Then the results of the previous Section 5.2 immediately yield the following assertions: 17) For Schr¨ odinger-Pauli operator (5.3.3) κ0 acquires factor D and κ1 = 0 because tr(as ) = 0.

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

449

Theorem 5.3.3. Let conditions (5.3.1), (5.3.4), (5.3.5), (5.3.6)1−3 and (5.3.7) be fulfilled and d ≥ 2. Let ϕ satisfy (4.2.73)–(4.2.75). Then as Q1 = I , Q2 = ψ(x) (i) The following estimate holds:  1−d    h  |RW + C  h1−d+δ ϑ ϕ,L | ≤ Ch L

(5.3.12)

where here and below C = C (d, c), K = K (d), δ  = δ  (d) > 0 is small enough and supx |ψ(x)| ≤ 1, C  = C ||ψ|| CK . (ii) Let Λ1 , ... , Λn be closed subsets of Στ = {(x, ξ) ∈ T ∗ B(0, 1), a(x, ξ) = τ } and let Λj ⊂ {γ 2 = |τ − V | + |∇x V |2 ≥ h1−δ } with δ > 0. Further, let us assume that through each point of Λj there passes a Hamiltonian trajectory of a(x, ξ) with t ∈ Jj , Jj = [0, Tj ] or Jj = [−Tj , 0], Tj ∈ [1, h−δ ] along which 

(5.3.13)1−3 |D α g jk | ≤ ch−δ ,



|D α Vj | ≤ ch−δ ,



|D α V | ≤ ch−δ ∀α : |α| ≤ K 

and    dist x(t), ∂X ≥ hδ ,   1 dist (x(t), ξ(t)), (x, ξ) ≥ |t|h 2 −δ

(5.3.14) (5.3.15)



and condition (2.4.29) holds with ω = h−δ with small enough exponent δ  = δ  (d, δ) > 0. Furthermore, let (5.2.101) be fulfilled with s ≥ 0. Then (5.3.16)

|RW ϕ,L |

 h1−d   h  1−d+δ  ≤C dxdξ : da + C h ϑ L T 1+s Λj 0≤j≤n j

where T0 = 1, Λ0 = Στ \ (Λ1 ∪ · · · ∪ Λn ), K  = K  (d, s, δ), C  = ||ψ|| CK  . Proof. Indeed, condition (5.2.20)+ d is fulfilled and therefore Statement (i) as d ≥ 2 and (ii) as d ≥ 3 follow automatically.

450

CHAPTER 5. SCALAR OPERATORS AND RESCALING

Further, as d = 2 for Statement (ii) we need to apply Proposition 5.2.6 but since as = 0 we have μτ = μτ . Note that in this case condition mes({V = τ¯, {z1 , z2 } = 0}) = 0 of Theorem 5.2.9(ii) translates into mes({V = τ¯, F12 = 0}) = 0

(5.3.17)

with Fjk defined by (5.3.8). Alternative proof. Alternatively, one can repeat more rigorously arguments of Subsection 5.1.1. Let us start from estimate (5.3.12), (5.3.16) which hold under microhyperbolicity condition (5.3.18) estimate

|V | + |∇V | ≥ 0

in B(0, 1),

(5.3.19)

|e(x, x, 0)| ≤ Chr

1 in B(0, ) 2

which holds with arbitrarily large exponent r under assumption V (x) ≥ 0

(5.3.20)

in B(0, 1)

and generic estimate (5.3.21)

|e(x, x, 0)| ≤ Ch−d

1 in B(0, ) 2

which holds with no extra condition. Rescaling them x → xγ −1 , hD → D and h →  = hρ−1 γ −1 we get estimate (5.3.22) under condition (5.3.18)∗ estimate (5.3.19)∗

|RW | ≤ Ch1−d ρd−1 γ −1 |V | + γ|∇V | ≥ 0 ρ2

in B(0, γ),

|e(x, x, 0)| ≤ Chr ρ−s γ −d−r

1 in B(0, γ) 2

which holds with arbitrarily large exponent r under assumption (5.3.20)∗

V (x) ≥ 0 ρ2

in B(0, γ)

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

451

and generic estimate (5.3.21)∗

|e(x, x, 0)| ≤ Ch−d ρd

1 in B(0, ) 2

which holds with no extra condition; here we assume that ργ ≥ h

(5.3.23) and replace (5.3.6)1−3 by (5.3.6)∗1−3 |D α g jk | ≤ cγ −|α| ,

|D α Vj | ≤ cργ −|α| ,

|D α V | ≤ cρ2 γ −|α|

∀x ∈ B(0, γ), ∀α : |α| ≤ K and (5.3.5) by (5.3.5)∗

B(0, γ) ⊂ X .

Further, assuming (5.2.101) with s ≥ 0 and also ρ ≤ γ we get estimate (5.3.24)

h 1−d d−1+s −s . ρ γ ϑ |RW ϕ,L | ≤ Ch L

Let us introduce scaling functions (5.3.25)

1 1  γ(x) =  |V | + |∇V |2 2 + γ¯ , 2

ρ=γ

1

with γ¯ = h 2

with small enough constant  and observe that conditions (5.3.6)∗1−3 are 1 1 fulfilled18) , as γ ≥ h 2 condition (5.3.18)∗ is fulfilled and as γ  h 2 all estimates are equally good (actually generic estimate even does not need spatial mollification). After summation over γ-admissible partition we arrive to Statement (i). Proof of Statement (ii) follows the same scheme but with the use of long time propagation results.

18) Actually condition to Vj may fail as α = 0 but we need to check it only as |α| ≥ 1 due to Remark 5.3.2.

452

CHAPTER 5. SCALAR OPERATORS AND RESCALING

Remark 5.3.4. Statement (i) as d ≥ 2 and Statement (ii) as d ≥ 3 remain true for Schr¨odinger-Pauli operator as well. As d = 2 (5.3.16) should be replaced by while for d = 2 (5.3.26)

|RW ϕ,L |

≤ C0 h

−1



1

0≤j≤n

Tj1+s

 Λ± j

dxdξ : da+  {≤hδ }

|F12 | dx + C  hδ



 h ϑ L

where we assume that  2 2 1−δ Λ± . j ⊂  := |V | + |∇V | ≥ h For the proof we refer to the proof of the Statement (ii) of Theorem 5.3.25 below where it is done for Dirac operator. Theorem 5.3.5. Let conditions (5.3.1), (5.3.4)–(5.3.7) be fulfilled and d = 1. Then (i) The following estimate holds (5.3.27)

|RW | ≤ Ch−δ .

(ii) Moreover, if either (5.3.28)



|D α V | ≥ 0

|α|≤n

or (5.2.101) holds with s > 0 then Statements (i) and (ii) of Theorem 5.3.3 remain true. However, for s ∈ N one should regularize the coefficient κ1+s and take it depending on h (and it is O(| log h|M ) in virtue of the analysis of the previous Section 5.2). Proof. Proof follows arguments to the previous section but with essential simplification. We leave details to the reader. Remark 5.3.6. Conditions (5.2.20)± 2 in this case mean exactly that V has non-degenerate minimum (maximum) in the point in question.

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

453

Asymptotics without Spatial Mollification Let us consider spectral asymptotics without spatial mollification. First of all, let us check ξ-microhyperbolicity condition; it is ∀x ∈ B(0, 1)

|V | ≥ 0

(5.3.29)

and we immediately arrive to Theorem 5.3.7. Let conditions (5.3.1), (5.3.4), (5.3.5), (5.3.6), (5.3.7) and (5.3.29) be fulfilled. Then h 1−d ϑ |RW x,ϕ,L | ≤ C0 h L

(5.3.30)

1 ∀x ∈ B(0, ). 2

Now let us use the same rescaling scheme as in the alternative proof of Theorem 5.3.3. Then we should replace (5.3.18)∗ by (5.3.29)∗

|V | ≥ 0 ρ2

∀x ∈ B(0, γ)

and therefore we need to pick up the scaling functions (5.3.31)

γ = |V | + γ¯ ,

1

ρ = γ2

1 1 2 with γ¯ = h 3 and ρ¯ = h 3 ; 2

2

we select the γ¯ equal h 3 to keep ργ ≥ h. Obviously |∇x γ| ≤ 12 . We immediately arrive to the estimate  hρ  1−d d−1 −1 (5.3.32) |RW  ρ γ ϑ x,ϕ,L | ≤ Ch Lγ ⎧ h 1 ⎪ ⎨Ch1−d |V |(d−3)/2 ϑ |V |− 2 L 2/3   ⎪ ⎩Ch− 23 d ϑ h L

2

as |V | ≥ h 3 , 2

as |V | ≤ h 3 .

So, we proved Theorem 5.3.8. Let conditions (5.3.1), (5.3.4), (5.3.5), (5.3.6) and (5.3.7) be fulfilled. Then estimate (5.3.32) holds. Corollary 5.3.9. Let conditions of Theorem 5.3.8 be fulfilled. Let us assume that

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454

(5.3.33) ϑ(τ )τ −s is a monotone decreasing function19) with some s ≥ 0. Then as d − 3 ≥ s estimate (5.3.30) holds and as d − 3 < s estimate (5.3.34)

h − 23 (d+s) |RW ϑ x,ϕ,L | ≤ C0 h L

1 ∀x ∈ B(0, ) 2

holds. The reader can reformulate Theorem 5.2.14 for the Schr¨odinger operator and improved asymptotics with the “no loop” condition. Finally the results of Section 5.2 yield Theorem 5.3.10. Let conditions (5.3.1), (5.3.4), (5.3.5), (5.3.6) and (5.3.7) be fulfilled. Then for τ ≤ V∗ = inf B(0,1) V the following estimate holds (5.3.35)

|e(x, y , τ )| ≤ C  h−d (1 +

|V∗ − τ | −l ) h

∀x, y ∈ B(0, 1 − )

where C  = C  (d, c, l, ) and l,  > 0 are arbitrary. 1

Here we use that |∇V | ≤ C |V − V∗ | 2 . Asymptotics without Spatial Mollification and Short Loops In this subsubsection we consider very special case ϑ = 1 and d = 1, 2 when 4 − 23 1−d estimate |RW fails replaced by |RW and Rx ≤ Ch− 3 as x | ≤ Ch x | ≤ Ch 2 1−d d = 1, 2. More precisely, as |V | ≥ h 3 estimates |RW |V |(d−3)/2 x || ≤ Ch holds. Our purpose is to improve these estimates, possibly adding a correction term associated with the short loop. As |∇x V | is our foe rather than our friend here, we assume that condition (5.3.18) holds. Later we will get rid of it by scaling. Our goal is to prove Theorem 5.3.11. Let conditions (5.3.1), (5.3.4), (5.3.5), (5.3.6), (5.3.7) and (5.3.18) be fulfilled. Then 1 2 2 d  (5.3.36) |e(x, x, 0) − κ0 (x)h−d − h− 3 d |∇V (x)|g3 Q W (x)h− 3 g | ≤ 1

Ch1−d γ(x) 2 (d−2) 19)

Before we assumed that it is increasing function.

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

455

where in the correction term W (x)  V (x) and Q will be defined by (5.3.62), and (5.A.11) respectively and (5.3.37)

Q(λ) = O(λ− 4 (d+3) ) as λ → ∞ 1

2

(see (5.A.12) for its asymptotics as λ → +∞) and γ(x)  |W (x)| + h 3 , (5.3.38)

|∇V |g =



g jk ∂j V · ∂k V

 12

.

j,k

The crucial step in the proof is Proposition 5.3.12. Let u(x, y , t) be the Schwartz kernel of e ih framework of Theorem 5.3.11 (5.3.39)

1

|Ft→h−1 τ χ¯T (t)Γx u| ≤ Ch1−d γ(x) 2 (d−2)

−1 A

. In the

∀τ : |τ | ≤ γ(x)

where T is the small constant. 1

Note that the rescaling arguments yield (5.3.39) with T = 0 γ(x) 2 and this would yield estimates O(h1−d γ (d−3)/2 ). Generalizing it to T = 0 leads to remainder estimates O(h1−d γ (d−2)/2 ) which are announced in Theorem 5.3.11, but with Tauberian main part. This Tauberian expression would have two contributors: one equal h−d κ0 (x) which is the main term from t = 0 and another equal to the correction term which from the loop and also collects all other terms from t = 0. Proof of Proposition 5.3.12. Consider first the left-hand expression of (5.3.39) with χ¯T (t) replaced with χT (t) supported in 12 T ≤ |t| ≤ T with T = T0 which is a small constant and with γ(x) ≤ 1 T ; then it would be less than Chs . Rescaling x → xnew = xT −2 , t → tnew = tT −1 , h → hnew = hT −3 and 1 multiplying operator by T −2 we reduce the general case of T ≥ γ(x) 2 to the previous one; so now the left-hand expression of (5.3.39) with χT (t) instead of χ¯T (t) does not exceed  h s CT −2d 3 T 1−2d −2d where T = T ×T and the first factor T appears from Fourier transform while T −2d appears because we have a density.

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456

1

Summation with respect to T ∈ [C0 γ 2 , T0 ] results in the same expression 1 as T = γ 2 i. e.  h s 1 C γ − 2 −d 3 γ2 which does not exceed the right-hand expression of (5.3.39). 1 Therefore we need to prove (5.3.39) with T = C0 γ 2 . Rescaling x → 1 3 xnew = xγ −1 , t → tnew = tγ − 2 , h → hnew = hγ − 2 and multiplying operator by γ −1 we reduce (5.3.39) to the case γ  1. 2 However, if originally γ(x) ≥ C0 h 3 then condition (5.3.29) is fulfilled after rescaling this estimate follows from (4.2.82). On the other hand, if 2 originally γ  h 3 then after rescaling hnew  1 and this estimate holds as well. Therefore due to Tauberian theorem we arrive to Corollary 5.3.13. In the framework of Theorem 5.3.11 (5.3.40)

h 1 |RTx,ϕ,L | ≤ Ch1−d γ(x) 2 (d−2) ϑ L

as NTx,ϕ,L is defined with T = T0 which is a small constant. Now our goal is to calculate NTx with the indicated error. Due to the same arguments as in the proof of Proposition 5.3.12 (5.3.41) Estimate (5.3.40) remains true as NTx,ϕ,L is defined with T =  1 1  max C γ 2 , h 3 −δ . Now let us calculate. We will do first the general calculations in the case of γ(x)  1 and then we rescale. To do this we need to prove Theorem 5.3.14. Let x¯, y¯ be fixed points and let t1 < t2 be of the same sign and t1  t2  (t2 − t1 )  1. Assume that (5.3.42) There exist only one Hamiltonian trajectory (x(t), ξ(t)) such that x(0) = y¯, x(t) = x¯ and t ∈ [t1 , t2 ]; let it happen as t = t¯, ξ(0) = η¯ and (t¯ − t1 )  (t2 − t¯)  1,

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

457

dx  dx     1 dt t=t¯ dt t=t¯

(5.3.43) and

(5.3.44) Map Σ0 ∩ Ty¯∗ X  η → πx Ψt (y , η) is nondegenerate in (¯ y , η¯). Then as t1 ≤ t¯ − , t2 ≥ t¯ + , x = x¯, y = y¯ (5.3.45) h−1



0 −∞

  1−d −1 Ft→h−1 τ χ¯ (t − T ∗ )u dτ ≡ e ih φ(x,y ) bn (x, y )h 2 +n n≥0

where  (5.3.46)

t¯

(x(t), ξ(t)) dt,

φ(x, y ) =

(x, ξ) := ∂ξ a, ξ.

0

Proof. First, let us rewrite the left-hand expression of (5.3.45) as    (5.3.47) i Ft→h−1 τ t −1 χ¯ (t − t¯)  . τ =0

On the other hand we know that (5.3.48)

− 12 (d+m)

u(x, y , t) ≡ (2πh)

 e ih

−1 ϕ(x,y ,θ,t)



bn (x, y , t, θ) dθ

n≥0

where θ is m-dimensional variable and ϕ(x, y , t, ϑ) is defined in the corresponding way. For example, one can take m = d, θ = η, (5.3.49)

ϕ(x, y , t, η) = −y , η + ψ(x, t, η)

where (5.3.50)

∂t ψ = −a(x, ∂x ψ),

ψ(x, 0, η) = x, η.

Let us apply stationary phase method with respect to θ, t; condition (5.3.42) and ∂t ψ = −a(x, ξ) imply that there is only one stationary point and it is (¯ η , t¯). Further, conditions (5.3.43) and (5.3.44) imply that this is non1 degenerate point. Thus we gain a factor h 2 (d + 1) and (5.3.45)–(5.3.46) are proven.

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458

Proof of Theorem 5.3.11. To apply Theorem 5.3.14 to our case we need just 3 rescale x → xγ −1 , h →  = hγ − 2 ; then as  ≤ hδ i. e. 2

γ ≥ h 3 (1−δ)

(5.3.51) we conclude that (5.3.52)

iφ(x,x) Nx,corr := NTx − NW x ≡ e



1

bn (x, x)γ 4 (−d−3−6n) h

1−d +n 2

n≥0

where we also multiplied by γ −d since we are dealing with densities. Thus (5.3.53) Correction term Nx,corr is of magnitude h 2 (1−d) γ − 4 (d+3) . 1

1

This implies a drastic difference between d = 1 when the correction term 1 is below remainder estimate h1−d γ(x) 2 (d−2) of (5.3.40) only as γ  1 and d ≥ 2 it is always so as γ ≥ γ¯1 := h(2d−2)/(3d−1) .

(5.3.54) Let us introduce (5.3.55)

X − = {x : V (x) < 0},

X 0 = ∂X − = {x : V (x) = 0},

Without any loss of the generality one can assume that X − = {x1 < 0}. Case d = 1.  We can assume without any loss of the generality that a(x, ξ) = β(x) ξ12 −V0 (x) (we can always get rid of V1 by gradient transform); then  23  3  x1 1 V0 (y1 ) 2 dy1 (5.3.56) W (x1 ) = 2 0 is a travel time from x1 ∈ X − to X 0 (on energy level 0, if we replace β by 1) and one can see easily that then (5.3.57) Under assumption (5.3.18) W (x)/V0 (x) is a smooth and disjoint from 0 function on X − ∪ X 0 .

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

459

We redefine x1 = W (x) and then a(x, ξ) will be in the same form as before but with different β and with W (x1 ) = x1 :   (5.3.58) a(x, ξ) = β(x) ξ12 − x1 Let us prove that in calculations one can replace β(x) by β(x) = 1 and assume that (5.3.59)

A = h2 D12 − x1

on R.

Really, let x¯ be a point where calculations are done, while x be a “running” point. Without any loss of the generality one can assume that β(¯ x ) = 1. Let us rescale as before. Then we can assume that we are at he point with γ(x) but on [−c, c] (5.3.60)

|D1j β(x)| ≤ cj εj

where ε is an original γ(¯ x ). Let us apply Theorem 5.3.14. Note that phase functions for the original operator and for the model operator coincide identically while amplitudes differ by O(ε) (where ε = γ); so an error is O(h1−d ε) and scaling back we 1 1 get an error estimate Ch1−d γ 2 (d−3) ε = Ch1−d γ 2 (d−1) which is actually better 1 by factor γ 2 than we need. Calculations for model operator (5.3.59) are produced in Appendix 5.A.1. Theorem 5.3.11 is proven as d = 1. Again without any loss of the generally one can assume Case d ≥ 2. that V (x) = −k(x)x1 with k(x) > 0 disjoint from 0. Let (5.3.61)

Θ = {(x, ξ) : V (x) = 0} ∩ Σ0 = {(x, ξ) : x1 = 0, ξj = Vj (x)}

parametrized by x  = (x2 , ... , xd ). Consider Hamiltonian trajectory passing through (0, x  ) ∈ Θ. One can select ξ1 as a natural parameter along this trajectory, so we get a ddimensional manifold Λ ⊂ Σ0 ; since Σ0 is (2d − 1)-dimensional, there exists (d − 1)-dimensional variable η: Λ = Σ0 ∩ {η = 0}. Assume first that (5.3.62) There is no magnetic field, i. e. V1 = V2 = ... = Vd = 0.

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CHAPTER 5. SCALAR OPERATORS AND RESCALING

x(t) = x(−t) x(0)

x(0)

(a) Original coordinates

x(t) = x(−t)

(b) Straighten coordinates

Figure 5.2: : Short loops Then trajectory passing through Θ as t = 0 is symmetric: x(−t) = x(t) and ξ(−t) = ξ(t). Then the set of loops coincides with Λ and x-projection of the loop is exactly as on the left picture below If we introduce new coordinates replacing x  by x  (0) we get picture as on the right. Note that due to the same arguments as for d = 1 one can assume that     (5.3.63) a(x, ξ) = β(x) ξ12 − x1 + g jk (x) ξj − αj (x)ξ1 ξk − αk (x)ξ1 j,k≥2

with positive definite matrix (g jk ). Then η = 0 if and only if ξj − αj (x)ξ1 = 0 ∀j = 2, ... , d. Also note that on Σ0 we must have η = 0 =⇒ {a, η} = 0 i. e. {ξ12 − x1 , ξj − αj (x)ξ1 } = 0 which easily yields that αj (x) = 0 ∀j = 2, ... , d i. e.   (5.3.64) a(x, ξ) = β(x) ξ12 − x1 + g jk (x)ξj ξk j,k≥2

and η = ξ  . One can see easily that in the special case g jk (x) = δjk trajectories are parabolas ξ1 = t, ξ  = const,

x1 = t 2 , x  = const + ξ  t (j ≥ 2)

and similarly looking in more general case.

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

461

x(t) x(0)

x(0) x(−t)

(a)

(b)

Figure 5.3: Selecting ξ  = 0 destroys the loop Again, as γ  1 we can estimate contribution of the loop by Ch1−d × 1 1 h = h− 2 (d−1) where extra factor h 2 (d−1) is due to Theorem 5.3.14. Scaling procedure leads to estimate (5.3.53). As g jk = δjk Appendix 5.A.1 implies that estimate (5.3.36) holds. Thus it holds for g jk = const. For variable g  jk we get instead of phase |ξ  |2 another phase |ξ  |2 + ω(x, ξ  ) with ω = O(γ|ξ  |2 ) which leads to an error 1 1 with an extra factor γ in the error estimate which becomes h− 2 (d−1) γ − 4 (d−1) and is less than the right hand expression of (5.3.36). Therefore under assumption (5.3.62) Theorem 5.3.11 is proven. We can consider operator (5.3.64) and add magnetic field to it. 1 (d−1) 2

x(0)

x(t) = x(−t)

(a) No magnetic field case

x(0)

x(t) x(−t)

(b) Magnetic field case

Figure 5.4: Short loop without and with magnetic field In this case equalities ξ(−t) = −ξ(t) and x(−t) = x(t) along trajectories passing through Θ fail, Λ loses is value and picture on Figure 5.2(b) is replaced by the picture on the left where actually the arcs are symmetric

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462

only asymptotically. However as before for each x (with 0 < x1 < ) there exists ξ = ξ(x) such that (x, ξ) ∈ Σ0 and Hamiltonian trajectory passing through (x, ξ) comes back to x. So, instead of few looping trajectories and each such trajectory being a loop for each of its points, we have now many looping trajectories but each of them serves only one point. Further, one can prove that then the phase is changed by O(ε) where 3 5 ε = γ and scaling back we get an extra term O(h−1 γ 2 × ε) = O(h−1 γ 2 ) 1 1 in the exponent which leads to the error not exceeding Ch− 2 (d+1) γ − 4 (d−7) 2 which does not exceed the right-hand expression of (5.3.36) as h 3 ≤ γ ≤ γ¯1 . Theorem 5.3.11 proven completely. Remark 5.3.15. Obviously (5.3.65)

4

W (x) = const  3 (x),

const =

 3  23 2

where (x) is the distance in the metrics g jk V (x)−1 from x to X 0 . Thus one can rewrite the correction term as   2 d 2 4 (5.3.66) h− 3 d |∇V (x)| 3 Q const h− 3  3 (x) . Finally, let us get rid of condition |∇V (x)| ≥ . To do this let us rewrite first the right-hand expression of (5.3.36) as (5.3.67)

 1 2  1 (d−2) Ch− 2 (d−1) max(|V |, h 3 ) 2

thus releasing notation γ.  1 1 Let us introduce a standard scale γ = |V |+|∇V |2 2 +h 2 but exclusively at point x¯ in question. Then applying the standard rescaling x → xγ −1 , h →  = hγ −2 , V → V γ −2 , ∇V → ∇V γ −1 we get instead of (5.3.67) expression   2 4  1 (d−2) 2 2  1 (d−2) Ch1−d γ d−2 max(|V |γ −2 , h 3 γ − 3 ) 2 = Ch1−d max(|V |, h 3 γ 3 ) 2 . As d ≥ 2 the latter expression can only increase as γ is replaced by 1. However as d = 1 situation is different and we arrive to 1 1 1  (5.3.68) C min(|V |− 2 , h− 3 γ − 3 ) . Thus we proved

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

463

Theorem 5.3.16. Let conditions (5.3.1), (5.3.4), (5.3.5), (5.3.6) and (5.3.7) be fulfilled. Then (i) As d = 2 asymptotics (5.3.36) holds; furthermore one can skip the correction term without penalty as hd−1 |∇V |d+1 ≤ |V |3d−1 . (ii) As d = 1 the left-hand expression of (5.3.36) does not exceed ⎧ 1 2 2 |V (x)|− 2 as h 3 |∇V (x)| 3 ≤ |V (x)| ≤ |∇V (x)|2 ⎪ ⎪ ⎪   ⎪ ⎨ or as |V (x)| ≥ max |∇V (x)|2 , h , (5.3.69) C 1 1 2 2 1 ⎪ h− 3 |∇V (x)|− 3 as |V (x)| ≤ h 3 |∇V (x)| 3 , |∇V (x)| ≥ h 2 , ⎪ ⎪ ⎪ ⎩ −1 1 h 2 as |V (x)| ≤ h, |∇V (x)| ≤ h 2 ;   further, as |V (x)| ≥ max |∇V (x)|2 , h one can skip the correction term without penalty. Problem 5.3.17. Consider averaged with respect to the spectral parameter correction term with s > 0 and, respectively, as d > s + 3 and generalize results we derived as s = 0, d = 1, 2.

5.3.2

Dirac Operator

Assumptions So, we consider Dirac operator (5.3.2) 1 σl (ω jl Pj + Pj ω jl ) + σ0 m + V · I A= 2 j,l where coefficients ω jl , the components of the vector-potential Vj and the scalar potential V are real-valued functions, σl are D × D-dimensional Pauli matrices (see Appendix 5.A.2), D ≥ 2, m = const ≥ 0 is a mass parameter, and h ∈ (0, 1). We can also consider massless Dirac operator 1 (5.3.70) A= σl (ω jl Pj + Pj ω jl ) + V · I 2 j,l which is not necessarily a special case of (5.3.2) because for a given dimension D there may be only d rather than (d + 1) of D × D-dimensional Pauli

CHAPTER 5. SCALAR OPERATORS AND RESCALING

464

matrices20) ; however unless otherwise specified we consider it under m = 0 blanket as well. Moreover, for the future applications we also consider the generalized Dirac operator A = J − 2 (A + W σ0 )J − 2 1

(5.3.71) with (5.3.72)

1

1 1 J = J+ (I + σ0 ) + J− (I − σ0 ) 2 2

where J± are positive-valued functions. Let us discuss operator (5.3.2) in the simple case of ω jl = δjl , Vj = V = 0: (5.3.73) a(x, ξ) = σj ξj + mσ0 + V (x). j

Then a(x, ξ) has eigenvalues (5.3.74)

1

λ± = ±(m2 + |ξ|2 ) 2 + V (x)

and observe that (5.3.75) For given x there is a spectral gap (−m + V (x), m + V (x)) which is empty as m = 0. As m  1 we get rather standard case of the matrix operator and actually we will improve results getting rid of the microhyperbolicity condition. On the other hand, as m  1 we get a huge spectral gap and to avoid it we need to take energy level close to its end; alternatively we can pick up energy level 0 but replace V (x) by ∓m + V0 (x). Then the “interesting” eigenvalue of a(x, ξ) becomes 1

(5.3.76) λ± = ±(m2 + |ξ|2 ) 2 ∓ m + V (x) ≈  1  2 1 |ξ|2 + V (x) = ± |ξ| ± 2mV (x) ± 2m 2m as |ξ|  1 and we arrive basically to the Schr¨odinger operator with the potential ±2mV (x) and we want it to be of the type we considered before. 20)

This can happen only for odd d.

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

465

The following array of assumptions codifies these two different situations for the generalized Dirac operator (5.3.71)–(5.3.72). Let us assume that |D α ω jl | ≤ c,

(5.3.77)5

|D α Vj | ≤ c ∀α : |α| ≤ K , 1 ∀α : 1 ≤ |α| ≤ K , |D α V | ≤ c min(1, ) m 1 (τj − m − V )− ≤ c min(1, ), m 1 (τj + m − V )+ ≤ c min(1, ), m 1 α ∀α : |α| ≤ K |D W | ≤ c min(1, ) m

and (5.3.77)6 (5.3.77)7

|D α Jς | ≤ c ∀α : |α| ≤ K , ς = ±,  0 ≤ Jς ≤ c

(5.3.77)1−2 (5.3.77)3 (5.3.77)− 4 (5.3.77)+ 4

where τ1 < τ2 are the energy levels in question. Moreover let us assume that (5.3.78) The metric tensor g jk =

(5.3.79)



ω jl ω kr δlr

l,r

satisfies (5.3.6)1 and (5.3.7). Let us recall that σ†l = σl ,

(5.3.80) (5.3.81)

σl σk + σk σl = 2δkl I

(k, l = 0, ... , d),

i. e., that σl are Hermitian, unitary and anti-commuting. The properties of Pauli matrices are presented more explicitly in Appendix 5.A.2. Calculations Lemma 5.3.18. For d ≥ 1 the principal symbol (of the Dirac operator) (5.3.82) a(x, ξ) = σl ω jl (ξj − Vj ) + σ0 m + V · I j,l

466

CHAPTER 5. SCALAR OPERATORS AND RESCALING

has two eigenvalues (5.3.83)

1

λ∓ = V ∓ (g jk (ξj − Vj )(ξk − Vk ) + m2 ) 2

of multiplicity 12 D. Proof. Obviously it is sufficient to treat the case V = V1 = ... = Vd = 0. Then (5.3.81) implies that a(x, ξ)2 = g jk ξj ξk + m2 and hence there is no eigenvalue of a different from λ± . Since λ± coincide only if ξ = 0, m = 0 then in the calculation of multiplicities one can assume that ξ = 0, m > 0. But in this case the multiplicities of λ± are equal to 12 D since σ−1 1 σ0 σ1 = −σ0 ; for the massless Dirac operator we can consider instead ξ = (1, 0, ... , 0). Remark 5.3.19. (i) Let d = 1; this is possible only for massless Dirac operator A = ω(D1 − V1 ) + V and the conclusion fails; however it is still valid for d ≥ 2 and (5.3.84) a(x, ξ) = σl ω jl (ξj − Vj ) + V · I . j,l

¯ := 2 2  (ii) Without any loss of the generality one can assume that D = D ¯ ×D ¯ matrices σ ¯k because σk = q † (¯ σk ⊗ ID )q with the unitary matrix q, D  −1 ¯ ¯ and D = DD . In particular, for d = 1, 2 we get D = 2 and the eigenvalues of a(x, ξ) are simple. d+1

(iii) Similarly for massless Dirac operator without any loss of the generality ¯ := 2 d2  . In particular, for d = 3 we get D ¯ =2 one can assume that D = D and the eigenvalues of a(x, ξ) are simple. As a corollary we obtain that for the Dirac operator  D −d dξ (5.3.85) tr(κ0 (x, τ1 , τ2 )) = (2π) 2 ε=± τ1 0 is a small constant and C  = C  (d, s, c, 0 , ). (ii) Moreover, the same estimate with an additional factor m−1 , m−1 , m−2 in the right-hand expression holds for the blocks u ∓,· (., ., .), u ·,∓ (., ., .) and u ∓,∓ (., ., .) respectively. 21) Under assumptions already made condition (5.3.117)± below is equivalent to (5.3.114) with the right-hand expression 0 r .

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Then the standard Tauberian arguments yield Corollary 5.3.23. (i) In the framework of Proposition 5.3.22 (5.3.119) |e(x, y , τ  , τ  )| ≤ C 

 h s r ∀x, y ∈ B(0, 1 − )]

∀τ  , τ  ∈ [− r ,  r ].

(ii) Moreover, the same estimate with a factor m−1 , m−1 , m−2 in the righthand expression holds for the blocks e ∓,· (x, y , τ  , τ  ), e ·,∓ (x, y , τ  , τ  ) and e ∓,∓ (x, y , τ  , τ  ) respectively. Remark 5.3.24. Corollary 5.3.23 yields that we can assume in what follows that m+W ±V ≤

(5.3.120)±

c . m

Then the following assertions similar to those for the Schr¨odinger operator are proven easily: Theorem 5.3.25 22) . Let assumptions (5.3.71)–(5.3.71), (5.3.77)1−7 , (5.3.78), (5.3.97) and (5.3.98)± , (5.3.120)± be fulfilled and let d ≥ 2. Let ψ ∈ C0K (B(0, 12 )), |ψ| ≤ 1 and let (4.2.73)–(4.2.75) be fulfilled and L ≥ m−1 h. Then (i) The following estimate holds (5.3.121)

 1−d   h  |RW . + C  h1−d+δ ϑ ϕ,L | ≤ Ch mL

± (ii) Moreover, let Λ± Σ± = {(x, ξ) : λ± = 0} 1 , ... , Λn be closed subsets of where λ± are given by (5.3.83),  2 2 2 1−δ Λ± j ⊂ ± := m| ± V + m + W | + m |∇(V ± W )| ≥ h

and suppose that through each point of the h 2 −δ -neighborhood of Λ± j there −1 passes a Hamiltonian trajectory generated by mJ± (x) λ± (x, ξ) with  t ∈ Jj = [0, Tj ] or Jj = [−Tj , 0], Tj ∈ [1, h−δ ], along which 1

22)

Cf. Theorem 5.3.3.

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(5.3.122)1,2 |D α ω jl | ≤ c1 h−δ , (5.3.122)3 (5.3.122)4 (5.3.122)5

|D α Vj | ≤ c1 h−δ , 1  |D α V | ≤ c1 h−δ min(1, ) (|α| ≥ 1) m 1  α −δ  |D W | ≤ c1 h min(1, ), |D α Jε | ≤ c1 h−δ m   1 hδ ≤ Jε ≤ c1 h−δ

∀α : |α| ≤ K  ,

and conditions (2.4.29), (5.3.14), (5.3.15) are fulfilled. Further, let (5.2.101) with s ≥ 0 be fulfilled. Then for d ≥ 3, d = 2 respectively estimates (5.3.16) and (5.3.26) hold with ϑ(h/L) replaced by ϑ(h/Lm) and a = λ± and γ = ± in the right-hand expressions. Proof. In this modification only the proof of Statement (ii) with d = 2 must be changed, only the part at which we consider a ball B(z  , γ1 ) with | Hess V ∗ | ≤ γ1 , |{z1 , z2 }|  γ3 where V ∗ = (m + W ± V )m, γ3 ≥ hδ3 with arbitrarily small δ3 > 0 and we refer to the proof of Theorem 5.2.7. The difference between magnetic Schr¨odinger and Schr¨odinger-Pauli or Dirac operators will be explored later. 1 Let us consider Dirac operator in the ball of radius γ4 = h 2 −δ4 with respect to x with a small enough exponent δ4 > 0. Let us introduce new coordinates xnew = xγ4−1 , divide the Dirac operator by γ4 γ3 and replace h, m by hnew = h/(γ42 γ3 ) < hδ5 , mnew = m/(γ3 γ4 ). One can easily check that we remain consistent with all the assumptions of this subsection provided V ∗ ≥ −c0 γ32 γ42 in the ball in question and in this case the contribution of this ball to the remainder estimate does not exceed 



−1 δ −1 δ −1 C0 hnew + C  hnew ≤ C0 h−1 γ42 |F12 | + C  hnew

where exponents and constants are changed if necessary; after summation with respect to all the balls of this type we obtain the second and the fourth terms of the right-hand expression in (5.2.26). On the other hand, balls with V ∗ ≤ −c0 γ32 γ42 are consistent with the 1 proof of Theorem 5.2.7: after rescaling we divide by (−V ∗ ) 2 and obtain the case in which the microhyperbolicity condition is fulfilled; so the total contribution of the balls of this type does not exceed the sum of the first and the fourth terms of the right-hand expression in (5.3.26).

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The similar proof works also for Schr¨odinger-Pauli operators as mentioned in Remark 5.3.4. Also one can prove easily Theorem 5.3.26 23) . In the framework of Theorem 5.3.25 but with d = 1 estimate (5.3.27) holds. Moreover, if either V ∗ satisfies (5.3.28) or (5.2.101) holds with s > 0 then Statements (i) and (ii) of Theorem 5.3.25 remain true. Also one can explore asymptotics without spatial mollification: Problem 5.3.27. (i) Prove statements similar to those of Theorems 5.3.7, 5.3.8, 5.3.10, Corollary 5.3.9, with the only difference that not L but Lm should be greater than h and ϑ(h/L) needs to be replaced by ϑ(h/Lm) in the right-hand expressions of estimates. (ii) Prove the statements similar to those of Theorems 5.3.11, 5.3.16. We believe that the complete proofs of these statements are worth to be published. Asymptotics for m ≤ c0 Let us now consider the opposite case (5.3.123)

m ≤ c0 .

Remark 5.3.28. Actually in the previous Subsubsection 5.3.2.4 Asymptotics for m ≤ c0 we used only ellipticity of one of the equations (5.3.102)± and one can achieve this by  -partition provided (5.3.124)

m + W ≥ 0

with the only difference that in sharp or sharper asymptotics both signs ± may contribute to the remainder estimate; so in (5.3.26) one needs to sum with respect to sign ±. So our main interest is when m + W  1. In this case the eigenvalues λ± of a(x, ξ) are not necessarily disjoint. 23)

Cf. Theorem 5.3.5.

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One can prove easily Proposition 5.3.29. (i) Operator (5.3.71)–(5.3.72) is microhyperbolic at energy level 0 in ξ-directions if and only if |m + W ± V | ≥ 0

(5.3.125) for both signs;

(ii) Operator (5.3.71)–(5.3.72) is microhyperbolic at energy level 0 if and only if (5.3.126)

|m + W ± V | + |∇(W ± V )| ≥ 0

for both signs and also (5.3.127)

|m + W | ≤ 0 =⇒ ∃ : |x | = 1, x (V ± W ) ≥ 0

for both signs. Then the general results of Section 4.4 imply Corollary 5.3.30. Let conditions (5.3.71)–(5.3.71), (5.3.77)1−7 , (5.3.78), and (5.3.123), (4.2.73)–(4.2.75) be fulfilled. (i) Further, let (5.3.126)–(5.3.127) be fulfilled. Then 1−d ϑ |RW ϕ,L | ≤ Ch

h . L

(ii) On the other hand, let (5.3.125) be fulfilled. Then 1−d |RW ϑ x,ϕ,L | ≤ Ch

h . L

We will discuss sharp and sharper estimates later. If we want to rescale our main obstacle is condition (5.2.128). The good news is that in our applications we can assume that either (5.2.135) holds or (5.3.128)

W =0

and since the first case is already covered we assume that (5.2.129) holds. Then condition (5.2.128) is equivalent to (5.2.127). As long we remain in the framework of m ≤ 0 there are two modes of rescaling:

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(a) If we are interested in estimates with the spatial averaging and magnetic field is 0, i.e. V1 = ... = Vd = 0 (5.3.129) we can take  1 (5.3.130) γ =  |V | + |∇V |2 ) 2 + γ¯ , ρ = γ 2,

 1 1 γ¯ = max h 3 , m 2

(b) If we are interested in estimates without the spatial averaging or magnetic field is 0 we should take  1  (5.3.131) γ = |V | + γ¯ , ρ = γ, γ¯ = max h 2 , m . Then rescaling x → xγ −1 and dividing by ρ we get to h → hρ−1 γ −1 , 1 m → mρ−1 , L → Lρ−1 ; here we want mnew = mρ−1 ≤ 1 i. e. γ  m 2 in the case (a) and γ  m in the case (b). We also want Lnew = Lρ−1 ≥ hnew = hρ−1 γ −1 i. e. γ ≥ hL−1 but we will ignore it since estimates hold without this condition as well. As a result the right-hand expressions with or without spatial averaging become  h  1−d ρd−1 ϑ dx, (5.3.132) Ch Lγ where dimmed elements are present only if there is a spatial averaging. Under assumption (5.3.33) we can estimate (5.3.132) by  h 1−d ρd−1 γ −s dx, (5.3.133) Ch ϑ L and it is where all our cases finally separate: Proposition 5.3.31. Let conditions (5.3.71)–(5.3.71), (5.3.77)1−7 , (5.3.78), and (5.3.123), (5.3.128), (4.2.73)–(4.2.75) be fulfilled. Then γ} (i) If condition (5.3.129) is fulfilled 24) the contribution of zone {γ ≥ 2¯ W with γ and γ¯ defined by (5.3.130) to Rϕ,L does not exceed ⎧ 1 if either 2d − 3 − s ≥ 0 ⎪ ⎪ ⎪ ⎨ h or 2d − 2 − s + r > 0, · (5.3.134) Ch1−d ⎪ if 2d − 3 − s + r = 0, L ⎪| log γ¯ | ⎪ ⎩ 2d−3−s+r γ¯ if 2d − 3 − s + r < 0. 24)

Then we plug ρ = γ 2 .

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Here and below r is defined by assumption (5.3.135)r Hess V has at least r eigenvalues disjoint from 0 as |∇V | ≤ . (ii) Otherwise 25) the contribution of zone by (5.3.131) to RW ϕ,L does not exceed ⎧ 1 ⎪ ⎪ ⎪ ⎨ h · (5.3.136) Ch1−d | log γ¯ | L ⎪ ⎪ ⎪ ⎩ d−2−s+r γ¯

{γ ≥ 2¯ γ } with γ and γ¯ defined if either d − 2 − s ≥ 0 or d − 2 − s + r > 0, if d − 2 − s + r = 0, if d − 2 − s + r < 0.

(iii) If |V (x)2 − m2 | ≥ γ¯ 2 with γ¯ defined by (5.3.131) then h 1−d |V (x)|d−1−s . ϑ (5.3.137) |RW x,ϕ,L | ≤ Ch L Now we need to consider zone {2¯ γ0 ≤ γ ≤ 2¯ γ } and either calculate its W contribution to RW or to R as γ ¯ ≤ |V (x)| ≤ γ¯ . There we need to 0 ϕ,L x,ϕ,L distinguish between γ¯ defined by h or by m. In the former case hnew  1 and we skip this step completely and define γ¯0 := γ¯ . In the latter case after rescaling we have mnew  1 and h ≤ 1. It means that we can apply directly results of the previous Subsubsection 5.3.2.3 Asymptotics for m ≥ c0 . However let us explain what this actually means. If we consider asymptotics with the spatial averaging we need to pick up here 1  1 ρ = γ, γ¯0 = h 2 γ =  |V ∗ | + |∇V ∗ |2 2 + γ¯0 , with V ∗ = V 2 − m2 and get hnew = hρ−1 γ −1 ,

Lnew = Lρ−2 .

But this would be true after original rescaling; so if we want to make proper definitions in the original settings we need to replace in these equations γ, ρ, V , m, γ¯0 , L and h by γ¯ γ −1 , ρ¯ ρ−1 , V ρ¯−1 , mρ¯−1 , γ¯0 γ¯ −1 , L¯ ρ−1 and h¯ γ −1 ρ¯−1 respectively thus arriving to  1 (5.3.138) γ = ¯ γ ρ¯−2 |V ∗ | + ρ¯−4 γ¯ 4 |∇V ∗ |2 2 + γ¯0 , ρ = ρ¯γ¯ −1 γ, γ¯0 = h 2 γ¯ 2 ρ¯− 2 , 1

25)

1

Then we plug ρ = γ.

1

hnew = h¯ γ ρ¯−1 γ −2 ,

Lnew = L¯ ρ−1 γ¯ 2 γ −2 .

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479 2

(i) Let us recall that in the case (a) we need to check if m ≥ h 3 and plug 1 γ¯ = m 2 , ρ¯ = m thus arriving to 1  (5.3.139) γ =  m−1 |V 2 − m2 | + m|∇V |2 2 + γ¯0 , γ¯0 = h 2 m− 4 , 1

1

ρ = m 2 γ,

hnew = hm− 2 γ −2 ,

1

Lnew = Lγ −2

1

where we actually slightly modified γ (without changing its magnitude). Therefore contribution of the zone {2¯ γ0 ≤ γ ≤ 2¯ γ } to RW ϕ,L does not exceed  h h 1 1 γ d−2 dx ≤ Ch1−d m 2 (2d−3+r −s) ϑ (5.3.140) Ch1−d md−1− 2 s ϑ L {γ(x)≤2¯γ } L provided d ≥ 2 26)

27)

. 1

(ii) Meanwhile in the case (b) we need check if m ≥ h 2 and to plug into (5.3.138) γ¯ = ρ¯ = m thus replacing (5.3.139) by 1  (5.3.141) γ =  |V 2 − m2 | + m4 |∇V |2 2 + γ¯0 , γ¯0 = h , and (5.3.140) by (5.3.142)

Ch

1−d

 h  ϑ Lm



ρ = γ,

hnew = hγ −2 ,

1 2

Lnew = Lmγ −2 .

h γ d−2 dx ≤ Ch1−d md−2+r −s ϑ L {γ(x)≤2¯ γ}

as d ≥ 2 26) . (iii) Finally, in the asymptotics without spatial mollification we need to 1 check again if m ≥ h 2 and then take γ = |V ∗ | + γ¯0 ,

1

ρ = γ2,

2

γ¯0 = h 3 ,

hnew = hγ − 2 , 3

Lnew = Lγ −1

after the first rescaling; to derive these relations in the original coordinates one needs the same replacement as in (ii) thus arriving to (5.3.143) γ = m−1 |V 2 − m2 | + γ¯0 ,

1

γ¯0 = h 3 m− 3 ,

1

2

ρ = γ2m2, hnew = hγ − 2 m− 2 , 3

26) 27)

1

1

Lnew = Lγ −1

It was proven that for d = 2 for Schr¨odinger operator there is no logarithmic factor. Case d = 1 needs to be reexamined anyway.

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480

and deriving estimate  1−d 12 (d−3) γ ϑ (5.3.144) |RW x,ϕ,L | ≤ Ch



h 1 2

Lγ m

1 2

≤

h 1 1 . Ch1−d |V 2 − m2 | 2 (d−3−s) m− 2 (d−3) ϑ L Let us recall that there is also zone {γ ≤ 2¯ γ0 } but in this zone hnew  1 and one can prove easily that contribution of this zone to the remainder does not exceed the estimate we derived for the first zone as γ¯0 = γ¯ and for the second zone as γ¯0 ≤ γ¯ (the second statement was actually proven as m ≥ 0 . Thus we arrive to Theorem 5.3.32. Let conditions (5.3.71)–(5.3.71), (5.3.77)1−7 , (5.3.78), (5.3.123), (5.3.128) and (4.2.73)–(4.2.75) be fulfilled. (i) Moreover let d ≥ 2 and (5.3.129) be fulfilled. Then |RW ϕ,L | does not exceed 1 1 (5.3.134) with γ¯ = max(m 2 , h 3 ). (ii) In the general case d ≥ 2 |RW ¯= ϕ,L | does not exceed (5.3.136) with γ 1 W 1−d max(m, h 2 ). In particular, |R | ≤ Ch as d ≥ 2. 1

2 2 (iii) RW ¯ 2 and (5.3.144) as m ≥ h 2 , x,ϕ,L satisfies (5.3.137) as |V (x) − m | ≥ γ 2 2 m2 ≥ |V (x)2 − m2 | ≥ h 3 m 3 . Furthermore

h 1 (1−d−s) 2 |RW ϑ x,ϕ,L | ≤ Ch L

(5.3.145)

1

as m + |V (x)| ≤ h 2 , and h 1 (−2d−s) − 61 (d−3+2s) 3 m ϑ |RW x,ϕ,L | ≤ Ch L

(5.3.146) 1

2

2

as m ≥ h 2 , |V (x)2 − m2 | ≤ h 3 m 3 . Remark 5.3.33. We did not formulate Theorem 5.3.32(i) as d = 1 only 1 because as we know contribution of zone {γ(x) ≤ cm 2 } is not necessarily estimated by C ϑ(h/L).

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Further Improvements Now our goal is to improve Theorem 5.3.32(i) especially as d = 1, s = 0. Theorem 5.3.34. Let conditions (5.3.71)–(5.3.71), (5.3.77)1−7 , (5.3.78), and (5.3.123), (5.3.128), (5.3.129) and (4.2.73)–(4.2.75) be fulfilled. Then 1−d (i) Estimate |RW ϑ(h/L) holds as 2d > s + 2. ϕ,L | ≤ Ch

(ii) Estimate (5.3.147)

1−d ϑ |RW ϕ,L | ≤ Ch

h | log(h + m)| L

holds as 2d = s + 2, d ≥ 2. (iii) Estimate |RW ϕ,L | ≤ C ϑ

(5.3.148)

h | log(h + m)|n−1 L

holds as d = 1, s = 0 and and both functions V ± m satisfy non-degeneracy condition (5.2.100)n . (iv) Estimate |RW ϕ,L | ≤ C ϑ

(5.3.149)

 h  −δ h L

holds as d = 1, s = 0. (v) Estimate (5.3.150)

 h   1 (2d−2−s) 1−d max(h2 , m3 ) 6 ϑ |RW ϕ,L | ≤ Ch L

holds as 2d < s + 2. Proof. Let us introduce scaling functions  (5.3.151) γn (x) = 

n

|∇α V | (n−|α|)

 n1

+ γ¯n ,

ρn = γnn ,

α:|α|≤n−1 1

1

γ¯n = max(h n+1 , m n ).

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482

(i) Assume that under assumption γn  1 Statement (i) is proven. This is the case as n = 2 because then the microhyperbolicity condition holds as m ≤  and as m ≥  we are in the framework of the previous Subsubsection 5.3.2.4 Asymptotics for m ≤ c0 . Let us consider ball B(x, γn (x)) with γn (x) ≥ C0 γ¯n . Rescaling it to B(0, 1) and simultaneously multiplying operator by ρ−1 n we find ourselves −1 −1 in the framework of γn  1 with h → hρ−1 n γn , L → Lρn and therefore the W contribution of such ball to Rϕ,L does not exceed C

 h 1−d  h  h ≤ Ch1−d ρd−1 ; ϑ γnd−1−s ϑ n ρn γ n Lγn L

then the total contribution of all such balls does not exceed   h h 1−d d−1 −1−s 1−d (5.3.152) Ch ϑ ρn γn γn(d−1)n−1−s dx. dx = Ch ϑ L L Meanwhile as m ≥ hn/(n+1) consider B(x, γn (x)) with γn (x)  γ¯n  m ; rescaling as before we arrive to m  1 and h → hρ¯−1 ¯n−1 , L → n γ −1 L¯ ρn . Applying results of the previous Subsubsection we conclude that the contribution of such ball to the remainder does not exceed the same expression as before and again the contribution of all such balls does not exceed h (5.3.153) Ch1−d γ¯n(d−1)n−s−1 ϑ mes({γ(x) ≤ γ¯n }). L 1/n

Finally, as m ≤ hn/(n+1) consider B(x, γn (x)) with γn (x)  γ¯n  h1/(n+1) ; rescaling as before we arrive to m ≤ 1 and h → 1 and again estimate (5.3.153) is achieved. Note that as (d −1)n > s under condition that γn+1  1 an integral in the right-hand expression of (5.3.152) converges and we estimate the right-hand expression of (5.3.152) by Ch1−d ϑ(h/L). In this case mes({γ(x) ≤ γ¯n })  γ¯n and we estimate (5.3.153) by Ch1−d ϑ(h/L) as well. So, we made the step of induction. This inequality (d − 1)n > s is fulfilled for n ≥ 3 provided it is fulfilled for n = 2 and d ≥ 2. So, everything works well and as (d − 1)n ≥ s + 1 integrand in (5.3.152) is bounded and we do not even need assumption γn+1  1 to get the proper estimate. Statement (i) is proven.

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483

(ii)–(iv) As d = s + 1 integral in (5.3.152) is logarithmically diverging for n = 2 only as d ≥ 2 and for every n as d = 1, s = 0; furthermore as d ≥ 2 for large enough n this integral converges even without condition γn+1  1 and thus Statements (ii) and (iii) are also proven. On the other hand, as d = 1 dropping condition γn+1  1 leads to an extra factor γ¯n−1 ≤ h−δ with δ = 1/(n + 1). This proves Statement (iv). Surely, one needs to consider also contribution of the zone {γn  γ¯n }; as m ≥ hn/(n+1) we get (5.3.153) as d ≥ 2, or as d = 1 and non-degeneracy condition is fulfilled and we get C γ¯n−1

 h −δ  h  ϑ γ¯nn+1 L

as d = 1 and we assume no non-degeneracy condition. Finally, as m ≤ hn/(n+1) the analysis of zone {γn  γ¯n } is also trivial. So, Statements (ii)–(iv) are proven completely. (v) As d < s + 1, n = 2 expression (5.3.152) has the same magnitude as (5.3.153) which is exactly the right-hand expression of (5.3.150). So, as γ3  1 estimate (5.3.150) is proven even for d = 1. Let us scale the right-hand expression of (5.3.150): i. e. plug m → mγ −n , h → hγ −(n+1) and multiply by γ −d ; we get  h   1 (2d−2−s) max(h2 γ −2(n+1) , m3 γ −3n ) 6 ≤ Lγ  h   1 (2d−2−s) 1 max(h2 , m3 γ −n−2 ) 6 Ch1−d γ −1+ 3 (d+s−1)(n−2) ϑ L

C (hγ −n−1 )1−d γ −d ϑ

and then integrating with respect to dx we get the same right-hand expression of (5.3.150) provided either γn+1  1 or n is large enough. This takes care of Statement (v). Let us improve Statement (iii) of Theorem 5.3.34: Theorem 5.3.35. Let d = 1 and conditions (5.3.71)–(5.3.71), (5.3.77)1−7 , (5.3.78), and (5.3.123), (5.3.128) (4.2.73)–(4.2.75) be fulfilled. Furthermore, let both V ± m satisfy non-degeneracy condition (5.2.100)n . Then estimate |RW | ≤ C holds.

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Proof. Let us repeat the proof of the previous theorem and treat the behavior of the Hamiltonian trajectories in the zone {ρ ≥ C0 m} (the only zone which yields the logarithmic factor)28) . Note that on each step of induction γ(x)  ε + |x1 − xˆ1 | with ε ≥ 0 where without any loss of the generality one can assume that xˆ1 = 0. Then on each step of the induction a logarithmic factor appears only due to the zone {|x1 | ≥ C0 γ¯n } with γ¯n increased to  1  1 (5.3.154) γ¯n := max ε, m n , h n+1 and an arbitrarily large constant C0 . But in this zone one should not apply the induction assumption but only the microhyperbolic theorem; so the power of logarithm in the right-hand expression of (5.3.148) is always 1. Moreover, let us note that after rescaling tnew = tγ −1 and therefore the only term to be treated is  1 dx1 (5.3.155) {|x1 |≥C0 γ ¯n } T (x1 ) and a logarithmic factor appears if we take T = 0 γ. Here on the energy level 0 1  (5.3.156) ξ1 = ± V 2 − m 2 2 . Let us consider the trajectory of the Hamiltonian system (after ξ1 is excluded)  2 1 V − m2 2 dx1 =± (5.3.157) dt V starting from the point x¯1 in the time direction in which |x1 | increases. Then the time until x1 remains in {|x1 | ≤ 0 } is disjoint from 0 and one can see easily that in this time interval |x1 (t)|  |x1 (0)| + |t| because as |x1 | increases |V (x)| also increases and stays larger than 2m. Then non-periodicity condition is fulfilled trivially. Further, one can prove easily that |dx1 (t)| ≤ C |dx1 (0)| + C (|x1 (0)| + |t|)r |x1 (0)|−r −n+1 |dτ | with large enough r and so in order to fulfill condition on the Jacobi matrix we should take T (x1 ) = 1 min(1, |x1 |1+δ γ¯n−δ ) with sufficiently small exponent δ > 0; we replaced here x1 (0) by x1 . But then integral (5.3.155) is bounded. 28)

Surely, these trajectories coincide geometrically with the trajectories of the Schr¨ odinger operator (depending on the energy level) but dynamically there is an essential difference.

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

485

Sharp Remainder Estimates Now our goal is to derive sharp spectral asymptotics. Let us first formulate them in microhyperbolic case; the following statement is due to Section 4.4: Theorem 5.3.36. Let conditions of Corollary 5.3.30 be fulfilled. ± Further, let Λ± j be closed subsets of Σ and suppose that through every ± point of Λj there passes a Hamiltonian trajectory generated by J±−1 λ± with 1  t ∈ Jj , Jj = [0, Tj ] or J= [−Tj , 0], Tj ∈ [1, h−δ ] in the h 2 −δ -neighborhood for which conditions (5.3.122)1−7 , (2.4.29), (5.3.14) are fulfilled and 

(m + W )2 + g jk (ξj − Vj )(ξk − Vk ) ≥ 1 h2δ .

(5.3.158)

(i) Moreover, let microhyperbolicity condition (5.3.126)-(5.3.127) be fulfilled and along trajectories non-periodicity condition (5.3.15) be fulfilled. Then estimate (5.3.159)

| tr(RW ϕ,L )| ≤ C

 h1−d   h  1−d+δ  dμ + C h ϑ L T 1+s Λj 0≤j≤n j

− ± holds with Λj = Λ+ j ∪ Λj , dμ := Jdxdξ : dλ± on Σ .

(ii) Moreover, let microhyperbolicity condition (5.3.125) be fulfilled and along trajectories non-looping condition   1 (5.3.160) dist x(t), x ≥ |t|h 2 −δ be fulfilled. W Then estimate (5.3.159) holds with RW ϕ,L , Λj and dμ replaced by Rx,ϕ,L , Λxj , dμx = Jdξ : dλ± respectively.

Remark 5.3.37. (i) Here condition (5.3.158) prevents eigenvalues λ± (x, ξ) from meeting one another. One can see easily that with μx = dx : dλ (5.3.161)

μx ({|ξ − V (x)|g ≤ ε}) ≤ C (m + W (x) + ε)εd−2

where |.|g is associated with matrix g jk and V = (V1 , ... , Vd ). (ii) Further, J+  J− and we can skip index here.

486

CHAPTER 5. SCALAR OPERATORS AND RESCALING

(iii) We took the trace to avoid dealing with matrices. Proposition 5.3.38. Let W = 0 and d ≥ 2 and suppose the condition  g jk ηj ηk ≥ hδ |η|2 ∀η ∈ Rd (5.3.162) j,k

holds along trajectories. Then in Theorem 5.3.36(i) one can drop condition (5.3.158). 

Proof. It is sufficient to prove that μ(Z) ≤ Chδ with small exponent δ  > 0. Here Z is the set of all points (x, ξ) ∈ Λ (we skip index j for the sake of simplicity) such that (5.3.158) is violated somewhere along trajectory. Let us first note that because of (5.3.161) this estimate holds for the measure of the set of all points of Σ± ⊂ T ∗ B(0, 1) at which |V (x)| ≤ ρ1 :=  hδ . Hence we should consider points in which |V (x)| ≥ hδ1 and therefore condition (5.3.158) holds but is violated somewhere along the trajectory. Let Z  be the set of these points. Without any loss of the generality we can assume that T ≤ h−δ1 and all inequalities along trajectories are fulfilled with δ1 instead of δ  ; so δ  which is arbitrarily small but much larger than δj is invoked only in definition of Z. Let X  be the union of x-projections of all the Hamiltonian trajectories of length not exceeding T passing through Z  . Then every such trajectory intersects Y1 where Yν = {(x, ξ) ∈ Σ, 2−ν ρ ≤ |V | ≤ 2ν ρ},

ρ = hδ



and one can assume that m ≤ 18 ρ. Along these trajectories (5.3.163)

|

dz | ≤ c1 h−δ2 ; dt

hence X  ⊂ B(0, h−δ3 ) and the trajectory intersecting Y1 belongs to Y2 in a time interval of length t0 = hδ4 ρ. Hence due to ability to select invariant measure equivalent to μ μ(Z  ) ≤

T μ(Y2 ) ≤ C  h−δ5 ρd−2 mes({x ∈ X  , |V | ≤ 4ρ}) ≤ C  h−δ6 ρd−2 t0

because μ(Y2 ) ≤ C ρd−1 h−δ5 due to (5.3.161), (5.3.162) and T ≤ h−δ1 . Then for d ≥ 3 we obtain the estimate in question as δ  > 2δ6 .

¨ 5.3. SCHRODINGER AND DIRAC OPERATORS

487

For d = 2 a more sophisticated analysis is necessary. Let Yν = {z ∈ Yν , |∇V | ≤ 2ν ρ1 },

Yν = {z ∈ Yν , |∇V | ≥ 2−ν ρ1 }.

with small enough δ1 > 0.  Then the previous arguments are applicable to Yν , t0 = hδ4 ρρ−1 1 , Yν and  δ4 t0 = h ρ and hence  μ(Z  ) ≤ C  h−δ5 ρd−2 ρ1 mes({x ∈ X  , |V | ≤ 4ρ})+

 3 mes({x ∈ X  , |V | ≤ 4ρ, |∇V | ≥ ρ1 }) ≤ C  h−δ6 ρd− 2

2 where we estimated the second measure by Ch−δ5 ρρ−1 1 and picked up ρ1 = ρ . 1

Theorem 5.3.39. Let conditions of Corollary 5.3.30 be fulfilled. ± Moreover, let Λ± Σ± = {(x, ξ) : λ± = 0} 1 , ... , Λn be closed subsets of where λ± are given by (5.3.83),  1 |V |2  Λ± ≥ h 2 −δ } j ⊂ {γ = |∇V | + min |V |, m and through every point of Λ± j there passes a Hamiltonian trajectory gen −1 erated by J± λ± with t ∈ Jj , J= [0, Tj ] or J= [−Tj , 0], Tj ∈ [1, h−δ ] in the 1 h 2 −δ -neighborhood for which conditions (5.3.122)1−5 and conditions (2.4.29), (5.3.14), (5.3.15) are fulfilled. Further, let (5.2.101) with s ≥ 0 be fulfilled. (i) Let either d > s + 2 or Vj = 0 and 2d > s + 2. Then estimate (5.3.16) − ± with Λj = Λ+ j ∪ Λj , da = dλ± at Σ remains true. (ii) Let d = 2, s = 0. Then estimate (5.3.26) holds where the notation of Theorem 5.3.3 is used. Proof. Proof of Statement (i) is trivial. Let us prove Statement (ii). We 1 need to consider only balls of radius γ1 = h 2 −δ in which (5.3.164)1,2,3



|V | ≤ c0 h 2 −δ , 1



|∇V | ≤ c0 h 2 −δ , 1

1 

m ≤ c0 h 2 − 4 δ ; 1

the last condition follows from the fact that γ ≥ 0 m in this proof. Let us consider various cases:

CHAPTER 5. SCALAR OPERATORS AND RESCALING

488

1 

(a) If |F12 (x l )| = μ ≥ h 4 δ and v = |V (x l )| ≤ μγ1 then |F12 |  μ and |V | ≤ cμγ1 in the ball B(x l , γ1 ). Let us change variables xnew = xγ1−1 , divide A by μγ1 and set hnew = h(μγ12 )−1 , mnew = m(μγ1 )−1 ; then we remain in the framework of our theorem and the same arguments as in the proof of Theorem 5.3.25 yield that the contribution of this ball to the remainder does not exceed  C γ12 μh−1 + C  γ12 hδ −1 + C  hr . 1 

(b) Let |F12 (x l )| = μ ≥ h 4 δ and v = |V (x l )| ≥ μγ1 . Let us change variables xnew = xγ1−1 , divide A by v and set hnew = h(v γ1 )−1 , mnew = mv −1 ; then we remain in the framework of this theorem and the microhyperbolicity condition is fulfilled; applying Theorem 5.3.25 we obtain that the contribution of this ball to the remainder does not exceed C γ1 vh−1 + C  γ12 hδ

 −1

+ C  hr .

1 

(c) If |F12 (x l )| ≤ μ = h 2 δ then the analysis of Parts (a) and (b) is applicable for v = |V (x l )| ≤ μγ1 and v = |V (x l )| ≥ μγ1 respectively and we obtain that the estimate of (b) remains true. Calculating the total contribution of all balls of this type we obtain (5.3.26). Remark 5.3.40. Thus againwe can easily derive asymptotics with the remainder estimate o h1−d ϑ(h/L) under assumption that the set of all periodic trajectories has measure 0; as m = 0 we should consider only trajectories which do not hit Θτ = {(x, ξ) : ξ1 − V1 (x) = ... = ξd − Vd (x) = 0, V (x) = τ }. As d = 2 we need an extra condition (5.3.165)

mes({x : V (x) = ±m, |F (x)| > 0}) = 0.

Remark 5.3.41. (i) Similar results could be proven for asymptotics without spatial mollification; however in this case we cannot get rid of condition (5.3.158) along trajectories but one could check that actual assumption should be with δ  replaced by 12 − δ  . (ii) It would be rather interesting to improve asymptotics without averaging as V1 = ... = Vd = 0 with addition of correction term, especially in the case of m = 0. However condition (5.3.158) seem to be an obstacle.

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

5.4 5.4.1

489

Operators with Irregular Coefficients Preliminary Remarks

Now we consider operators with irregular coefficients (as in Sections 4.6 and 4.7) and for such scalar operators we are going to extend to a certain degree results of the previous sections of this chapter. We consider only asymptotics with the spatial averaging due to the following reason: Analysis of asymptotics without spatial averaging is much more straightforward and thus simpler. Really, our base points for rescaling procedure used to be results of Sections 4.4 and 4.5 for microhyperbolic and ξ-microhyperbolic operators respectively as we consider asymptotics with or without spatial averaging. However the results Sections 4.6 and 4.7 require ξ-microhyperbolicity and only in the case of ridiculously large roughness parameter29) we could derive them under microhyperbolicity condition. So our first goal is to recover these results under microhyperbolicity condition (plus some extra assumptions). On the other hand, as analysis of Section 5.3 shows, for a Schr¨odinger operator h2 |D|2 +V (x) exactly points with V = 0 and |∇V |  1 are the main obstacle for pointwise spectral asymptotics. Still, it is not that simple: as this analysis shows the remainder estimate obtained just by rescaling is not necessary optimal even if we consider only asymptotics without correction term. Therefore we formulate the following problem which we consider really challenging, especially its second part: Problem 5.4.1. (i) Extend results of Subsection 5.3.1 (i.e. only for Schr¨odinger operator) partially (i.e. without correction term and respectively with the remainder at least of the same magnitude as a correction term) to operators with irregular coefficients. (ii) Extend results of Subsection 5.3.1 completely (i.e. with correction term) to operators with irregular coefficients. Recall that our goal are operators with irregular coefficients rather than the “rough” operators which are our main objects of study and we either bracket irregular operators between rough ones but this bracketing has implications only for asymptotics of the number of eigenvalues which will 29)

1

ε = C (h| log h|) 2 as opposed to ε = Ch| log h|.

CHAPTER 5. SCALAR OPERATORS AND RESCALING

490

be assembled from (micro)local asymptotics with the spatial averaging or use the method of perturbation as in Sections 4.6 and 4.7 respectively. Let us discuss in more detail bracketing method. Recall that applying our rescaling technique we have two scaling functions γ(x) and ρ(x). However, applying results of Section 4.6 we need to remember about roughness parameter ε which becomes a roughness function which is (much) smaller than γ(x) and usually is defined to satisfy logarithmic uncertainty principle (see (5.4.14)). Applications: we construct approximations which provide a rather small error in Weyl expressions and also rather small remainder estimate. Further, for semi-bounded operators we can select two approximations of this type 30) + A± such that A− ε ε ≤ A ≤ Aε and thus numbers of negative eigenvalues − − have opposite order: N (Aε ) ≥ N− (A) ≥ N− (A+ ε ). For non-semibounded operators we are looking for number of eigenvalues between τ1 and τ2 and thus for number of negative eigenvalues of 2 2  1 B = A − (τ2 + τ1 ) − (τ2 − τ1 )2 2 4

(5.4.1)

− + and one can construct two operators A± ε such that Bε ≤ A ≤ Bε . We will do it when we consider applications (so much later).

5.4.2

Schr¨ odinger Operator

Reminder We start from the Schr¨odinger operator as a toy-model but we consider the general one (5.3.1). Our first goal is to replace ξ-microhyperbolicity condition (5.4.2)

|V (x)| ≥ 0 ,

by microhyperbolicity ocondition (5.4.3)

|V (x)| + |∇V (x)| ≥ 0 ,

where V (x) is the potential. Since such condition makes sense only as original operator had at least C1 -coefficients we assume this. In fact we assume that 30) Where subscript ε means not the actual selection of the parameter function but simply fact that we are talking about approximations

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

491

(5.4.4) g jk , Vj , V are Cνl -functions with t l ≤ ν(t) ≤ t l+1 , l = 1, 2, ... (see Definition 4.6.1). Consider operator Aε with g jk , Vj , V replaced by gεjk , Vjε , Vε which are their ε-mollifications. Then under condition (5.4.2) we derived asymptotics (5.4.5)

h 1−d |RW ϑ x,ϕ,L,ε | ≤ Ch L

and exactly the same estimate for RW ϕ,L,ε . Let us recall that approximation −d error was Ch ν(ε). Let us recall that we could take ε pretty small: (5.4.6)

ε = Ch| log h|.

Getting Rid of Condition (5.4.2) First we consider the case of low smoothness: (5.4.7) Let ν(t) be monotone increasing function such that ν(0) = 0 and (5.4.8) and also (5.4.9)

ν(t) ≥ t,

d (ν(t)t −1 ) ≤ 0 dt

ν(C1 t) ≥ c1 ν(t) ∀t ∈ (0, 1)

with c1 > 1, C1 > 1.

Here if (5.4.9) holds for some c1 > 1 and C1 > 1 then it holds for every c1 > 1 and C1 = C1 (c1 ). Then γ(x) defined from   (5.4.10) ν(γ(x)) = max |V (x)|, ν(¯ γ) satisfies |x − y | ≤ γ(x) =⇒ C −1 γ(x) ≤ γ(y ) ≤ C γ(x) (5.4.11) and thus is a proper scaling function. Really, |x − y | ≤ γ implies |V (x) − V (y )| ≤ ν(γ). Then |x − y | ≤ γ(x) implies that |V (y )−V (x)| ≤ 1 max(|V (x)|, ν(¯ γ ) which due to (5.4.9)–(5.4.10) implies that |γ(x) − γ(y )| ≤ 2 γ(x).

CHAPTER 5. SCALAR OPERATORS AND RESCALING

492

Further, let 1

ρ(x) = ν(γ(x)) 2

(5.4.12)

and therefore we should define γ¯ , ρ¯ from (5.4.13)

1

γ¯ ν(¯ γ ) 2 = 2h,

1

ρ¯ = ν(¯ γ) 2 .

Let us consider an element B(y , γ) with γ = γ(y ) etc and let us scale x → xγ −1 , ξ → ξρ−1 , h → hρ−1 γ −1 . This also scales ε → εγ −1 and we need to satisfy ε = Ch| log h| after rescaling; thus we need to pick up h h ε = C | log | ρ ργ

(5.4.14)

Now let us notice that ν(t) scales as well: (5.4.15)

νnew (t) = ρ−2 ν(γt) = ν(γ)−1 ν(γt)

which satisfies (5.4.7). We know (see Theorem 4.6.21) that under condition (5.4.2) −d |RW ν(ε); x,ε | ≤ Ch

(5.4.16) there ε = Ch| log h|. Then (5.4.16) scales to (5.4.17)

−d d−2 |RW ρ ν x,ε | ≤ Ch

h h  | log | ρ=ν(γ) 12 . ρ ργ

Let us consider the right-hand expression of (5.4.17) as a function of ρ (so γ = γ(ρ) is defined from ν(γ) = ρ2 ). As d ≤ 2 it is clearly monotone decreasing and therefore it does not exceed its value as ρ = ρ¯ which is (5.4.18)

 1d −d |RW ν γ¯ 2  C γ¯ −d . x,ε | ≤ Ch

On the other  hand, the right-hand expression of (5.4.17) does not exceed Ch−d ρd−2 ν hρ−1 | log h| which for d ≥ 3 is a monotone increasing function of ρ due to (5.4.8) and therefore the right-hand expression of (5.4.17) does not exceed its value as ρ = 1. Thus we arrive to

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

493

Theorem 5.4.2. Let (5.4.7)–(5.4.9) be fulfilled and let us drop condition (5.4.2). Then for described above operator Aε estimate (5.4.17) holds. −d In particular, as d ≥ 3 estimate |RW ν(h| log h|) still holds and x,ε | ≤ Ch W −d as d = 1, 2 estimate |Rx,ε | ≤ C γ¯ holds with γ¯ defined from (5.4.13). Remark 5.4.3. We conclude immediately that in the framework of Theorem 5.4.2 for ψ with compact support31) . (i) As d ≥ 3 estimate (5.4.16) holds for RW ε with ε = Ch| log h|; (ii) As d = 1, 2 estimate |RW ¯ −d ε | ≤ cγ

(5.4.19)

holds with γ¯ defined by (5.4.13). (iii) One can easily improve (5.4.19) under assumption mes(Z (t)) ≤ σ(t) where Z (t) = {x : |V (x)| ≤ t}; (iv) So far the same estimates hold for bracketing-approximation errors W (actually estimates for |RW ε |, |Rx,ε | are marginally better). Indeed, we take ± Aε = Aε + C0 ν(ε). Thus we got estimates for the combined error. (v) Due to low smoothness we did not apply results of Section 4.7. Example 5.4.4. Let ν(t) = t r | log t|σ where (5.4.20)

r ∈ [0, 1],

r = 0 =⇒ σ < 0,

r = 1 =⇒ σ ≥ 0.

Then (i) If either (5.4.2) is fulfilled or d ≥ 3 then (5.4.21)

−d+r |RW | log h|r +σ ; ε | ≤ Ch

(ii) If d = 1, 2 and (5.4.2) fails then (5.4.22)

σd

− r +2 |RW | log h| r +2 . ε | ≤ Ch 2d

We leave to the reader Problem 5.4.5. Generalize these results to the spectral means averages. 31) One can easily get rid of this assumption and take ψ = 1 in Rd provided {x : V (x) ≤ 0 } is compact.

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From Condition (5.4.2) to Condition (5.4.3) Now we change our assumptions: in addition to (5.4.7) we assume that and (5.4.23)

ν(t) ≤ t,

d (ν(t)t −1 ) ≥ 0, dt

ν(t)t −1 → 0 as t → +0.

1−d Let us recall that then under condition (5.4.2) |RW and we are x,ε | ≤ Ch not talking about approximation error now. Then we can introduce scaling functions as they were introduced in the smooth case when we wanted to get rid of condition (5.4.2):

(5.4.24)

γ(x) :=  max(|V (x)|, γ¯ ),

1

2

ρ(x) := γ(x) 2 ,

γ¯ = C1 h 3

(which coincides with the definition of the previous subsubsection as ν(t) = t) and rescale as we used to; then h → hnew = hρ−1 γ −1 and ε → εγ −1 ; we define ε by (5.4.14). Then after rescaling νnew (t) satisfies νnew (t) ≤ t. Then remainder estimate scales to 1−d+s −d γ × ρ2s = Ch1−d+s ρd−1+s γ −1−s = (5.4.25) |RW x,s,ε | ≤ C (h/ργ) 1

Ch1−d+s γ 2 (d−s−3) s because ρ2 = γ and RW x,s,ε is a spectral averaging with ϕ(τ ) = τ+ , s ≥ 0 (in W W particular, Rx,0,ε = Rx,ε ). So far all as it used to be in the smooth case when we study local spectral asymptotics without averaging. Therefore  1 W 1−d+s γ 2 (d−s−3) dx (5.4.26) |Rx,s,ε | ≤ Ch

and now condition (5.4.3) changes the rules of the game: Proposition 5.4.6. For Schr¨odinger operator under conditions (5.4.3), (5.4.7) and (5.4.23)

(5.4.27)

⎧ 1−d+s ⎪ ⎨h W |Rx,s,ε | ≤ C h1−d+s | log h| ⎪ ⎩ 2 (−d+s+1) h3

as d > s + 1, as d = s + 1, as d < s + 1.

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

495

So, a proper remainder estimate is recovered for d > s + 1; while for d = s + 1 logarithmic factor appears. Meanwhile due to (5.4.3) bracketingapproximation error does not exceed   −d d−2+2s −d ρ ρd−1+2s ν(ε(ρ)) dρ. ν(ε(ρ)) dx  Ch (5.4.28) Ch Therefore Theorem 5.4.7. Let conditions (5.4.3), (5.4.7) and (5.4.23) be fulfilled. 30) − Then for appropriate bracketing operators A± (A+ ε ε ≥ A ≥ Aε ) the combined Weyl and bracketing-approximation error does not exceed (5.4.27) + 1 (5.4.28) 32) with ε(ρ) = C ρ−1 h| log(hρ−3 ) and ρ running from ρ¯ = Ch 3 to a small constant. Remark 5.4.8. (i) (5.4.28) is O(h1−d+s ) provided either d + s ≥ 2 and ν(t) = t s+1 | log t|−s−1 or d = 1, s = 0 and ν(t) = t| log t|−2−δ . (ii) As d ≤ s+1 we are rather unhappy with Weyl error because in arguments leading to Proposition 5.4.6 no smoothness requirement can produce such remainder O(h1−d+s ). Case d ≤ s + 1 Reloaded Now our goal is to improve arguments leading to Theorem 5.4.7 and to 1−d prove |RW as d ≤ s + 1 without significantly s,ε | ≤ Ch   increasing ε. We need to better estimate the contribution of B y , γ(y ) to the remainder. To do so we need to study a long term propagation of wave front sets. For simplicity of notations we assume that V = 0, g jk = δjk . Let us introduce 2

(5.4.29) γ(y ) = 0 |V (y )| + C0 (h| log h|) 3 ,

1

ρ(y ) = C0 γ(x) 2 , ε(y ) = Ch| log h|ρ−1 .

Let us consider point y and operator Q with the symbol supported in {(x, ξ) : |x − y | ≤ γ, ±ξ, ∇V (y ) ≥ −ρ} with γ, ρ calculated at x and sufficiently small constant . 32)

Here and in what follows (5.4.28) denotes the right-hand expression only.

CHAPTER 5. SCALAR OPERATORS AND RESCALING

496 Then (5.4.30)

WF(uε tQ) ∩ {|τ | ≤  ρ2 } ∩ {C0 γ 2 ≤ ∓t ≤ 0 γ 2 } ⊂ 1

1

{(x, ξ) : |x − y | ≤ C 0 γ, ±ξ, ∇V (y ) ≥ C ρ, x − y , ∇V (y ) ≥ C0 2 γ}. Indeed, it follows after rescaling due from standard propagation results. 1 Here t is an “absolute time”, after rescaling it is divided by γ 2 . Here and −1 below uε is the Schwartz kernel of Uε (t) = e ih tAε . Then one can extend it for larger t: (5.4.31)

WF(uε tQ) ∩ {|τ | ≤  ρ2 } ∩ {T ≤ ∓t ≤ 2T } ⊂

{(x, ξ) : |x − y | ≤ C 0 T 2 , ±ξ, ∇V (y ) ≥ 1 T , x − y , ∇V (y ) ≥ 21 γ} 1

as 0 γ 2 ≤ T ≤ 0 . Indeed, one can prove it easily by recursion T → 2T . Then, using property of Γ we conclude   (5.4.32) Let ψ be supported in {|V (x)| ≤ γ}. Then Ft →h−1 τ χT (t)Γ uε ψ is 1 negligible for T∗ ≤ T ≤ T ∗ with T∗ = γ 2 and T ∗ = , as |τ | ≤  γ. As usual χ ∈ C0∞ ([−1, − 12 ] ∪ [ 12 , 1]) and therefore in the Tauberian 1 estimates we can replace T  γ 2 by T  1. This works not only as 1 γ ≥ 2C0 (h| log h|) 2 where we use ξ-microhyperbolicity but also as γ ≤ 1 2C0 (h| log h|) 2 where we use microhyperbolicity (5.4.3) which is possible because there ε  γ and after rescaling h →  = C0−1 | log h|−1 . So, contribution of B(x, γ(x)) to RTs,ε does not exceed  h 1−d+s  T∗ 1+s × × ρ2s  Ch1−d+s ρd−2 γ d ργ T∗  which sums to Ch1−d+s ρd−2 dx which is does not exceed Ch1−d+s even as d = 1 in virtue of (5.4.3). So, in the current framework |RTs,ε | ≤ Ch1−d+s . (5.4.33)

C

Remark 5.4.9. Now we need to pass from Tauberian to Weyl expression and there is a twist: we need to take sufficiently many terms. (i) If d = s+1 there was only logarithmic divergence and we need to take only terms with h−d+n with 0 ≤ n ≤ s + 1 (the term where logarithmicdivergence was because “out of the box” this term does not exceed C γ −1 dx  C | log  h|5 and remainder estimate should be O(1); next term does not exceed Ch γ − 2 dx = o(1).

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

497

(ii) If d < s + 1 there was a power divergence and we need to take many terms, and only if (5.4.29)∗ γ(y ) = 0 |V (y )| + C0 h 3 −δ , 2

1

ρ(y ) = C0 γ(x) 2 , ε(y ) = Ch| log h|ρ−1 .

Recall that the bracketig error is delivered by (5.4.28). Thus we arrive to Theorem 5.4.10. Let d ≤ s +1 and conditions (5.4.3), (5.4.7) and (5.4.23) 30) − be fulfilled. Then for appropriate bracketing operators A± (A+ ε ε ≥ A ≥ Aε ) the combined Weyl and bracketing-approximation error does not exceed (i) Ch−d+s+1 + (5.4.28) 32) with ρ running from ρ¯ = (h| log h|) 3 to a small constant as d = s + 1. 1

(ii) Ch−d+s+1 + (5.4.28) + (5.4.34) with the third term Ch−d ρ¯d+2s ν(¯ ε)

(5.4.34)

and ρ running from ρ¯ = h 3 −δ to a small constant and and ε¯ = ρ¯2 as d < s +1. 1

Example 5.4.11. In the framework of Theorem 5.4.10 let ν(t) = t l | log t|−σ . (i) Then the combined error is O(h−d+s+1 ) provided l = s + 1 and either d + s > 1, σ = s + 1 or = 1, s = 0, σ = 2. Recall that l = s + 1, σ = −1 needed for this remainder estimate to work under condition (5.4.2). (ii) Other cases are left to the reader as well as ν(t) = t l | log t|σ . The following problem seems to be challenging, at least Part (ii): Problem 5.4.12. (i) Prove that in the framework of Theorem 5.4.10 one can preserve in the Weyl expression NW s,ε only terms containing h to the power lesser than −d + s + 1, and then one can replace Weyl expression for W approximation NW s,ε by Weyl expression for the operator itself Ns . Recall that Weyl coefficients appear first as distributions with respect to τ . (ii) Prove that Statement (i) of Theorem 5.4.10 holds even as d < s + 1.

498

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Dropping Condition (5.4.3) Recall that as d ≥ s +3 we simply dropped condition (4.5.2) but as d < s +3 we traded it to (4.5.3). Now we want to get rid of the latter. So, let d < s +3. Let ν(t) satisfy (5.4.7), (5.4.23) as before and let us replace (5.4.9) by a stronger assumption (5.4.35)

ν(C1 t) ≥ C1 c1 ν(t) ∀t ∈ (0, 1)

with c1 > 1, C1 > 1

which means exactly that ν(t)t −1 also satisfies (5.4.9). We also assume that (5.4.36) t −2 ν(t) is a monotone decreasing function so that we do not assume more smoothness than C2 . Let us introduce γ, ρ by the following formula similar to (5.4.10) (5.4.37) γ = max(γ  , γ¯ ),

γ  =  min{t : |V | ≤ ν(t), |∇V | ≤ t −1 ν(t)}, 1

with γ¯ defined from γ¯ ν(¯ γ) 2 = h Then γ is a scaling function i.e. (5.4.11) holds. Indeed, if |x − y | ≤ γ then |∇V (x) − ∇V (y )| ≤ ν(|x − y |)|x − y |−1 ≤ 1 ν(γ)γ −1 and therefore |∇V (y )| ≤ 2ν(γ)γ −1 . Then |V (x) − V (y )| ≤ 2ν(γ)γ −1 |x − y | + ν(|x − y |) ≤ 1 ν(γ). Combining these two inequalities and using (5.4.9), (5.4.35) we conclude that γ(x) satisfies (5.4.11). Let us define ρ, γ¯ by (5.4.12), (5.4.13). Then after our usual rescaling, we obtain an operator satisfying (5.4.3) as γ ≥ 2¯ γ . Then h−d+s+1 is replaced by 1 (d+s−1) −d+s+1 −s−1 Ch γ dx which in the current framework we cannot ν(γ) 2 estimate by anything better than (5.4.38)

Ch−d+s+1

sup γ:γν(γ)1/2 ≥h

ν(γ) 2 (d+s−1) γ −s−1 . 1

One can observe easily that in this scaling procedure h → h/(ργ), ν(ε) → ν(εγ)/ρ2 with ρ = ν(γ)1/2 and terms (5.4.28) and (5.4.34) (in the framework of Theorem 5.4.10(ii)) do not increase. Also one can observe that the contribution of zone {γν(γ)1/2 ≤ C0 h} does not exceed (5.4.28). Therefore we arrive to

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

499

Theorem 5.4.13. Let d ≤ s + 3 and conditions (5.4.7) and (5.4.23), (5.4.35), (5.4.36) be fulfilled. Then for appropriate bracketing operators 30) − A± (A+ ε ε ≥ A ≥ Aε ) the combined Weyl and bracketing-approximation error does not exceed 1

(i) (5.4.38) + (5.4.28) 32) with ρ running from ρ¯ = (h| log h|) 3 to a small constant as d = s + 1. (ii) (5.4.38) + (5.4.28) + (5.4.34) with ρ running from ρ¯ = h 3 −δ to a small constant and and ε¯ = ρ¯2 as d < s + 1. 1

Remark 5.4.14. (i) Let d ≥ 2 and ν(γ) ≤ γ l with l = l(s) = 2(s + 1)/(d + s − 1). Then expression (5.4.38) is  Ch−d+s+1 ). Here l(s) = 2 as d = 2 and l(s) < 2 as d ≥ 3. (ii) Let d = 1, l = 2 expression (5.4.38) is  Ch−d+s+ 2 ). Assuming that ν(t) = o(t l ) and ν(t) ≥ t l+1 with l = 2, 3, ... , one can introduce a scaling function 1

(5.4.37) γ = max(γ  , γ¯ ),

γ  =  min{t : |∇j V | ≤ t −j ν(t) ∀j = 1, ... , l} 1

with γ¯ defined from γ¯ ν(¯ γ) 2 = h and as l = 2 recover remainder estimate O(1) or O(| log h|) provided V has a single non-degenerate minimum or maximum. One can even recover remainder estimate O(1) if either V has a single non-degenerate minimum or s > 0, ν(t) = O(t 2 | log t|−σ , σ > 2. The one can continue as in Subsection 5.2.3. Sharp Remainder Estimates So far the best possible combined remainder estimate we derived is O(h−d+1+s ): Theorem 5.4.15. (a) In the framework of Theorem 5.4.7 assume that d > s + 1 and expression (5.4.28) is O(h1−d+s ). (b) In the framework of Theorem 5.4.10(i) assume that expression (5.4.28) is O(h1−d+s ). (c) In the framework of Theorem 5.4.10(ii) assume that expressions (5.4.28) and (5.4.34) are O(hs+1 ).

500

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(d) In the framework of Theorem 5.4.13(i) assume that expressions (5.4.28) and (5.4.38) are O(h1−d+s ). (e) In the framework of Theorem 5.4.13(ii) assume that expressions (5.4.28), (5.4.34) and (5.4.38) are O(h1−d+s ). Here all terms are calculated in the frameworks of the corresponding theorems. Then the combined error is O(h−d+1 ) Could we do better than this? Observe that all terms in the estimates we derived which contain ν estimate approximating-bracketing errors while Ch1−d+s estimates semiclassical error33) . Only the latter could be improved. Therefore Theorem 5.4.16. (a) In the framework of Theorem 5.4.7 assume that d > s + 1 and expression (5.4.28) is o(h1−d+s ). (b) In the framework of Theorem 5.4.10(i) assume that expression (5.4.28) is o(h1−d+s ). (c) In the framework of Theorem 5.4.10(ii) assume that expression (5.4.28) is o(hs+1 ) and (5.4.39)

sup γ 0. 35) Let us recall we call point z periodic if z ∈ Ψt (z) for some t : ±t > 0. Let us recall that to dismiss point as dead-end or periodic we are free to chose (the same) sign ±. 34)

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

501

Proof. The easy proof is left to the reader. Applying Aesults of Section 4.7 We know from Section 4.7 that under assumption (5.4.2) (5.4.40) and (5.4.41)

1−d+s |RW + Ch−d ν(ε) 2 (s+1) x,s | ≤ Ch 1

1−d+s |RW + Ch−d ν(ε) 3 (s+1) s | ≤ Ch 2

with ε = h. Then applying the arguments of the proof of Theorem 5.4.7 we arrive to Therefore Theorem 5.4.17. Let conditions (5.4.3), (5.4.7) and (5.4.23) be fulfilled. Then |RW s | does not exceed (5.4.27) + (5.4.42) with the second term  1 2 2 −d (5.4.42) Ch ρd− 3 + 3 s ν(ε(ρ)) 3 (s+1) dρ, ε(ρ) = C ρ−1 h and ρ running from ρ¯ = Ch 3 to a small constant. 1

Applying Results of Section 4.7. II To improve (5.4.27) as d ≤ s + 1 we can apply arguments of the proof of Theorem 5.4.10 but we need to recall how estimate (5.4.41) was derived to understand how term (5.4.42) should be modified. However this improvement 2 3 would not make any sense if T := (hν(ε(ρ))− 3 < ρ, i.e. ν(ε) ≥ ε 2 . So we assume that (5.4.43)

ν(ε) = εl ,

3 l> . 2

The trouble is that we consider propagation for time larger than ρ and then the magnitude ε(ρ) (and ν(ε(ρ)) changes) and it may increase or decrease. So (5.4.44) Ch−d ρd−s−1 γ d−s−1 × ρ2s × ρ−2s−1 T 3 ν(ε(ρ))2  Ch−d ρd−s−3 γ d−s−1 × T 3 ν(ε(ρ))2

502

CHAPTER 5. SCALAR OPERATORS AND RESCALING

is not necessarily the correct estimate of the third term in the successive approximation (on the given γ-element, ρ2  γ) an it seems that we need 2 to replace here ε(ρ) by ε¯ = h 3 | log h| but it would be too bad in the end. While we cannot completely cure it, we can actually do significantly better. Recall that for each h-pseudodifferential partition element Q we need to consider Tr(G ± Q) which for the third term translates into Tr(G ± Aε Gε± Aε Gε± Q) where G ± and Gε± are parametrices for perturbed (original) and unperturbed (mollified) operators and Aε is a perturbation. If for given Q propagator Gε± increases ρ and thus decreases ε(ρ) we can indeed preserve one copy of ν(ε(ρ)). Furthermore, since we actually  have ν(ε(ρ(t))) dt in virtue of (5.4.43) we can replace here T by ρ < T . Application of Aε forces propagator to “lose direction”. However in front of Aε we can insert Q1± of the same nature as Q ± and in the case “Q ± meets Q1± ” we can do the same, but in the case “Q ± meets Q1∓ ” we need to take ρ¯ν(¯ ε) instead. Now we need to understand what to do in the opposite case, Q = Q ∓ . Observe that in successive approximation we can put G ± to the left as we did or to the right as we will do now and the choice is ours. But because of the trace we can move Q ∓ to the left. And finally, because the result is real we can complex conjugate the result, thus take adjoint operator under the trace which restores this term to its original form but with a twist: while Q ± and Aε are preserved, G ± and Gε± are replaced by G ∓ and Gε∓ and are now in the accordance with Q ∓ and the same estimate as before holds. Further, instead of three term approximation we can make many term approximation, each time gaining factor h−1 ρ¯ν(¯ ε) ≤ hδ due to (5.4.3). Therefore we can consider only “regular terms” with only Gε± and therefore the last copy of T could be replaced by ρ since we need to return to ρ2 -vicinity of supp(ψ). Thus we estimate the total error on the element Q by (5.4.45)

Ch−d+s+1 ρd−2 γ d + Ch−d−2 ρd+2s γ d (ρν(ε(ρ)))(¯ ρν(¯ ε))

and summation over partition in ρ ≤ r } returns (5.4.46)

Ch−d+s+1 + Ch−d+ 3 (l−1)−δ r d+2s+3−l 5

because to avoid troubles with Weyl terms we may need to take ε¯ = h 3 −δ . Here the first term is actually smaller but it would not make any difference. 2

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

503

However taking r = 1 we get rather poor estimate as the last term usually 2 would exceed both the first term and Ch−d+ 3 (s+1)l whicjh is the second term in (5.4.41). Therefore we apply arguments of the previous subsubsection in {r ≤ ρ  1} and estimate its contribution by the second line in the following expression (5.4.47) Ch−d+ 3 (l−1)−δ r d+2s+3−l + 5

Ch−d+s+1 r d−s−1 + Ch−d+ 3 (s+1)l r d− 3 l+ 3 s−2sl 2

2

2

1

and optimizing by r : h 3 ≤ r ≤ 1 we arrive to Theorem 5.4.18. Let d ≤ s +1 and conditions (5.4.3), (5.4.7) and (5.4.43) be fulfilled. Then the total error does not exceed espression (5.4.47) minimized 1 over r : h 3 ≤ r ≤ 1. We leave to the reader Problem 5.4.19. (i) Get rid of condition (5.4.3) like in Subsubsection 5.4.2.5. (ii) Analyze separately d = 1 as in this dimension there {V = 0} is a point and the movement could be either to it or from it which probably allows to replace one factor ρ by a factor ρ−2 h| log h|. (iii) Consider case of more regular V and irregular magnetic potential (the case we will deal in Chapters 27 and 28 and use the fact that a factor ρ pops-up in ν(ε(ρ)).

5.4.3

Dirac Operator

Let us consider Dirac operator now. We assume that m ≤ 1; otherwise it brings nothing new in comparison with Schr¨odinger operator. As |V (x)| ≥ 2m we can apply the same approach as before but with the obvious modifications described below. Getting Rid of Condition (5.4.2) Assume that (5.4.7)–(5.4.9) hold and define γ(x) by (5.4.10). Then (5.4.11) holds but now (5.4.12)–(5.4.13) are replaced respectively by (5.4.48)

ρ(x) = ν(γ(x))

CHAPTER 5. SCALAR OPERATORS AND RESCALING

504 and (5.4.49)

γ¯ ν(¯ γ ) = h.

This is a correct definition as long as ν(¯ γ ) ≥ 2m which is equivalent to (5.4.50)

ν

h ≥ 2m. m

Let us define ε by (5.4.14). Let us recall that the combined error does not exceed Ch−d ν(h| log h|) under condition (5.4.2). Let us scale x → xγ −1 , ξ → ξρ−1 , h → hρ−1 γ −1 , ε → εγ −1 and ν(t) → νnew (t) := ρ−1 ν(γt). Then we conclude that (5.4.51)

|RxW | ≤ Ch−d ρd−1 ν

h h  | log | ρ ργ

and this expression reaches its maximum as ρ  1 as d ≥ 2 and as ρ = ρ¯ as d = 1. Thus we arrive to Theorem 5.4.20. Let (5.4.7)–(5.4.9) and (5.4.50) be fulfilled and let us drop condition (5.4.2). Then (i) For d ≥ 2 the combined error still does not exceed Ch−d ν(h| log h|). ¯ −1 holds with γ¯ defined from (5.4.49). (ii) For d = 1 estimate |RW x | ≤ Cγ On the other hand if (5.4.50) fails, i. e. if (5.4.52)

ν

h ≤ 2m m

then in the zone {γ(x) ≤ γ¯1 } we find ourselves in the framework of the Schr¨odinger operator with V (x) replaced by V (x)/2m and therefore with ν(t) replaced by m−1 ν(¯ γ1 t); here γ¯1 is defined defined by (5.4.53)

ν(¯ γ1 ) = 2m.

Then as d ≥ 3 we have still the same estimates rescaled; in particular, −d |RW ν(h| log h|). x | ≤ Ch ¯−d with However for d = 1, 2 we need to rescale estimate C |RW x | ≤ t t¯ defined after rescaling by (5.4.13); the latter equation after rescaling is

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

505

 1 t¯ m−1 ν(¯ γ1 t¯) 2 = h(m¯ γ1 )−1 . Thus we arrive to estimate |RW ¯1−d t¯−d . x | ≤ Cγ W −d Denoting γ¯1 t¯ by γ¯ we arrive to |Rx | ≤ C γ¯ with γ¯ defined by (5.4.54)

γ¯ ν(¯ γ ) 2 = hm− 2 . 1

1

Thus we proved Theorem 5.4.21. Let (5.4.7)–(5.4.9) and (5.4.52) be fulfilled and let us drop condition (5.4.2). Then (i) For d ≥ 3 the combined error still does not exceed Ch−d ν(h| log h|); (ii) For d ≥ 1, 2 estimate |RW ¯ −d holds with γ¯ defined from (5.4.54). x | ≤ Cγ From Condition (5.4.2) to Condition (5.4.3) Let (5.4.7), (5.4.23) be fulfilled. Define γ(x) by (5.4.37). Then we need 1 1 to define γ¯1 = m and replace (5.4.50), (5.4.52) by m ≤ h 2 and m ≥ h 2 respectively. However in both cases we arrive to Theorem 5.4.22. Let (5.4.7), (5.4.23) be fulfilled and let us drop condition 1 (5.4.2). Let either d ≥ 3 or d = 2 and m ≤ h 2 or condition (5.4.3) be fulfilled. Then |RW | ≤ Ch1−d while an approximation error does not exceed Ch−d ν(h| log h|) as d ≥ 2 and (5.4.27) as d = 1 (and s = 0). Dropping Condition (5.4.3) 1

We are left with the case d = 1, 2 and m ≥ h 2 only. Otherwise we arrive to estimate Ch1−d for the combined error with a single exception of d = 1 when this estimate is replaced by C γ¯ −1 with γ¯ defined by (5.4.49). Using scaling defined by (5.4.37) we arrive to Theorem 5.4.23. Let d = 1, 2; let (5.4.7), (5.4.23), (5.4.35) and (5.4.36) 1 be fulfilled and let us drop condition (5.4.3). Let either d = 2 and m ≥ h 2 or d = 1. Then the combined error does not exceed C γ¯ −d where γ¯ is defined by 1 (5.4.54) as m ≥ h 2 and by (5.4.49) otherwise (which happens only as d = 1).

506

CHAPTER 5. SCALAR OPERATORS AND RESCALING

Sharp Remainder Estimate Obviously we arrive Theorem 5.4.24. Let (5.4.7) and (5.4.23) be fulfilled. Further, let d ≥ 3. Alternatively let d = 2, mes(Z (0)) = 0 and either (5.4.3) be fulfilled or 1 m ≤ h 2 or ν(t) = t 2 . Assume that the measure of the set of all dead-end34) or periodic points35) is 0. Then |RW | = o(h1−d ) and the same is true for approximation error provided (5.4.28) holds. We leave details to the reader. Let us observe that if m → m ¯ we need to investigate Hamiltonian system corresponding to m ¯ only. We leave to the reader Problem 5.4.25. (i) Consider RW s (see also Subsubsection 5.3.2.4). (ii) Consider Riesz means using also methods and results of Section 4.7.

5.4.4

General Operators

In this subsection we consider general differential operators with irregular coefficients. We approximate them by operators with the “rough” coefficients and estimate both Weyl remainder estimate and the approximation error in Weyl expression. The applications to asymptotics will come later. Dropping Condition |∇ξ a| =  0 Again, instead of the microhyperbolicity condition (5.4.55)

|a| + |∇x,ξ a| ≥ 0

as in the smooth case our “reference theorem” requires a stronger ξ-microhyperbolicity condition (5.4.56)

|a| + |∇ξ a| ≥ 0

which also works for asymptotics without spatial mollification.

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

507

Assume first that ν(t) satisfies (5.4.7)–(5.4.9) and define scaling functions from equations   (5.4.57) ν(γ) =  max |a| + |∇ξ a|2 , ν(¯ γ) ,

1

ρ = ν(γ) 2 ,

hnew =

h ργ

with γ¯ defined from equation (5.4.13) and a roughness function h h ε = C | log |. ρ ργ

(5.4.58)

Then instead of (5.4.11) we have (5.4.59) |x − y | ≤ γ(x), C

−1

|ξ − η| ≤ ρ =⇒

γ(x, ξ) ≤ γ(y , η) ≤ C γ(x, ξ),

C −1 ρ(x, ξ) ≤ ρ(y , η) ≤ C ρ(x, ξ).

Indeed, |aη (y , η) − aξ (x, ξ)| ≤ |aη (y , η) − aη (x, η)| + |aη (x, η) − aξ (x, ξ)| ≤ C0 ν(|x − y |) + C0 |ξ − η| (where aξ = ∇ξ a etc) which implies that |aη (y , η) − aξ (x, ξ)| ≤ ρ and also |η(y , η) − a(x, ξ)| ≤ |a(y , η) − a(x, η)| + |a(x, η) − a(x, ξ)| ≤ C0 ν(|x − y |) + C0 |aξ (x, ξ)| · |ξ − η| + C0 |ξ − η|2 ≤ ρ2 . These two inequalities imply (5.4.59). So, (γ, ρ) is a scaling vector-function. Consider first asymptotics without spatial mollification. Scaling estimate Ch−d ν(h| log h|) of the combined error we estimate the contribution of (γ, ρ) element by h h  Ch−d ρd−2 ν | log | ρ ργ (let us recall that νnew (t) = ρ−2 ν(γt))) and the total combined error by  h h  (5.4.60) Ch−d ρ−2 ν | log | dξ. ρ ργ

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CHAPTER 5. SCALAR OPERATORS AND RESCALING

Let us assume that one of two following conditions holds: (5.4.61)+ r aξξ has r eigenvalues of the same sign with absolute values larger than 1 ; (5.4.61)r aξξ has r eigenvalues with absolute values larger than 1 where here and below aξξ , aξx , axξ and axx denote the matrices of the second derivatives (we don’t use the last one since it does not necessarily exist). Observe that under assumption (5.4.61)r (5.4.62) a = Q(z; ξ  − μ(z)) + W (z), ξ  = (ξ1 , ... , ξr ), z = (x, ξ  ), ξ  = (xr +1 , ... , ξd ) where is Q a non-degenerate quadratic form with respect to ξ  −μ(z); further, under condition (5.4.61)+ r this form is either positive or negative definite. Then ρ  |ξ  − μ(z)| + |V (z)| and (5.4.60) equals   h h     −d ρ−2 ν | log dξ dξ (5.4.63) Ch ρ ργ where the integrand does not exceed C ρ−3 ν(h| log h|). Then the inner integral in expression (5.4.63) and thus the whole expression (5.4.63) do not exceed Ch−d ν(h| log h|) as either (5.4.61)4 or (5.4.61)+ 3 is fulfilled. The same is true under condition (5.4.61)3 provided ν(r )r −l is monotone decreasing with some l < 1.  Really, the inner integral does not exceed ρ−l−2 dρr with integral generally taken over ρ¯ ≤ ρ ≤ c and with l = 1; hence we need r = 4 to have this integral bounded; l < 1 relaxes it to r = 3; finally, under condition 1  2 (5.4.61)+ r integration is taken over set |ξ − μ(z)|  |W (z)| and therefore as r = 3 this integral is also bounded. On the other hand, under weaker assumptions we get weaker estimates: Theorem 5.4.26. Let ν satisfy (5.4.7)–(5.4.9). (i) Let either (5.4.61)4 or (5.4.61)+ 3 be fulfilled or (5.4.61)3 be fulfilled and ν(r )r −l be monotone decreasing with some l < 1. Then the combined error (for both N and Nx ) does not exceed Ch−d ν(h| log h|). (ii) Let (5.4.61)3 be fulfilled. Then the combined error (for both N and Nx ) does not exceed Ch−d ν(h| log h|).

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

509

(iii) Let condition (5.4.61)r with r = 1, 2 be fulfilled. Then the combined error (for both N and Nx ) does not exceed Ch−d ν(h| log h|) does not exceed Ch−d ρ¯r with ρ¯, γ¯ defined by (5.4.13). Case ν(t) = o(t) Assuming instead that ν(t) satisfies (5.4.7), (5.4.23) and using scaling (5.4.57) associated with ν(t) = t, i. e. (5.4.64)

  γ =  max |a| + |∇ξ a|2 , γ¯ ,

1

ρ = γ2,

hnew = hγ − 2 3

one arrives instantly to the following Theorem 5.4.27. Let ν satisfy (5.4.7), (5.4.23). Then the (i) Let either (5.4.61)+ 3 or (5.4.61)4 be fulfilled.   combined error −d ν(h| log h|) + h ; (for both N and Nx ) does not exceed Ch Then the combined error (for both N and Nx ) (ii) Let (5.4.61)3 be fulfilled.   does not exceed Ch−d ν(h| log h|) + h| log h| ; (iii) Let (5.4.55) and (5.4.61)  2 be fulfilled. Then the combined error (for N only) does not exceed Ch−d ν(h| log h|) + h| log h| ; (iv) Let (5.4.55) and (5.4.61)1 be fulfilled. Then the combined error (for N  1 only) does not exceed Ch−d ν˜(h| log h|) + h 3 | log h|) where  (5.4.65)

1

ν˜ =

ν(ρ−1 | log ρ−3 h|)ρ−1 dρ.

h1/3

Problem 5.4.28. Following the proof of Theorem 5.4.10 one   can prove that in the framework of Statement (iv) estimate is actually Ch−d ν˜(h| log h|)+h . Now we encounter strange new possibilities not available for Schr¨odinger operator. First, it may happen that (5.4.55) holds but (5.4.56) and (5.4.61)1 fail. Second, it may happen that (5.4.55) also fails and rank(aξξ ) is not large enough, but rank(aξξ aξx ) helps. Therefore let us assume that

CHAPTER 5. SCALAR OPERATORS AND RESCALING

510

(5.4.66)m The following (d + 1) × 2d-matrix has rank at ⎛ ⎞ ⎛ da a ξ1 ... aξd a x1 ... ⎜daξ ⎟ ⎜aξ ξ ... aξ ξ aξ x ... 1 d 1 1 ⎜ 1⎟ ⎜ 1 1 (5.4.67) ⎜ .. ⎟ = ⎜ .. .. .. .. .. ⎝ . ⎠ ⎝ . . . . . aξd ξ1 ... aξd ξd aξd x1 ... daξd

least m 36) : ⎞ axd a ξ1 x d ⎟ ⎟ .. ⎟ . ⎠ a ξd x d

Then we can prove easily Theorem 5.4.29. Let ν satisfy (5.4.7) and (5.4.23). (i) Let either (5.4.55) and (5.4.66)  2 or (5.4.66)4 be fulfilled; then the combined error does not exceed Ch−d ν(h| log h| + h . (ii) Let either (5.4.55) and (5.4.66)  1 or (5.4.66)3 be fulfilled; then the combined error does not exceed Ch−d ν(h| log h| + h | log h|. (iii) Let either (5.4.55) or (5.4.66)2 be fulfilled; then the combined error does 2 not exceed Ch−d+ 3 . (iv) Let (5.4.66)1 be fulfilled; then the combined error does not exceed 1 Ch−d+ 3 . Proof. Let us introduce γ and ρ by (5.4.64) again (only in the vicinity of {∇ax = 0}). Then the total combined error does not exceed again    h 3 h  −d (5.4.68) Ch γ −1 ν 1 | log 3 | + Chγ − 2 dxdξ γ2 γ2 1

where we divided the previous expression by ρd γ d and plugged ρ = γ 2 . Note that under condition ((5.4.66))m in an appropriate non-symplectic coordinates γ zj2 + γ¯ 1≤j≤m

and under conditions (5.4.55) and ((5.4.66))m γ  |z1 | + zj2 + γ¯ 2≤j≤m

which leads to the corresponding estimates of integral (5.4.68). 36)

Which means that there exists m-minor with an absolute value not less than 1 .

5.4. OPERATORS WITH IRREGULAR COEFFICIENTS

511

Case ν(t) = o(t). II Basically we are left with the cases when (5.4.55) or (5.4.66)2 or (5.4.66)1 is fulfilled and we hope to improve remainder estimate as ν(t) ≤ t l with l ∈ (1, 2]. Problem 5.4.30. Following the proof of Theorem 5.2.14 prove that under  condition (5.4.55) the combined error does not exceed Ch−d ν˜(h1−δ ) + h1−δ with arbitrarily small δ > 0. Now we need to cover case when only (5.4.66)m with m = 1, 2 is fulfilled. Assuming that (5.4.35), (5.4.36) are fulfilled we can introduce γ, ρ by the following formula similar to (5.4.37) (5.4.69) γ = max(γ  , γ¯ ),

γ  =  min{t : |a| + |aξ |2 ≤ ν(t), |ax | ≤ t −1 ν(t)}.

with ρ, γ¯ defined by (5.4.12) and (5.4.13). One can prove easily that (5.4.59) holds. Theorem 5.4.31. Let continuity modulus ν satisfy (5.4.7), (5.4.23), (5.4.35), (5.4.36) and (5.4.66)m be fulfilled with m = 1, 2. Then the combined error is O(h−d ρ¯m + h1−δ−d ) with γ¯ , ρ¯ defined by (5.4.13). Proof. Let us consider (ρ, γ)-element. (a) As γ ≥ c0 γ¯ and |a| + |∇ξ a|2  ρ2 there its contribution to the remainder does not exceed Ch1−d ρd−1 γ d−1 and contribution of all these elements does not exceed  1−d Ch ρ−1 γ −1 dxdξ. (b) As γ ≥ c0 γ¯ and |∇x a|  ν(γ)γ −1 there its contribution to the remainder does not exceed due to problem 5.4.30 Ch1−δ−d ρd−1+δ γ d−1+δ and contribution of all these elements does not exceed  Ch1−δ−d ρδ−1 γ δ−1 dxdξ. Thus, we got the second integral anyway. Then due to condition (5.4.66)m mes({z : ρ(z)  ρ})  ρm and this integral does not exceed  (5.4.70) Ch1−δ−d ρm+δ−2 γ(ρ)δ−1 dρ and if

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(5.4.71) M(ρ) = ρm+δ−1 γ(ρ)δ−1 satisfies M(kρ) ≥ k −σ M(ρ) with σ > 0 as k≤1 we conclude that this integral does not exceed Ch1−δ−d ρ¯m+δ−1 γ¯ δ−1 = Ch−d ρ¯m . Obviously (5.4.71) holds as m = 1, δ = 13 for sure, we do not need even Problem 5.4.30 but can refer to Theorem 5.4.29(ii). If m = 2 situation becomes more complicated and slightly varying δ we can achieve either (5.4.71) or M(kρ) ≤ k σ M(ρ); in the latter case (5.4.70) does not exceed Ch1−d+δ .

Problem 5.4.32. (i) Assuming that ν(t) = o(t 2 ) introduce axx into consideration and replace assumption (5.4.66)m by a weaker assumption (5.4.72)m The following (2d + 1) × 2d-matrix has rank at least m 36) : ⎞ ⎛ a ξ1 da ⎜ daξ ⎟ ⎜ aξ ξ ⎜ 1⎟ ⎜ 1 1 ⎜ .. ⎟ = ⎜ .. ⎝ . ⎠ ⎝ . ⎛ (5.4.73)

daxd

a x d ξ1

⎞ ... aξd a x1 ... axd ... aξ1 ξd aξ1 x1 ... aξ1 xd ⎟ ⎟ .. .. ⎟ . .. ... ... . . ⎠ . ... axd ξd axd x1 ... axd xd 1

2 introduce (ii) Alternatively   ε = C (h| log h|) , derive the combined estimate 1 −d ν((h| log h|) 2 ) + h under microhyperbolicity condition and scale it. Ch

Problem 5.4.33. (i) Consider RW s (see also Subsubsection 5.3.2.4). (ii) Consider Riesz means using also methods and results of Section 4.7. Sharp Remainder Estimates As long as the contribution  of {z : γ(z) ≤ η} to the remainder estimate does not exceed μ(η) + oη (1) h1−d as h → +0 with μ(η) = o(1) as η → +0, under standard non-periodicity condition one can derive remainder estimate RW = o(h1−d ). Moreover if an approximation error is o(h1−d ) then the combined error is o(h1−d ) as well.

5.A. APPENDICES

5.A 5.A.1

513

Appendices Spectral Kernel Calculations for some Model Operators

Case d = 1 Consider operator (5.3.59). Note first that B = −x1 =⇒ ∂τ eB (x, y , τ ) = δ(x1 − y1 )δ(x1 + τ ).

(5.A.1)

Then making unitary h-Fourier transform we conclude that for   (5.A.2) B = hDξ1 =⇒ ∂τ eB (ξ1 , η1 , τ ) = (2πh)−1 exp ih−1 (ξ1 − η1 )τ . Therefore since T ∗ hDξ1 T = hDξ1 + ξ12 for T = exp(ih−1 13 ξ13 ) we conclude that (5.A.3) B = hDξ1 + ξ12 =⇒

 

1 ∂τ eB (ξ1 , η1 , τ ) = (2πh)−1 exp ih−1 (ξ1 − η1 )τ − (ξ13 − η13 ) . 3

Finally making inverse Fourier transform we conclude that for A defined by (5.3.59) (5.A.4) ∂τ e(x1 , y1 , τ ) =  

 1 exp ih−1 (ξ1 − η1 )τ + x1 ξ1 − y1 η1 − (ξ13 − η13 ) dξ1 dη1 (2πh)−2 3 and therefore (5.A.5) ∂τ e(x1 , x1 , τ ) =   

1 −2 exp ih−1 (ξ1 − η1 )(τ + x1 ) − (ξ13 − η13 ) dξ1 dη1 . (2πh) 3 One can rewrite it as (5.A.6)

e(x1 , x1 , τ ) = h

− 23



2

(x1 +τ )h− 3

F (τ ) dτ , −∞

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CHAPTER 5. SCALAR OPERATORS AND RESCALING

with (5.A.7)



  1 exp i (ξ1 − η1 )t − (ξ13 − η13 ) dξ1 dη1 = 3   

2 exp i 2βt − β 3 − 2βα2 dα dβ = 2(2π)−2 3 

 3 1 2  π (2π)− 2 |β|− 2 exp −i sgn β + i 2βt − β 3 dβ 4 3

F (t) = (2π)

−2

where we first substituted ξ1 = α + β, η1 = α − β and then integrated with respect to α. 1 The remaining integral has stationary points β = ±t 2 and a singular point β = 0. Decomposing for t > 0 F (t) = F1 (t) + F2 (t),    π 3 1 1 1 (5.A.8) F1 (t) = (2π)− 2 |β|− 2 exp −i sgn β + 2iβt dβ = κt − 2 4 2 with  ∞ 1 π − 32 (5.A.9) β − 2 cos(2β − ) dβ, κ = 2(2π) 4 0 and (5.A.10) F2 (t) = (2π)

− 32



   2   π 1 |β|− 2 exp −i sgn β + 2iβt exp − iβ 3 − 1 dβ. 4 3

Then primitives of F1 (t) and F2 (t) produce exactly main term and correction term in (5.3.36):  t F2 (t  ) dt  = (5.A.11) Q(t) = ∞     2  3 3 π −2 − 2 (2π)− 2 |β|− 2 i sgn β + 2iβt exp − iβ 3 − 1 dβ. 4 3 In this integral singularity at β = 0 gives a relatively small contribution 3 (one can prove it is O(t − 2 ) as t → ∞); the main contribution comes from 1 3 the stationary points β = ±t 2 and modulo O(t − 2 ) (5.A.12) Q(t) ≡ −2−1 (2π)−1

1

 π 4  |β|−2 exp i sgn β + iβt = 2 3

β=±t 2

(2π)−1 t −1 sin

4 3  t2 . 3

5.A. APPENDICES

515

Case d = 2 Now we consider operator A = h2 Dx21 + h2 Dx22 − x1 leading to B = ξ12 + ξ22 + hDξ1 and then instead of (5.A.5) we have (5.A.13) ∂τ e(x1 , x1 , τ ) =  

 1 −3 exp ih−1 (ξ1 − η1 )(τ + x1 − ξ22 ) − (ξ13 − η13 ) dξ1 dξ2 dη1 . (2πh) 3 and all previous formulae are adjusting accordingly. Then (5.A.6)–(5.A.7) are replaced by (5.A.14) e(x1 , x1 , τ ) = h

− 43



2

(x1 +τ )h− 3

F (τ ) dτ , −∞

with

   2 exp i 2β(t + γ 2 ) − β 3 − 2βα2 dαdβdγ = (5.A.15) F (t) = 2(2π)−3 3  

 2 (2π)−2 |β|−1 exp i 2βt − β 3 dβ 3 and (5.A.8)–(5.A.10) are replaced by    −2 |β|−1 exp 2iβt dβ = κ + κ1 t −3 (5.A.16) F1 (t) = (2π)

with (5.A.17)

κ = 2(2π)

− 32



∞ 0

β − 2 cos(2β − 1

π ) dβ, 4

where integral was understood in the sense of distributions and      2  (5.A.18) F2 (t) = (2π)−2 i −1 |β|−1 exp 2iβt exp − iβ 3 − 1 dβ. 3 Then (5.A.11) is replaced by  t F2 (t  ) dt  = (5.A.19) Q(t) = ∞      2  −1 −2 − 2 (2π) |β|−2 sgn β exp 2iβt exp − iβ 3 − 1 dβ. 3

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516

In this integral singularity at β = 0 gives a small contribution; really,  decomposing exp − 23 iβ 3 − 1 into powers of β 3 we get that he leading term 1 is O(t −2 ). The main contribution comes from the stationary points β = ±t 2 3 and modulo O(t − 2 ) 4 3  5 (5.A.20) Q(t) ≡ (2π)−1 t − 4 sin t 2 . 3

5.A.2

On Pauli Matrices

The results of this appendix are important for Section 4.6 and for later applications. Definition 5.A.1. Pauli matrices 37) are D × D matrices σ0 , ... , σd such that (5.A.21) and (5.A.22)

σ†j = σj , σj σk + σk σj = 2δjk I

j, k = 0, ... , d.

We have the key statement Theorem 5.A.2. (i) A set σ0 , ... , σd of D × D Pauli matrices exists if and  ¯ where D ¯ = 2 d+1 2 only if D ∈ DZ . ¯ and d is odd then all sets of Pauli matrices are (ii) Moreover, if D = D 38) ¯ and even d there exist unitarily equivalent On the other hand, for D = D precisely two sets of Pauli matrices which are not unitarily equivalent. ¯ and d is even then (iii) If D = D (5.A.23)

d

σ0 · · · σd = ςi 2 I

with ς = ±1

¯ and d is odd then and two sets are equivalent if and only if ς = ς  . If D = D σ0 , ... , σd , σd+1 with (5.A.24)

σd+1 = ς(−i)

d+1 2

are also Pauli matrices. 37)

As D = 4 they are also called Dirac matrices. The sets σ0 , ... , σd and σ0 , ... , σd are unitarily equivalent if σj = q † σj q ∀j = 0, ... , d with appropriate unitary matrix q. Obviously the transition to a unitarily equivalent set preserves properties (5.A.21) and (5.A.22). 38)

5.A. APPENDICES

517

¯ ∈ Z and d is odd then every set of Pauli matrices is unitarily (iv) If p = D/D ¯ j ⊗ Ip j = 0, ... , d where {¯ equivalent to the set σj = σ σj j = 0, ... , d} is a ¯ standard set in dimension D. ¯ ∈ Z and d is even then every set of Pauli matrices Further, if p = D/D ¯ j ⊗ Jp,q j = 0, ... , d where Jp,q = is unitarily equivalent to the set σj = σ diag(1, ... , 1, −1, ... , −1) is a diagonal p × p matrix, q = 0, ... , p. Here sets with different q are not unitarily equivalent. ¯ So a set of Pauli matrices is irreducible if and only if D = D; (v) The property “Every matrix commuting with all the Pauli matrices is ¯ scalar” is fulfilled if and only if D = D. Proof. Property (5.A.22) yields that σ2j = I and hence Spec σj ⊂ {−1, 1}. Let us reduce σ0 to the diagonal form; then for d = 0 a set of Pauli matrices is unitarily equivalent to the set consisting of one matrix   I 0 (5.A.25) σ0 = 0 −I ¯ =1 and hence for d = 0 we obtain all the statements of the theorem and D and for D = 1 σ0 = ε = ±1. Thus without any loss of the generality one can assume that (5.A.25) holds and for d ≥ 1 (5.A.21) and (5.A.22) yield that −σ0 = σ−1 1 σ0 σ1 and hence the multiplicities of the eigenvalues ±1 are equal to 12 D. Moreover, (5.A.21) and (5.A.22) yield that   0 αj for j = 1, ... , d (5.A.26) σj = α†j 0   I 0 where αj are unitary. Let q = where q  is a unitary 12 D × 12 D matrix; 0 q then q † σ0 q = σ0 and σj = q † σj q (j ≥ 1) have the same form as σj but with αj replaced by αj = αj q. Hence without loss of generality one can assume that α1 = iI , i. e. , that   0 iI . (5.A.27) σ1 = −iI 0 ¯ = 2. So we obtain Statements (i) and (ii) for d = 1 with D

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Moreover, (5.A.21) and (5.A.22) yield that if σ0 and σ1 have the indicated form then the matrices αj with j = 2, ... , d also satisfy (5.A.21) and (5.A.22). The proof is complete. Lemma 5.A.3. If d ≥ 2 and conditions (5.A.21) and (5.A.22) are fulfilled then by a unitary transformation the set σ0 , ... , σd can be reduced to the form       I 0 0 iI 0 αj , σ1 = , σj = (5.A.28) σ0 = 0 −I −iI 0 αj 0 (j = 2, ... , d) where α2 , ... , αd are 12 D × 12 D matrices satisfying (5.A.21) and (5.A.22) also. Conversely, if α2 , ... , αd are 12 D × 12 D matrices satisfying (5.A.21) and (5.A.22) then the D × D matrices σ0 , ... , σd introduced by (5.A.28) way also satisfy (5.A.21) and (5.A.22). Moreover if σ0 , ... , σd satisfy (5.A.23) if and only if α2 , ... , αd do with the same ς and with d replaced by d − 2. Theorem 5.A.2 can be easily proven by induction with respect to d with step 2. ¯ = 2 and We have obtained in particular that for d = 2 D       1 0 0 i 0 1 , σ1 = , σj = ς (5.A.29) σ0 = 0 −1 −i 0 1 0 modulo unitary equivalence; we have obtained the Pauli matrices; moreover, (5.A.30)

[σj , σk ] = 2iς ε¯jkl σl

where ε¯jkl is an absolutely skew-symmetric tensor with ε¯123 = 1 (we do not discuss the change of coordinates) and we write σ3 instead of σ0 here. ¯ = 4 and we have a method to We have also obtained that for d = 3, 4 D construct the corresponding sets. In particular, for d = 3 one can assume that     I 0 0 αj , σj = j = 1, 2, 3 (5.A.31) σ0 = 0 −I αj 0 where α3 = α0 , α1 , α2 are Pauli matrices. We do not prove the two following simple assertions here:

5.A. APPENDICES

519

Proposition 5.A.4. Let σ0 , ... , σd be Pauli matrices with d ≥ 2. Let K± = Ker(σ1 ± iσ2 ). Then (i) dim K± = 12 D, K+ and K− are orthogonal, and K+ ⊕ K− = H; (ii) σj K± = K± for j = 1, 2 and σj K± = K∓ for j = 1, 2. (iii) If (5.A.23) is fulfilled then σ0 · · · σd |K± = ±ςi

(5.A.32)

d−2 2

I.

Proposition 5.A.5. Let σ0 , ... , σd be Pauli matrices and (ω jk )j,k=0,...,d be a real orthogonal matrix. Then σj =



ω jk σk

j = 0, ... , d

0≤k≤d

are also Pauli matrices. Moreover, (5.A.23) for σ0 , ... , σd yields the same equality for σ0 , ... , σd with ς  = ς sgn det(ω jk ). Moreover the following assertion is useful: Proposition 5.A.6. Let σ0 , ... , σd be Pauli matrices. Let us introduce Hermitian matrices βj = 2i1 [σ2j−2 , σ2j−1 ] with j = 1, ... , r ≤ q = " d+1 #. Then 2 (i) βj2 = I ,

β1 , ... , βr commute.

(ii) For every set (ε1 , ... , εr ) ∈ {−1, 1}r (5.A.33)

dim



 Ker(βj − εj ) = 2−r D. 1≤j≤r

Proof. Statement (i) is obvious; Statement (ii) is easily proven by induction with respect to r with a constant d − 2r where the inductive construction of the proof of Theorem 5.A.2 is used. Propositions 5.A.5 and 5.A.5 yield

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Corollary 5.A.7. Let σ0 , ... , σd be Dirac matrices, (Fjk )j,k=0,...,d be a skewsymmetric real matrix and ±ifj , j = 1, ... , q be its non-zero eigenvalues (we take multiplicities into account), fj > 0. Then    Fjk [σj , σk ] = νε = εj fj , ε ∈ {−1, 1}r . (5.A.34) Spec j,k

1≤j≤r

Further, the multiplicity of each eigenvalue on the left coincides with the multiplicity of each number on the right multiplied by 2−r D, assuming that if νε and νε coincide then their multiplicities are combined.

Chapter 6 Operators in the Interior of Domain. Esoteric Theory This chapter is devoted to even more specialized results than the previous one. In Section 6.1 we extend some of the above results to conjoint spectral asymptotics for families of commuting operators (general or scalar). Then in Section 6.2 we consider scalar operators with periodic Hamiltonian flow of the principal symbol (or some perturbations of such operators) and under certain conditions we obtain spectral asymptotics with highly accurate remainder estimates; these asymptotics are not necessarily completely Weyl-type. In the case of the cluster character of the spectrum we get the asymptotic distribution of the eigenvalues inside the clusters. Our main tools here are very long term propagation of singularities (the main result is that singularities propagate along periodic trajectories which drift along the Hamiltonian field of some auxiliary symbol) and conjoint spectral asymptotics of two commuting operators (namely, the operator in question can be represented as the sum of an operator A0 with spectrum {2πhT0−1 n, n ∈ Z} and a perturbing operator μB commuting with A0 , where μ is a small parameter). Next, in Section 6.4 we consider asymptotics of  ω(x, y )|e(x, y , 0)|2 dxdy with ω(x, y ) singular at the diagonal x = y .

© Springer Nature Switzerland AG 2019 V. Ivrii, Microlocal Analysis, Sharp Spectral Asymptotics and Applications I, https://doi.org/10.1007/978-3-030-30557-4_6

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522

Finally, in Section 6.5 we consider mollification with respect to nonspectral parameters and establish that under corresponding microhyperbolicity condition it would lead to the same effect as mollification with respect to spectral parameter; we will need it much later. As usual we have few appendices.

6.1

Spectral Asymptotics for Families of Commuting Operators

In this section we generalize the results of the previous sections to the case of families of commuting operators.

6.1.1

Introduction

Let Aj = Aj (x, hD, h) (j = 1, ... , l) be h-pseudodifferential operators on a manifold X . We assume that (6.1.1) Either X = Rd or X is a compact closed CK manifold or X is a compact CK manifold with boundary ∂X ∈ CK with large enough K = K (d), d = dim X . In the third case Aj are differential operators and their original domains are given by some boundary value conditions. Moreover, let us assume that (6.1.2) The operators Aj are self-adjoint operators in L2 (X , H) where H is a Hermitian space1) and commute. Let us recall that Definition 6.1.1. Unbounded self-adjoint operators A1 , ... , Al commute if 2) (i) Their spectral projectors E1 (τ1 ), ... , El (τl ) commute; (ii) Their propagators U1 (t1 ), ... , Ul (tl ) commute; 1) 2)

Surely, one can consider operators acting in the Hermitian bundles. These conditions are equivalent.

6.1. FAMILIES OF COMMUTING OPERATORS

523

(iii) Their resolvents (z1 − A1 )−1 , ... , (zl − Al )−1 commute. Then  (6.1.3)

Aj = Rl

τj dτ E (τ )

where τ = (τ1 , · · · , τl ) ∈ Rl , E (τ ) = E1 (τ1 ) ... El (τl ) is the spectral projector of the family of commuting operators {A1 , ... , Al } and Ej (τj ) is the spectral projector of Aj . The standard question is how to get the semiclassical asymptotics of the number of conjoint eigenvalues τ lying in the (compact) domain Λ ⊂ Rl . Usually, however, the case of a domain with the smooth boundary is treated when Λ = {τ , φ(τ ) ≤ 0} with φ ∈ CK and |φ| + |∇φ| disjoint from 0. But results of this type are due to the standard theory applied to the operator A = φ(A1 , ... , Al ). The real problem arises when ∂Λ is not smooth. These settings are too general to get a meaningful answer. We assume that (6.1.4)

Λ = [λ1 , τ1 ] × [λ2 , τ2 ] × · · · × [λl , τl ]

with τj < τj .

Remark 6.1.2. Then we immediately will be able to generalize to the case when Λ = {τ : φ(τ ) ∈ Λ } where Λ rather than λ is defined by (6.1.4) with l replaced by m and φ : Rl → Rm ⊃ Λ is a smooth map (dimensions of Λ and Λ do not necessarily coincide. Really, if φ = (φ1 , ... , φm ) one can consider family (A1 , ... , Am ) = φ1 (A1 , ... , Al ), ... , φm (A1 , ... , Al ). However conditions we will use may be not preserved when we pass from family (A1 , ... , Al ) to (A1 , ... , Am ) unless m = l and φ is a diffeomorphism. Therefore we assume that (6.1.4) is fulfilled i.e. that Λ is the box. In this case one can consider the dimensions of this box as parameters as well. Thus we are going to treat the semiclassical asymptotics of Tr(E (τ  ; τ ))Q where     E (λ; τ ) = E1 (τ1 ) − E1 (λ1 ) · · · El (τl ) − El (λl ) and Q is an appropriate h-pseudodifferential operator. To give a more precise description of the operators in question we assume that

CHAPTER 6. ESOTERIC THEORY

524

(mj ) ˜ h,ρ,γ,h (6.1.5) Either Aj are h-differential operators of orders mj or Aj ∈ Ψ ˜ (m) where the classes of h-pseudodifferential operators Ψh,ρ,γ and Ψ h,ρ,γ were introduced in Subsection 6.4.2 for ρ, γ ∈ (0, 1], ργ ≥ h1−δ , δ > 0 is arbitrarily small.

To attack this problem let us consider the propagator of the family of commuting operators {A1 , ... , Al } (6.1.6) U(t) = exp ih−1 (t1 A1 + · · · + tl Al ) = exp(ih−1 t1 A1 ) · · · exp(ih−1 tl Al ) with t = (t1 , ... , tl ) ∈ Rl . Then

 e ih

U(t) =

(6.1.7)

−1 t,τ

Rl

dτ E (τ )

and its Schwartz kernel u(x, y , t) satisfies the “Cauchy” problem (6.1.8)

(hDtj − Aj )u = 0 (j = 1, ... , l),

u|t=0 = δ(x − y )I .

Since Aj commute we conclude that (6.1.9) The principal symbols of Aj commute as well: [aj , ak ] = 0.

6.1.2

General Theory

Elliptic Arguments Applying routine elliptic arguments one can easily prove the following Proposition 6.1.3. Let (A1 , ... , Al ) be a system of commuting operators of (6.1.5) type and let the ellipticity condition |aj (z)v | ≥ 2|v | ∀v ∈ H (6.1.10) 1≤j≤l

be fulfilled in the γ-neighborhood U of z¯ ∈ T ∗ Rd with γ ≥ h 2 −δ ,  ≥ γ 2 where aj are the principal symbols of Aj . Let Qk ∈ Ψh,γ,γ and their symbols be supported in U. Then 1

(6.1.11) |Ft→h−1 τ χT (t)Q1x u tQ2y | ≤ Chs T1 · · · Tl ∀τ : |τ | ≤ 

∀x, y ∈ Rd

provided Tj  ≥ h1−δ ∀j where χ ∈ C0K (Bl (0, 1)) is any fixed function, subscript l means dimension here, T = diag(T1 , ... , Tl ), χT (t) = χ(T −1 t).

6.1. FAMILIES OF COMMUTING OPERATORS

525

Successive Approximations As in Section 4.3 let us first construct Ft→h−1 τ χT (t) Tr(U(t)Q)

(6.1.12)

1

with χT (t) = χ1T1 (t1 ) · · · χlTl (tl ) and h1−δ ≤ T ≤ h 2 +δ by the successive approximation method. To do it we do not need to repeat the construction of that section. An easy induction on l with Proposition 4.3.7 at each step gives us the following Proposition 6.1.4. Let (A1 , ... , Al ) be a system of commuting operators of (6.1.5) type and let χj ∈ C0K [−1, 1] and 1

Tj ∈ [h1−δ , h 2 +δ ]

(6.1.13)

∀j.

Let Qk ∈ Ψh (Rd ) be operators with compactly supported symbols 3) . Then (6.1.14) ||Ft→h−1 τ χT (t)Q1x u(x, y , t) tQy −  −1 −d e ih x−y ,ξ χ(h ˆ −1 T (τ − τ  ))F(y , ξ, τ  , h) dξdτ  || ≤ Chs h T1 · · · T l where (6.1.15)

F(y , ξ, τ  , h) = ResR F (y , ξ, τ  , h)

with l-dimensional operation ResR and (6.1.16) F is the symbol of the operator (2π)−d Q1 (τ1 −A1 )−1 · · · (τl −Al )−1 Q2 constructed in the framework of the formal symbol calculus and s is arbitrarily large. Therefore (6.1.17)

F =



Fn (y , ξ, τ )

0≤n≤M

where M = M(s) and Fn are sums of terms of the sandwich type  −1  −1 (6.1.18) bj0 (y , ξ) τj1 − aj1 (y , ξ) bj1 (y , ξ) τj2 − aj2 (y , ξ) ···  −1 bjr −1 (y , ξ) τjr − ajr (y , ξ) bjr (y , ξ) 3)

If X = Rd we replace Qk by Qk ψk with ψ ∈ C0K (B(¯ x , 12 )) where B(¯ x , 1) ⊂ X .

CHAPTER 6. ESOTERIC THEORY

526

and b∗ are pseudodifferential symbols supported in supp(q1 ) ∪ supp(q2 ) as l + 1 ≤ r ≤ l + 2n for n ≥ 1. In particular, (6.1.19)

F0 = q10 (τ1 − a1 )−1 · · · (τl − al )−1 q20

and (6.1.20) F0 = (2π)−d q10 δ(τ1 − a1 ) · · · δ(τl − al )q20 = ∂τ1 · · · ∂τl (2π)−d q10 θ(τ1 − a1 ) · · · θ(τl − al )q20 where qk are symbols of Qk . Basic Results (General Case) Let us start from the really rough estimates: Theorem 6.1.5. Let (A1 , ... , Al ) be a system of commuting operators of (6.1.5) type and let Qk be operators with compactly supported symbols. Then (i) The following estimate holds: (6.1.21)

|(Q1x e tQ2y )(x, y , τ )| ≤ Ch−d .

(ii) In the framework of Proposition 6.1.3 (i.e. if system (A1 , ... , Al ) is elliptic on either supp(q1 ) or supp(q2 )) the following estimate holds: (6.1.22)

|(Q1x e tQ2y )(x, y , τ  , τ )| ≤ Chs

∀τ  , τ : |τ | ≤ ρ, |τ  | ≤ ρ.

Proof. Let + us recall that e(x, y , λ, τ ) is a Schwartz kernel of operator E (λ, τ ) = j (E (τj ) − E (λj )) (which is a projector as τ ≥ λ where τ ≥ λ means exactly that τj ≥ λj for j = 1, ... , l. Then estimates (6.1.21), (6.1.22) follow from the similar estimates for Γx (Q1x e tQ2y )(τ , λ) replaced by Ft→h−1 τ φT (t)Γx (Q1x u tQ2y ) (where the first esxtimate is obvious and the second one is due to Proposition 6.1.3) as we pick T  1 and l-dimensional H¨ormander function φ (which + can be taken as a product of l ordinary H¨ormander functions: φ(τ ) = j φ(τj )). We get rid of Γx and assumption that Q1 = Q2∗ following the same scheme as in Section 4.2.

6.1. FAMILIES OF COMMUTING OPERATORS

527

Theorem 6.1.6. Let (A1 , ... , Al ) be a system of commuting operators of (6.1.5) type and let Qk be operators with compactly supported symbols. Let ϕ ∈ C0K (R). Then estimate     −1  (6.1.23) ϕ(hL (¯ τ − τ )) dτ (Q1x e tQ2y )(x, x, τ )dx−   −d F(x, ξ, τ )dxdξdτ  ≤ Chs h holds provided L = diag(L1 , ... , Ll ) with h 2 −δ ≤ Lj ≤ h−n ∀j. 1

Proof. We just apply ϕ(L−1 (hDt − τ¯ )) to (6.1.14) (inside absolute value) and then take t = 0. The Microhyperbolicity Condition Let us replace our standard microhyperbolicity condition by the similar condition for a system (a1 , ... , al ): (or ξ-microDefinition 6.1.7. We call system (a1 , ... , al ) microhyperbolic  hyperbolic) in Ω if for each ζ ∈ Sl symbol aζ = 1≤j≤l ζj aj (x, ξ) is microhyperbolic (or ξ-microhyperbolic respectively) in Ω. Note that in this definition direction of (ξ-)microhyperbolicity of aζ may depend on ζ. Let us repeat arguments of Section 2.1.5. The routine propagation of singularities arguments (combined with elliptic arguments) yield Proposition 6.1.8. Let (A1 , ... , Al ) be a system of commuting operators of (6.1.5) type. (i) Let symbol aζ be ξ-microhyperbolic at the point z¯ as ζ = ζ¯ and let χ ∈ C0K be supported in the -vicinity of ζ¯ with sufficiently small  > 0. Then (6.1.24)

||Ft→h−1 τ χT (t)Γx (Q1x u tQ2y )|| ≤ Chs

∀x

∀τ : |τ | ≤ 

with any scalar T ∈ [1 , T0 ] and with Qj ∈ Ψh with symbols supported in the -neighborhood of z¯ where 1 > 0 is arbitrarily small, and T0 is a sufficiently small constant which does not depend on 1 or the choice of χ.

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528

(aj )j∈J be ξ-microhyperbolic at point (ii) Let J ⊂ {1,  ... , l} and let system  z¯. Let χ ∈ C0K Br (0, 1) \ Br (0, 12 ) where r = #J, T ∈ [1 , T0 ]. Further, let φ ∈ C0K Bl−r (0, 1) , T  ∈ [h1−δ , 0 T ]. Then (6.1.24) holds with χT (t) replaced by χT (t  )φT  (t  ), t  := (tj )j∈J , t := (tj )j ∈J / . 

Then we immediately arrive to Corollary 6.1.9. Let (A1 , ... , Al ) be a system of commuting operators of (6.1.5) type. Let J ⊂ {1, ... , l} and let system (aj )j∈J be ξ-microhyperbolic at point z¯. Let Qk be operators with symbols supported in vicinity of z¯. Then (i) Estimate (6.1.14) holds provided (6.1.25)

Tj ∈ [h1−δ , T0 ]

∀j ∈ J,

1

Tj ∈ [h1−δ , h 2 +δ ]

∀j ∈ / J.

Lj ∈ [h 2 −δ , h−n ]

∀j ∈ / J.

(ii) Estimate (6.1.23) holds provided (6.1.26)

Lj ∈ [h1−δ , h−n ]

1

∀j ∈ J,

Exploiting Microhyperbolicity Repeating rescaling arguments of Theorem 2.1.19 and remembering that in scaling t → T −1 t brings factor T l rather than T as it was then (when l = 1) we arrive to Proposition 6.1.10. In the framework of Proposition 6.1.8(ii)  h s T ∀τ : |τ | ≤  ∀T ∈ [h, T0 ].

(6.1.27) ||Ft→h−1 τ χT (t  )φT  (t  )Γx (Q1x u tQ2y )|| ≤ Chl−d ∀x

Proposition 6.1.11. Let (A1 , ... , Al ) be a system of commuting operators of (6.1.5) type. Let (a1 , ... , al ) be ξ-microhyperbolic at point z¯ and let Qk be operators with symbols supported in vicinity of z¯. Then (i) Estimate (6.1.28) ||Ft→h−1 τ φT (t)Γx (Q1x u tQ2y )|| ≤ Chl−d

  Tj  +1 T0 1≤j≤l

∀x holds as φ ∈

C0K (B(0, 1)).

∀τ : |τ | ≤  ∀Tj ≥ h

6.1. FAMILIES OF COMMUTING OPERATORS

529

(ii) the following estimate holds: (6.1.29) ||Γx (Q1x e tQ2y )(τ  , τ )|| ≤ Chl−d T0−l



1+

1≤j≤l

 T0 |τj − λj | h

∀τ  , τ : |τ  | ≤ 0 , |τ | ≤ 0 . Proof. (a) Statement (i) for Tj ∈ [h, T0 ] follows from Proposition 6.1.10 after we pick up an appropriate partition of unity with respect to τ . (b) Let us pick up T = T0 and multidimensional H¨ormander function φ. Then we derive immediately from Proposition 6.1.10 the following estimate ||Γx (e tQy )(τ  , τ )|| ≤ Chl−d T0−l

∀τ  , τ : |τ | ≤ 0 , |τ − λ| ≤ C

h T0

which immediately implies (6.1.29). We can get rid of Γx and assumption that Q1 = Q2∗ following the same scheme as in Section 4.2. (c) After (6.1.29) is proven we can get prove (6.1.28) in the full generality as it was done in Section 4.2. Remark 6.1.12. (i) Let us recall that T0  1 but in certain settings this could change. (ii) One can replace ξ-microhyperbolicity condition by microhyperbolicity condition and Γx by Γ. (iii) Finally, if we know only that (aj )j∈J is ξ-microhyperbolic where J ⊂ {1, ... , l}, #J = r then we conclude that (6.1.30) ||Ft→h−1 τ φT (t)Γx (Q1x u tQ2y || ≤ Chr −d

 T j j∈J

∀x

T0

  +1 · (Tj + 1) j ∈J /

∀τ : |τ | ≤  ∀Tj ≥ h

and (6.1.31) ||Γx (Q1x e tQ2y )(τ  , τ )|| ≤ Chr −d T0−r

 j∈J

1+

 T0 |τj − λj | h

∀τ  , τ : |τ  | ≤ 0 , |τ | ≤ 0 .

CHAPTER 6. ESOTERIC THEORY

530 Tauberian Arguments Let

Nx,y (λ, τ ) := Q1x e(x, y , λ, τ ) tQ2y

(6.1.32) and (6.1.33)

Nx,y ,ϕ1 ,...,ϕl ;L1 ,...,Ll (τ ) :=

 

ϕj (L−1 j (τj − λj ))dλ Nx,y (λ)

1≤j≤l

(where Nx,y (τ ) = Nx,y (−∞, τ ) etc). Now we want to approximate e(x, y , τ ) by    T −l (6.1.34) e (x, y , τ ) := h ¯ −1 t)u(x, y , t) dλ Ft→h−1 λ χ(T τ +R−,l

with matrix T = diag(T1 , ... , Tl ). Namely, (6.1.35) Let NTx,y (λ, τ ) and NTx,y ,ϕ1 ,...,ϕl ;L1 ,...,Ll be defined by (6.1.32) and (6.1.33) with e(x, y , τ ) replaced by e T (x, y , τ ). Let R∗ := N∗ − NT∗ denote the Tauberian error. Theorem 6.1.13. Let (A1 , ... , Al ) be a system of commuting operators of (6.1.5) type. (i) Assume that the following modification of condition (6.1.29) (6.1.36) ||(Q1x e tQ2y )(τ  , τ )|| ≤ Chl−d

1  1 + |τj − λj | Tj h 1≤j≤l ∀τ  , τ : |τ  | ≤ 0 , |τ | ≤ 0

holds where Tj ≥ h. Then (6.1.37)

RTx,y (λ, τ ) ≺ Ch−d

h Tk 1≤k≤l





|τj − λj | +

1≤j≤k:j =k

h Tj

and if ϕk satisfy (4.2.73)–(4.2.75) and Lj Tj ≥ h for all j (6.1.38)

Rx,y ,ϕ1 ,...,ϕl ;L1 ,...,Ll ≺ Ch−d L1 · · · Ll

1≤k≤l

 h  h ϑk Lk Tk Lk Tk

6.1. FAMILIES OF COMMUTING OPERATORS

531

(ii) Further, if (6.1.36) holds (only) after application of Γx or Γ then the same is true for (6.1.37) and (6.1.38). Note that (6.1.37) is a special case of (6.1.38) with ϕk which are characteristic functions of [0, 1]. Proof. Proof is trivial: we just apply l times Tauberian Lemma 4.2.13 with respect to variable τj with j running from 1 to l. Principal Results Now assuming that system (a1 , ... , al ) is ξ-microhyperbolic we can apply Theorem 6.1.13 with Tj  1. However due to ξ-microhyperbolicity with the negligible error we can replace in the Tauberian expressions NTx,x,∗ (as x = y ) Tj  1 by any Tj ≥ h1−δ thus arriving to Weyl expression   W −d (6.1.39) e (x, y , τ ) = (2π) F(x, y , λ)e ix−y ,ξ dλdξ. τ +R− l

Thus we proved Theorem 6.1.14. Let (A1 , ... , Al ) be a system of commuting operators of (6.1.5) type and let Q1 , Q2 be operators with symbols compactly supported in Ω. (i) Let system (a1 , ... , al ) be ξ-microhyperbolic in Ω. Then (6.1.40) RW x (τ , λ) := Nx (τ , λ)− (2πh)−d

 q10 (x, ξ)e(τ , λ, x, ξ)q20 (x, ξ) dξ ≺    |τj − λj | + h Ch1−d 1≤k≤l 1≤j≤l:j =k

where (6.1.41)

   e(τ , λ, x, ξ) = θ(τ1 − a1 (x, ξ) · · · (τl − al (x, ξ)

and skipped subscript y as usual means that y = x.

CHAPTER 6. ESOTERIC THEORY

532

Further, assume that ϕk satisfy (4.2.73)–(4.2.75) and Lk ≥ h; then h h −d ≺ Ch L · · · L ϑk (6.1.42) RW 1 l x,ϕ1 ,L1 ,...,ϕl ,Ll Lk Lk 1≤k≤l where the left-hand expression is the difference between Nx,ϕ1 ,L1 ,...,ϕl ,Ll and W NW x,ϕ1 ,L1 ,...,ϕl ,Ll obtained by plugging e (x, y , τ ) instead of e(x, y , τ ) in the definition of Nx,ϕ1 ,L1 ,...,ϕl ,Ll . (ii) Let system (a1 , ... , al ) be microhyperbolic in Ω. Then all above estimates hold after integration with respect to x. In particular (6.1.43) RW (τ , λ) := N(τ , λ)− −d

(2πh)

 q10 (x, ξ)e(τ , λ, x, ξ)q20 (x, ξ) dxdξ ≺    |τj − λj | + h . Ch1−d 1≤k≤l 1≤j≤k:j =k

Possible Generalizations, Remarks One can easily generalize Theorem 6.1.14 to the case when only subsystem is (ξ-)microhyperbolic: Theorem 6.1.15. Let (A1 , ... , Al ) be a system of commuting operators of (6.1.5) type and let Q1 , Q2 be operators with symbols compactly supported in 1 Ω. Let ϕj ∈ C0K , Lj ≥ h 2 −δ for all j ∈ / J where J ⊂ {1, ... , l}. (i) Let system (aj )j∈J be ξ-microhyperbolic in Ω. Then     := (6.1.44) RW (τ , λ ) ϕjLj (τj ) dτ  RW x,(ϕj ,Lj )j ∈J x (τ , λ) ≺ / j ∈J /

Ch1−d



 

 |τj − λj | + h + Chs

k∈J j∈J:j =k

with arbitrarily large exponent s. Further, assume that ϕk satisfy (4.2.73)–(4.2.75) and Lk ≥ h for all k ∈ J; then  h h −d + Chs . Lj ϑk (6.1.45) RW x,ϕ1 ,L1 ,...,ϕl ,Ll ≺ Ch L L k k k∈J k∈J

6.1. FAMILIES OF COMMUTING OPERATORS

533

(ii) Let system (aj )j∈J be microhyperbolic in Ω. Then all above estimates hold after integration with respect to x. In particular,     := (6.1.46) RW (τ , λ ) ϕjLj (τj ) dτ  RW (τ , λ) ≺ (ϕj ,Lj )j ∈J / j ∈J /

Ch1−d



 

 |τj − λj | + h + Chs .

k∈J j∈J:j =k

Problem 6.1.16. Get sharp remainder estimates (namely with right-hand expressions (6.1.37) or (6.1.38) with Tj ≥ 1. To do this one needs to introduce multitime generalized Hamiltonian flow exactly as we introduced in Section 2.2 generalized Hamiltonian flow and then to define loop or periodic points for such flow. However since a1 , ... , al commute one can introduce such flow as a product of generalized Hamiltonian flows: (6.1.47)

Ψt,a1 ,...,al = Ψt1 ,a1 ◦ Ψt2 ,a2 ◦ · · · ◦ Ψtl ,al

(but one needs to show that these flows commute). We will do it only in the case of scalar symbols a1 , ... , al in the next subsection. The general construction is left to the reader and we believe that full exposition deserves to be published. Remark 6.1.17. We do not need large smoothness; in particular we can assume that operators A1 , ... , Al satisfy regularity conditions of Section 4.6 (but then we need ξ-microhyperbolicity). However our main target are operators with irregular coefficients rather than rough operators. So if we have commuting irregular operators A1 , ... , Al one should be able to approximate them by rough operators A1ε , ... , Alε in such way that these operators also commute. While in some special cases it is possible but there is no general theory and it even does not seem to be possible. Remark 6.1.18. Let a1 , ... , am , m ≤ l be scalar symbols. Then (a1 , ... , al ) is microhyperbolic if and only if for all J ⊂ {1, ... , m} (a) (daj )j∈J are linearly independent at (aj )j∈J = 0 and therefore ΣJ := {(aj )j∈J = 0} is a smooth manifold in T ∗ X and (b) as m < l (am+1 , ... , al ) restricted to Σm is microhyperbolic. One can describe ξ-microhyperbolicity as microhyperbolicity for fixed x and therefore the above description works (with fixed x) for ξ-microhyperbolicity.

CHAPTER 6. ESOTERIC THEORY

534

6.1.3

Operators with Scalar Symbols

Preliminary Remarks Let us assume now that (6.1.48) All the symbols a1 , ... , al are scalar. Then since (A1 , ... , Al ) is a system of commuting operators {aj , ak } = 0

(6.1.49)

∀j, k = 1, ... , l

and assuming that at z¯ a subsystem (a1 , ... , ar ) is microhyperbolic and no its subsystem is elliptic one can introduce symplectic coordinates in the vicinity of z¯ such that aj = ξj for j = 1, ... , m there. Furthermore one can prove easily Proposition 6.1.19. Let (A1 , ... , Al ) be a system of commuting operators. Assuming that at z¯ a subsystem (a1 , ... , am ) is microhyperbolic and no its subsystem is elliptic one can find a h-Fourier integral operator F such that FF ∗ ≡ I in vicinity of z¯ and (6.1.50) F ∗ F ≡ I ,

F ∗ Aj F ≡ hDj

∀j = 1, ... , m

in vicinity of 0.

Furthermore (6.1.51)

F ∗ Aj F ≡ Aj (x  , hD, h)

∀j = m + 1, ... , l

in vicinity of 0

with (x  ; x  ) = (x1 , ... , xm ; xm+1 , ... , xl ) etc. Asymptotics with Mollifications Our first task is to replace condition Lj ≥ h 2 −δ by the condition Lj ≥ h1−δ in Theorems 6.1.6 and 6.1.15. 1

Theorem 6.1.20. Theorems 6.1.6 (with mollification with respect to x) and 6.1.15(ii) hold under extra condition (6.1.48) if we relax conditions to Lj to Lj ≥ h1−δ for all j = 1, ... , l (in Theorem 6.1.6) and or all j = m + 1, ... , l (in Theorem 6.1.15) respectively.

6.1. FAMILIES OF COMMUTING OPERATORS

535

Proof. Let us apply induction with respect to (r , m) where r is the number of j = 1, ... , l such that Lj ≤ h1−δ . Note that as r = 0 condition Lj  1 is fulfilled for all j and we can apply Theorems 6.1.6 and 6.1.15; meanwhile as m = l system (a1 , ... , al ) is microhyperbolic and we can apply Theorem 6.1.15. Due to Proposition 6.1.19 one can assume without any loss of the generality that z¯ = 0 and in its vicinity Aj = hDj for j = 1, ... , m and Aj = Aj (x  , hD  , hD  ) for j = m + 1, ... , l. Let us introduce the scaling function (6.1.52)

γ=



(|bj | + |∇bj |2 ) + |ξ  |

 12

+ γ¯

m+1≤j≤l

with bj = aj (x  , 0, ξ  ) and (6.1.53)

1

|L| = max Lj

γ¯ = |L| 2 ,

1≤j≤l



and a small constant  > 0. Let us recall that |L| ≥ h 2 −δ unless m = l. Then |∇γ| ≤ 12 . Let us make a (, γ, γ, γ 2 )-admissible partition with respect to (x  , x  , ξ  , ξ  ) and consider the contribution of one element of this  partition. Obviously, we need to rescale x  → xnew = x  γ −1 and introduce   hnew = hγ −2 . This yields the rescaling ξ  → ξnew = ξ  γ −2 , ξ  → ξnew = ξ  γ −1 . 2 Moreover, let us divide the operators Aj by γ for all j = 1, ... , l; so, Lj → Lj,new = Lj γ −2 and h/Lj are preserved. First, let us consider elements of the partition with |ξ  | ≥ 0 γ 2 and γ ≥ C1 γ¯ . Then system is elliptic near 0 and Theorem 6.1.5(ii) yields that the contribution of such elements to the principal part is 0 and the contribution to the remainder is O(hs ) provided C1 is large enough. Now, if |L| ≥ 0 γ 2 then after this rescaling procedure we are in the case (r − 1, m) 4) and due to induction assumption we obtain that the contribution of such an element does not exceed the rescaled right-hand expression of estimate (6.1.45) which is 1

(6.1.54)

 C γ 2d−2m h−d

 k=1,...,m

4)

Lj

 h h  ϑk + hs Lk Lk k=1,...,m

Really, Ljnew  1 for at least one number j.



CHAPTER 6. ESOTERIC THEORY

536

where s  is another but still arbitrarily large exponent. On the remaining elements with γ ≥ C γ¯ microhyperbolicity condition is fulfilled for a subsystem with (m+1) elements after rescaling and permutation of Aj with j = m + 1, ... , l. Therefore we can also apply the induction hypothesis. We obtain that the contribution of such element to the remainder also does not exceed the rescaled right-hand expression of estimate (6.1.45) which is (6.1.54). Now let us restrict γ to Λ := {x  = ξ  = 0} and consider contribution of B(z  , γ(z  )) × R2m ∩ Ω where z  = (x  , ξ  ); due to the previous arguments it does not exceed (6.1.54) and therefore the total contribution of such balls does not exceed (6.1.54) multiplied by γ −2d+2m (since dim Λ = 2d − 2m) and integrated over Λ; this is exactly (6.1.45). Principal Results Now we can get rid of the mollification. Theorem 6.1.21. Let (A1 , ... , Al ) be a system of commuting operators of (6.1.5) type and condition (6.1.48) be fulfilled. Moreover, let (a1 , ... , al−1 ) be microhyperbolic in the vicinity of z¯ (in which case Σ := {a1 = ... = al−1 = 0} is a smooth manifold). Assume that ϕj satisfy (4.2.73)–(4.2.75) and Lj ≥ h. (i) Let b = al |Σ satisfy condition (5.2.20)r (6.1.55)r |b(x  , ξ  ) − τl | + |∇x  ,ξ b(x  , ξ  )| ≤ 0 =⇒ Hess b(x  , ξ  ) has r eigenvalues (counting multiplicities) f1 , ... , fr such that |f1 | ≥ 0 , ... , |fr | ≥ 0 5) with r ≥ 3 or (6.1.55)+ |b(x  , ξ  ) − τl | + |∇x  ,ξ b(x  , ξ  )| ≤ 0 =⇒ Hess b(x  , ξ  ) has 2 2 eigenvalues (counting multiplicities) f1 , f2 of the same sign such that |f1 | ≥ 0 , |f2 | ≥ 0 5) . Then estimates (6.1.43) and (6.1.56)

−d RW L1 · · · L l ϕ1 ,L1 ,...,ϕl ,Ll ≺ Ch

hold. 5)

The other eigenvalues are arbitrary.

h h ϑk L Lk k 1≤k≤l

6.1. FAMILIES OF COMMUTING OPERATORS

537

(ii) Let b satisfy condition (6.1.55)2 . Then estimates    RW ≺ Ch1−d |τj − λj | + h | log h| (6.1.57) 1≤k≤l 1≤j≤k:j =k

and (6.1.58)

−d RW L1 · · · L l ϕ1 ,L1 ,...,ϕl ,Ll ≺ Ch

h h ϑk | log h| Lk Lk 1≤k≤l

hold. (iii) Let b satisfy condition (6.1.55)1 . Then estimates    1 RW ≺ Ch 2 −d |τj − λj | + h (6.1.59) 1≤k≤l 1≤j≤k:j =k

and (6.1.60)

− 2 −d RW L1 · · · L l ϕ1 ,L1 ,...,ϕl ,Ll ≺ Ch 1

h h ϑk Lk Lk 1≤k≤l

hold. 1

Proof. Let us introduce scaling function γ(z) by (6.1.52) with γ¯ = h 2 . Let us rescale as in the proof of Theorem 6.1.20. Then the right-hand expressions of (6.1.43) and (6.1.57) scale to the same expressions but with an extra factor γ 2d−2l and therefore the contribution of B(z  , γ(z  )) × R2m ∩ Ω to the remainder does not exceed γ 2d−2l R where R denotes the original expression, z  ∈ Σ, dim Σ = 2d − 2l + 2 and we again restricted γ to Σ.  Then the total contribution to the remainder does not exceed γ −2 dz  × R and we arrive to all estimates; under condition (6.1.55)+ 2 we introduce coordinates z  = (z1 , z2 ; z  ) on Σ such that γ(z  )  |z1 | + |z2 | and |z1 | + |z2 | has a constant magnitude as z  is constant and ellipticity is violated (as we did in the proof of Theorem 5.2.2). Problem 6.1.22. (i) It would be very interesting to carry out the analysis of Section 5.2 but it requires largetl which is impossible due to terms with |α| = 1 in the decomposition b ∼ bα (x  , ξ  )ξ α . Really, these terms mean the propagation with respect to x  with the speed bα (x, ξ  ). On the other hand, if bα = const we would be able to replace Al by Al := Al − α bα Aα where α ∈ Z+ l−1 and Aα := Aα1 1 · Al−1 Aαl−1 and analysis of this new system (A1 , ... , Al−1 , Al ) would lead us to the desired results. However bα are variable and this is a difficulty we could not tackle.

CHAPTER 6. ESOTERIC THEORY

538

(ii) It would be even more challenging to consider few degenerating symbols a1 , ... , al . We strongly believe that solution of any of these two problems deserves to be published. Example 6.1.23. We leave to the reader the hosts of examples related to spherically symmetric operators: e. g. A1 = h2 Δ on Sd (d ≥ 3) or A1 = h2 Δ + V (|x|) in Rd while A2 = x2 hD1 − x1 hD2 is a component of the angular momentum.

6.2

Scalar Operators with Periodic Hamiltonian Flow

In this section we treat scalar operators with the periodic Hamiltonian flow and weak perturbations of such operators. We consider very long-time propagation of the singularities. Then under some restrictions we prove a microlocal version of a theorem concerning spectral gaps (i.e. lacunas in the spectrum), and under other restrictions we obtain highly accurate semiclassical spectral asymptotics which may contain a non-Weyl term. Thus periodic trajectories which were so far the worst enemy of sharp spectral asymptotics are tamed and become their best friend.

6.2.1

Introduction

Let us assume that (6.2.1) A ∈ Ψh (X , K) is a self-adjoint operator with principal symbol a(x, ξ). Assume for a while that a is a scalar symbol; later we abandon this condition but then we will return to it. Let τ¯ be a fixed energy level (usually for a sake of simplicity τ¯ = 0), and let Ω be a connected domain in T ∗ Rd . Let us assume that (6.2.2) a is microhyperbolic in Ω on the energy level τ¯ which for scalar symbol means that (6.2.3)

|∇(x,ξ) a(x, ξ)| + |a(x, ξ) − τ¯| ≥ 0

in Ω,

6.2. OPERATORS WITH PERIODIC HAMILTONIAN FLOW

539

and that all Hamiltonian trajectories are periodic in Ω i.e. (6.2.4)

Ψa,T (x, ξ) = (x, ξ) ∀(x, ξ) ∈ Ω

for T = T (x, ξ)

where Ψa,t is a Hamiltonian flow generated by a. Then one can replace Ω by 



Ψa,t (x, ξ)

(x,ξ)∈Ω t∈[0,T (x,ξ)]

and hence without any loss of the generality we can assume that Ψa,t (Ω) = Ω

(6.2.5)

∀t.

It is well known that (6.2.4) then holds with T (x, ξ) = T (a(x, ξ)) (i.e., the period depends only on the energy level) with a smooth function T (τ ) as τ ∈ [τ1 , τ2 ] provided this interval is contained in the set of values of a(x, ξ) at Ω 6) . Remark 6.2.1. (i) There can be subperiodic points (x, ξ) with T (x, ξ) = 1 T (a(x, ξ)), m ∈ Z+ but condition (6.2.2) forbids stationary points of Ψa,t . m Further, the set of subperiodic points is always a symplectic submanifold (see Proposition 6.3.1). (ii) This conclusion would be wrong if we assume only that all the trajectories on the given energy level τ¯ are periodic. Really, let τ¯ = 0; then replacing 6) More precisely, if we can select T = T (z) as C1 function then T (z) is constant on each connected componentof {a(z) = τ } provided ∇a(z) is disjoint from 0 here. Indeed, let us consider dH(z) ∧ dt(z) over

Σ = {z = Φt (x(s)), 0 ≤ s ≤ 1, 0 ≤ t ≤ t(x(s))} where x(s) (0 ≤ s ≤ 1) is a C1 -curve {a(z) = E }. Then with an arbitrary constant c  0=

 da(z) ∧ dT (z) =

  (a + c)dT |s=1 s=0 = (E + c) T (x(1)) − T (x(0)) =⇒ T (x(1)) = T (x(0)).

Obviously, if da = 0 and T (z) is bounded in the neighbourhood of some point z¯, then one can select T such that ΦT (z) (z) = z; however at some points T (z) could be the multiple of the shortest period.

CHAPTER 6. ESOTERIC THEORY

540

a(x, ξ) by a (x, ξ) = b(x, ξ)a(x, ξ) with b = 0 we do not change trajectories on {a(x, ξ) = 0}, so they remain periodic but their periods are now 



T (x,ξ)

T (x, ξ) =

b −1 (Ψa,t (x, ξ)) dt

0

which is not necessarily constant on Στ . Further, it is well known that the quantity (6.2.6)

1 1 κ1 = T T



T

(−ξdx + adt) 0

(where the integral is taken along the Hamiltonian trajectory (x(t), ξ(t)) = Ψa,t (x, ξ)) is a constant. Here κ1 is a (classical) action. Furthermore, we assume that (6.2.7)

T (τ ) = T0 ,

i.e. Ψa,T0 = I ;

in Ω

then κ1 will also is a constant. Remark 6.2.2. Starting from (6.2.4) one can always reach (6.2.7) with T0 = 1 replacing A by f (A) with  τ T (τ  ) dτ  . (6.2.8) f (τ ) = However we will formulate main theorems with general T0 . So, we assumed that the classical dynamics is periodic (with period T0 ). −1 What can we say about quantum dynamics? We know that e ih tA is an h-Fourier integral operator (uniformly with respect to t : |t| ≤ c). The corresponding symplectomorphism is Ψa,−t which is an identity as t = T0 due to (6.2.7). Therefore we conclude that it is actually h-pseudodifferential operator (at least in Ω). More precisely Theorem 6.2.3. (i) Let conditions (6.2.1), (6.2.2) and (6.2.7) be fulfilled. Then (6.2.9)

e ih

−1 T

0A

≡ e ih

−1 κ

G

in Ω

6.2. OPERATORS WITH PERIODIC HAMILTONIAN FLOW

541

where G ∈ Ψh (X , K) is an operator with the principal symbol V (x, ξ, T0 ) where V (x, ξ, t) ∈ S(X , K) is a solution to Cauchy problem d V (z, t) = ias ◦ Ψa,−t (z)V (z), dt

(6.2.10)

V (z, 0) = I ,

as is subprincipal symbol of A and κ = κ1 − 12 πκ0 h is the Maslov constant, κ0 is the Maslov index of the periodic trajectory Ψt,a (x, ξ) (which is also a constant and κ1 is an action. (ii) In particular, if as is also a scalar symbol e ih

(6.2.11)

−1 T

0A

≡ e ih

−1 κ+iB

where B is an operator with the principal symbol  T0 b := (6.2.12) as ◦ Ψa,−t (z)dt. 0

Proof. Proof easily follows from the construction of Subsection 1.2.2. Also one can find the proof at Y. Colin de Verdiere [2]. Now for the purpose of introduction assume that Ω = T ∗ X . First of all, assume that G = e iκ2 and (6.2.9) holds precisely rather than modulo negligible operator (6.2.13)

e ih

−1 T

0A



= e iκ ,

1 κ = h−1 κ1 + κ2 − πκ0 , 2

κ ∈ R;

then we conclude that Spec(h−1 T0 A) ⊂ 2πZ + κ and equivalently (6.2.14)

Spec(A) ⊂

2πh h Z + κ ; T0 T0

let us recall that T0 is a period. This ratio hT0−1 plays important role in all statements. In this case we get pure point spectrum; if X is not compact manifold it could be of the infinite multiplicity. So, there are spectral gaps of the length 2πhT0−1 . Further, the results of Chapter 4 imply that if X is a compact manifold and then for “typical” n ∈ Z such that (6.2.15)

τn =

2πh h n + κ T0 T0

CHAPTER 6. ESOTERIC THEORY

542

belongs to [−0 , 0 ] multiplicity of τn as an eigenvalue of A is  h1−d (provided a is not elliptic). This in turn implies that we cannot reasonably expect asymptotics with the remainder estimate better than O(h1−d ), however we can reformulate the problem in this case in the opposite way: Calculate precisely multiplicity Nn of τn . This problem we will be able to solve. If T0 is not a minimal period then replacing T0 by mT0 we conclude that for n ∈ Z \ mZ formula (6.2.15) gives us fake possible eigenvalues. So it is crucial to assume during this section that (6.2.16) T0 is a minimal period of the Hamiltonian flow Ψa,t ; then we will prove that every τn given by (6.2.15) is actually an eigenvalue and its multiplicity is actually  h1−d . It would not preclude from subperiodic points but their contribution to τn would be of magnitude h1−d+2r where 2r is a codimension of the manifold of subperiodic points. Their effect would be basically to shift some eigenvalues from τn with n ∈ / mZ to τn with n ∈ mZ. More generally, instead of (6.2.13) let us assume that (6.2.11) holds with the bounded Hermitean operator B; then instead of (6.2.14) we conclude that (6.2.17)

Spec(A) ⊂

2πh h h Z + κ + Spec(B). T0 T0 T0

in particular if Spec B ⊂ [−ηπ, ηπ] with η < 1 we conclude that Spec(A) is consisting in spectral clusters contained in η-vicinities of τn defined by (6.2.15). So, a single eigenvalue τn of large multiplicity Nn would be split into eigenvalues of the total multiplicity of Nn and our calculation of Nn will be still useful. However the new and more challenging question arrises: How eigenvalues are distributed inside the cluster? which boils down to asymptotics of N(τ , h) again; under certain assumptions we will be able to give an answer with the remainder estimate O(h2−d ). In this case η may be a constant or a small parameter (hl ≤ η < 1), in the latter case the spectral clusters will be narrow. The task to find sharp asymptotics of N(τ , h) is not restricted to the case of η < 1 even if otherwise the spectral gaps disappear and clusters overlap.

6.2. OPERATORS WITH PERIODIC HAMILTONIAN FLOW

543

We will add some twist replacing A by A + εA with ε  1 we will arrive to (6.2.17) with B replaced by ε h−1 B where typically ε = ε. Note that ε h−1 could be a large parameter. In this case under certain assumptions we will be able to give an answer with the remainder estimate O(εh1−d ). Let us describe the method. Let us replace (6.2.11) by e ih

(6.2.18)

−1 T

0A

= e iεh

−1 B

where (6.2.19) A and B commute. We reached κ = 0 by replacing A by A − hT0−1 κ and included εh−1 in B explicitly. We will justify these assumptions later (modulo negligible). Then e ih

−1 nT

0A

= e inh

−1 t  A

−1 nT

−1 εB

n ∈ Z, |n| ≤ h−l



and therefore (6.2.20) e ih

−1 tA

= e ih

e ih

0A

= e ih

−1 t  A

e ih

−1 εnB

−1





= e ih (t A+t B) as t = nT0 + t  , t  = nε.

Here the right-hand expression is h-Fourier integral operator uniformly with respect to |t  | ≤ c, |t  | ≤ c; picking up n = "tT0−1 # we rewrite this assumption as (6.2.21)

|t| ≤ T := cε−1 .

Therefore instead of one-parameter long time evolution we get two-parameter normal time evolution described by Hamiltonian flow Ψa,b,t  ,t  := Ψa,t  Ψb,t  provided B has a scalar principal symbol b; in the general case Ψb,t  would be generalized Hamiltonian flow. Instead of one time t we have a normal time t  and a slow time t  . Our classical dynamics contains periodic movement and a drift controlled by b. So, if the drift breaks periodicity instantly we get Weyl asymptotics with the remainder estimate CT −1 h1−d  C εh1−d where T is defined by (6.2.21) with c replaced by 0 . Periodicity is broken instantly in the classical dynamics provided (a, b) satisfies microhyperbolicity condition (see Definition 6.1.7). We will deal with a more degenerate scenario of Subsection 6.1.3 later.

544

CHAPTER 6. ESOTERIC THEORY

However, situation is more complicated for a quantum dynamics: the drift for time t  shifts trajectory by  |t  | and therefore the shift is observable only as |t  | ≥ Ch| log h| (due to logarithmic uncertainty principle) or as |n|ε ≥ Ch| log h|.

Figure 6.1: Periodicity classical and quantum: the shift between two consecutive windings may be quantum observable only after many windings. So, the shift is observable instantly (i.e. after the first period) only as ε ≥ Ch| log h| (or ε ≥ C | log h|) and our conclusion stays only then. Otherwise we need to wait time T  := Chε−1 | log h| and then (6.2.22)

||Ft→h−1 τ χ¯T (t)Γ(u tQ)| ≤ CT  h1−d = C ε−1 h2−d | log h|.

In fact, a more accurate analysis allows us to prove a better estimate (6.2.23)

||Ft→h−1 τ χ¯T (t)Γ(u tQ)| ≤ C (ε−1 + h−1 )h2−d

valid for ε ≤ 1; so as ε ≤ h effectively T   ε−1 h which corresponds to the standard uncertainty principle. Then (6.2.23) leads to the Tauberian estimate (6.2.24)

RT ≺ CT −1 (ε−1 + h−1 )h2−d  C (ε + h)h1−d .

Remark 6.2.4. (i) We can apply logarithmic uncertainty principle only under analyticity assumption; in the general case one needs to replace | log h| by h−δ but we are getting rid of both factors anyway.

6.2. OPERATORS WITH PERIODIC HAMILTONIAN FLOW

545

(ii) However instant transition from Tauberian asymptotics to the Weyl one is possible only as ε ≥ Ch| log h| (or ε ≥ h1−δ ); in the general case one needs to calculate the correction term which is due to contribution of the windings with 1 ≤ |n| ≤ ε−1 T  .

6.2.2

Operator B

Let us analyze the crucial assumption (6.2.18)–(6.2.19). In fact, it is the only assumption we made so far which we really need. We do not need even assumptions that a is scalar and Hamiltonian flow periodic as long as (6.2.18)–(6.2.19) hold. However we need to discover how realistic this assumption is. So, let us assume that a is a scalar symbol and it generates the Hamiltonian flow which is 1-periodic in Ω. Scalar Case Let as be a scalar symbol as well. Then (6.2.11) holds (at least modulo O(h)) with operator B with the scalar principal symbol b defined in Ω by (6.2.12); this symbol obviously is Ψa,t -invariant and therefore {a, b} = 0 there. We can always extend it to the smooth real-valued symbol. Consider B = Opw (b); this operator commutes with A modulo O(h) and (6.2.11) holds modulo O(h). However we can fix it due to the following procedure described in more general settings. Assume first that (6.2.18) holds. Let R ∈ Ψh (X , K). Consider averaging of R over period equal 1:  1 −1  −1  ¯ (6.2.25) R := e −ih t A Re ih t A dt  . 0 −1

−1

Consider R¯t := e −ih tA Re ih tA ; it is also given by (6.2.25) but with an integral taken from t to 1 + t and thus (6.2.26) R¯t − R¯ =  t  −1  −1  −1 −1 e −ih (t +1)A Re ih (t +1)A − e −ih tA Re ih tA dt  = 0  t  −1   −1  −1 −1 e −ih t A e −ih A Re ih A − R e ih t A dt  0

CHAPTER 6. ESOTERIC THEORY

546

and the latter expression vanishes as R = B. So, B¯ commutes with A modulo an operator negligible in Ω. ¯ Due to Theorem 6.2.3 their principal symbols Let us compare B and B. coincide and B − B¯ ∈ hΨh (X , K). Then in Ω e ih

−1 A

¯ ¯ ¯ ¯ ¯ + R, ≡ e iB ≡ e i B+i(B−B) ≡ e i B + ie i B (B − B)

R ∈ h2 Ψh (X , K).

−1 Since e ih A and B¯ commute with A, after averaging according to (6.2.26) we arrive to

(6.2.27)

e ih

−1 A

¯

¯ ≡ e i B + R,

R¯ ∈ hk Ψh (X , K)

with k = 2. Let us assume that (6.2.27) holds with some B¯ and R¯ which commute with A (modulo negligible in Ω). Then replacing B¯ by B¯  := ¯ B¯ + e −i B R¯ we get the same (6.2.27) but with k replaced by k + 1. Here again all operators commute with A. Iterating and adjusting B¯ we can achieve (6.2.18)–(6.2.19). Therefore we arrive to Proposition 6.2.5. Let conditions (6.2.1)–(6.2.4) and (6.2.7) be fulfilled and let as be scalar symbol. Then one can find operator B ∈ Ψh (X , K) with the scalar principal symbol b defined by (6.2.12) and satisfying (6.2.18)– (6.2.19). Then we conclude that Corollary 6.2.6. In the framework of Proposition 6.2.5 operator A0 := A − hB commutes (modulo negligible) with A and B and (6.2.28)

e ih

−1 T

0 A0

≡I

modulo an operator which is negligible in Ω. So A could be considered as perturbation of A0 by εB with ε = h. Let us forget about (6.2.18)–(6.2.19), consider scalar unperturbed operator A0 ∈ Ψh (X ) satisfying (6.2.28) and perturb it by operator εP where P ∈ Ψh (X , K) and it does not necessarily commute with A0 anymore.

6.2. OPERATORS WITH PERIODIC HAMILTONIAN FLOW

547

Proposition 6.2.7. Let conditions (6.2.1)–(6.2.4) and (6.2.7) be fulfilled and let A0 ∈ Ψh (X ) be an operator with the principal symbol a satisfying (6.2.28). Let P ∈ Ψh (X ) and (6.2.29)

ε ∈ [hl , hδ ].

A(ε) = A0 + εP,

Then there exists an operator

B(ε) ∼

(6.2.30)

B k εk

k≥0

commuting with A(ε) and satisfying e ih

(6.2.31)

−1 T

≡ e ih

0 A(ε)

−1 εB(ε)

in Ω. Furthermore the principal symbol of B0 is defined by  T0  b ◦ Ψa,−t (z) dt. (6.2.32) b0 = (x, ξ) = 0 −1

−1

Proof. Let U0 (t) = e ith A0 and U(t) = e ith A ; so in fact objects we study depend on ε. Let us consider operator V (t) = U(t)U0 (−t); then (6.2.33) with (6.2.34)

Dt V (t) = h−1 U(t)PU0 (−t) = εh−1 V (t)L(t),

V (0) = I

L(t) := U0 (t)PU0 (−t).

Our purpose is to construct V (t). Let us construct a self-adjoint operator R(t) ∈ Ψh (X ) such that   −1 t  V (t) ≡ exp iεh R(t  ) dt  (6.2.35) in Ω. 0

To construct R(t) let us consider (6.2.36)

 W (σ, t) := exp iεh−1 σ



t

 R(t  ) dt  .

0

Then ∂σ W − iεh

−1



t 0

R(t  )dt  · W = 0

CHAPTER 6. ESOTERIC THEORY

548

and differentiating with respect to t we obtain that  t   −1 R(t  ) dt  (∂t W ) = iεh−1 R(t)W . ∂σ − iεh 0

Then ∂t W (σ, t) = iεh

−1



σ

W (σ − σ  , t)R(t)W (σ  , t) dσ  =

0

iεh−1 R  (σ, t)W (σ, t) due to definition (6.2.36) and equality ∂t W = 0 as σ = 0; here we define  σ W (σ  , t)R(t)W (−σ  , t) dσ  . R  (σ, t) = 0

Setting σ = 1 we obtain that ∂t W1 (t) = iεh−1 R  (1, t)W1 (t)

(6.2.37)

for W1 (t) = W (1, t) and 



1

R (1, t) =

(6.2.38)

W (σ, t)R(t)W (−σ, t)dσ. 0

Comparing (6.2.35) and (6.2.37)–(6.2.38) and recalling that V (0) = W1 (0) = I we conclude that (6.2.35) is fulfilled provided R  (1, t) ≡ L(t) ∀t ∈ [−T0 , T0 ] ¯ Thus we need to modulo negligible operators in a fixed neighborhood of Ω. construct R(t) such that  1 (6.2.39) R  (t) := W (σ, t)R(t)W (−σ, t) dσ ≡ L(t) in Ω 0

∀t ∈ [−T0 , T0 ] where we denote this neighborhood by Ω again. Theorem 2.4.10 yields that R  and L belong to Ψh (X ) provided R and P belong to this class; moreover R  (t) ≡ R(t) mod εΨh (X ). Further, (6.2.39) is equivalent to (6.2.40)



m  −1 1 iεh Ad t R(t) dt R(t) ≡ L(t) 0 (m + 1)! m≥0

in Ω ∀t ∈ [−T0 , T0 ].

6.2. OPERATORS WITH PERIODIC HAMILTONIAN FLOW

549

Really, the left-hand expressions of (6.2.39) and (6.2.40) coincide as ε ≤ hδ . This allows to fulfill (6.2.40) and in the end of the day (6.2.35) recurrently by Rn (t)εn−1 . (6.2.41) R(t, ε) = n≥1

Namely, as R1 (t), ... , Rn−1 (t) are defined we can calculate the corresponding partial sum R(n−1) (t, ε) of (6.2.41), plug it instead of R(t) in any term with m ≥ 1 in (6.2.40) and drop all the terms with power of ε greater than (n − 1); finally we plug R(t) = R(n−1) (t, ε) + Rn (t)εn−1 to the term with m = 0. Then terms with the power of ε less than (n − 1) will be reduced and we get

 Rn (t) = R R1 (t), ... , Rn−1 (t) + δn1 L(t) where R is non-linear operator, differential with respect to (x, ξ) but containing integrations with respect to t. One can see easily that all operators Rk (t) are self-adjoint. In particular (6.2.42) (6.2.43)

R1 (t) = L(t) = U0 (t)PU0 (−t),  1 −1 t [U0 (t  )PU0 (−t  ), U0 (t)PU0 (−t)] dt  . R2 (t) = − ih 2 0

Finally, setting t = T0 in (6.2.35) we arrive to U(t)U0 (−1) = V (T0 ) ≡ exp(iεh−1 B) and therefore since U0 (−T0 ) ≡ I in Ω we arrive to (6.2.31) with (6.2.44) B(ε) = Bn εn−1 , n≥0



T0

Bn =

(6.2.45)

Rn (t) dt. 0

In particular, (6.2.42), (6.2.43) yield  (6.2.46)

T0

B1 =

U0 (t)PU0 (−t) dt,  

1 −1 B2 = − ih U0 (t  )PU0 (−t  ), U0 (t)PU0 (−t) dt  dt 2 0≤t  ≤t≤T0 0

(6.2.47)

CHAPTER 6. ESOTERIC THEORY

550

with the principal symbols  T0  (6.2.48) b ◦ Ψa,t dt b1 = 0   1 b2 = − (6.2.49) b ◦ Ψa,t  , b ◦ Ψa,t dt  dt. 2 0≤t  ≤t≤T0 Proposition 6.2.8. In the framework of Proposition 6.2.7 [A, B] ≡ 0

(6.2.50)

in Ω.

Proof. Let us assume that C = ih−1 [A, B] ∈ hp Ψh (X ) in Ω, p ≥ 0. Let us apply formula  t −1  −1  −iεh−1 tB iεh−1 tB −1 Ae = A + iεh e −iεh t B [A, B]e iεh t B dt  (6.2.51) e 0

to C1 instead of A. We then obtain that (in Ω) e −iεh

−1 tB

C1 e iεh

−1 tB

≡ C1 + εtC2

with C2 = ih−1 [C1 , B] ∈ hp Ψh (X ). Applying (6.2.51) to A we then obtain that e −iεh

−1 tB

Ae iεh

−1 tB

=≡ A + εtC1 −1

mod ε2 Ψh (X ). −1

Taking t = T0 and replacing e ±iεh B by e ±ih T0 A we obtain that C1 ≡ 0 mod εΨh (X ) and hence p can be replaced by p+δ because ε ≤ hδ ; meanwhile division by ε is justified by ε ≥ hl . Induction with respect to p concludes this analysis. Remark 6.2.9. (i) Surely if the right-hand expression in (6.2.48) vanishes then B(ε) = max(ε, h)B  with B  Ψh (X ) and therefore effectively ε is redefined as εnew = ε max(ε, h) etc. (ii) Most likely Propositions 6.2.7–6.2.8 hold for ε ∈ [hδ , ε0 ] with small constant ε0 and the proof would be very interesting. (iii) However we will investigate the case ε ∈ [hδ , ε0 ] by using results of Section 4.5.

6.2. OPERATORS WITH PERIODIC HAMILTONIAN FLOW

551

Matrix Case The case of matrix operator P is more complicated. However in this case we need to assume that ε ≤ ch because otherwise the nature of operator −1 e ih (A0 +εP) is not clear unless we assume that b has eigenvalues of constant multiplicities. Unfortunately we were unable to prove existence of B(ε) with desired properties unless a bit stronger assumption is made: Proposition 6.2.10. Let conditions (6.2.1)–(6.2.4) and (6.2.7) be fulfilled and let A0 ∈ Ψh (X ) be an operator with the principal symbol a satisfying (6.2.28). Let P ∈ Ψh (X , K) and (6.2.52)

A(ε) = A0 + εP,

ε ∈ [hl , 0 h]

Then there exists an operator (6.2.53)

B(ε) ∼



Bk (h−1 ε)k

k≥0

commuting with A(ε) and satisfying (6.2.31) in Ω. −1

Proof. Consider operator e ih t(A0 +ηhP) . According to Subsection 1.2.2 its parametrix is a h-Fourier integral operator with a phase function independent −1 on η and a symbol analytic with respect to η. Therefore F (η) := e ih t(A0 +ηhP) is (in Ω) unitary h-pseudodifferential operator with the symbol analytic with respect to η. Also it is I as η = 0. Then as F (η) − I  ≤ 1 −  (which is fulfilled as η ≤ 0 ) one can define log F (η) =

1 n≥1

n

 n (−1)n−1 F (η) − I

analytic with respect to η; further this function is skew-symmetric, vanishes as η = 0 (again, modulo negligible in Ω) and thus B := −iη −1 F (η) is a required operator. Problem 6.2.11. Extend the statement above to 0 h ≤  ≤ ch. Example 6.2.12. (i) Operator (6.2.54) A= σij (xi hDj − xj hDi ) + mσ0 1≤i 0 (see Theorem 4.6.16). (ii) Irregularity of the boundary which leads to the remainder estimate at least R2 = Ch−d ν(h) with ν(t) = mes(Yt ) even for a constant coefficients. Our goal is to recover in the general case the remainder estimate R1 + R2 or close to it. We will assume that (7.5.76) ν(t)t −1 is monotonically decreasing function of t. Then using constant scale ε-approximation with ε ≥ η = Ch| log h| we −d get contribution of  that the  total approximation error is h ϑ(ε) and the −1 B x, γ(x) to the semiclassical error after rescaling x → x , (7.5.77)

 ε   = min γ(x), ν(ε)

(see Subsection 4.6.5 for analysis inside domain) does not exceed   (7.5.78) Ch1−d γ d ε−1 ν(ε) + γ −1

732

CHAPTER 7. STANDARD THEORY NEAR THE BOUNDARY

and then the contribution of Xh to the semiclassical error does not exceed  1 1−d −1 1−d t −1 dt ν(t) (7.5.79) Ch ε ν(ε) + Ch h

while the total contribution of the strip Yh does not exceed R2 = Ch−d ν(h) according to Chapter 9. The first term in (7.5.79) is less than R1 as ε = η and only the second term could be a problem. However, if the boundary is quite irregular, namely ν(t) ≥ t σ with σ < 1, then  1 t −1 dt ν(t) ≤ Cs −1 ν(s) s

and then this term does not exceed R2 . Therefore we have proven Theorem 7.5.22. (i) Remainder does not exceed R1 + R2 unless (7.5.80)

ν(t) := mes(Yt ) ≤ Cδ t 1−δ

with some exponent δ > 0. (ii) If (7.5.80) holds, then the remainder does not exceed R1 + R2 + R2 where R2 is the second term in (7.5.79). Therefore we need to reconsider only case when (7.5.80) holds and we need to assume that (7.5.81) ν satisfies condition (7.5.80) as well: ν(t) ≤ Ct 1−δ ; otherwise R1 will be larger than R2 and then we pick up ε = η again. We will apply  thenour more sophisticated variable scale ε-approximation with ε = min ηγ −σ , γˆ as in Subsections 7.5.1–7.5.3. In this case approximation error does not exceed     −d −σ −d −δ ν(η0 t ) dt ν(t) ≤ Ch ν(η)t dt ν(t) = R1 t −σ dt ν(t) Ch provided (7.5.82) ν(ε)ε−1−σ1 is decreasing function of ε

7.A. APPENDICES

733

and σ  = σ(1 + σ1 ). Assuming that exponents σ and σ1 are small enough, we see that this integral does not exceed   −σ  C ν(s)s + C ν(t)t −σ −1 dt ≤ C and so an approximation error does not exceed R1 . To estimate a semiclassical error we apply the same arguments as in the proof of Proposition 7.5.6 and find that the contribution of the strip   := {x : β ≤ γ(x) ≤ β 1−δ } to the semiclassical error does not exceed X(β)   δ−1 1−d 1−d Ch γ dx + Ch dμτ   X(β)

 ×Rd X(β)

Στ

with the standard measure μτ on Στ . One can see easily that the second integral does not exceed dν(β)/dβ and then the total semiclassical error does not exceed  dν(β)  1−d (7.5.83) Ch γ δ−1 dx + Ch1−d k . dβ β=e −e Xh k:0≤k≤C log | log h|

Replacing this sum by an integral and changing variable in the integral we arrive to  1 1 1−d Ch dt ν(t) h t| log t| which is larger than the first term in (7.5.83). Therefore we proved Theorem 7.5.23. Let ν, ν satisfy (7.5.49). Then the remainder does not exceed R1 + R2 + R2 with   1−d γ −1 | log γ|−1 dx. (7.5.84) R2 = Ch Xh

7.A 7.A.1

Appendices Sets Associated with the Microhyperbolicity Condition

In this section we study the structure of certain sets associated with the microhyperbolicity condition; namely let   σ(x, ξ) = Spec J −1 (x)a(x, ξ) (7.A.1)

CHAPTER 7. STANDARD THEORY NEAR THE BOUNDARY

734

be the set of all the eigenvalues of J −1 (x)a(x, ξ), (7.A.2)

Στ = {(x, ξ) ∈ T ∗ X¯ , τ ∈ σ(x, ξ)},

(7.A.3)

Στ b = ιΣτ

where ι : T ∗ X¯ |Y → T ∗ Y is a natural map. Let Ξ0 =

(7.A.4)



σ(x, ξ).

T ∗ X¯

Further, let (x  , ξ  ) ∈ T ∗ Y and let σd (x  , ξ  ) be the discrete spectrum of the problem (7.A.5)

(a(x  , D1 , ξ  ) − τ J(x  ))v = 0, )

(7.A.6)

ðb(x  , D1 , ξ  )v = 0, v = o(1)

(7.A.7)

as x1 → +∞;

and let Στ b = {(x  , ξ  ) ∈ T ∗ Y , τ ∈ σd (x  , ξ  )},

(7.A.8)

Ξ1 =



σd (x, ξ)

T ∗Y

where we skip x1 = 0 in the notation for a and J. Let Στ b = Στ b ∪ Στ b . Moreover, let us introduce the sets  (7.A.9) Y0 = τ ∈ R, Στ contains at least one point at which the symbol τ J(x) − a(x, ξ) is microhyperbolic in no direction  ∈ T (T ∗ X¯ ) , and  Στ b contains at least one point at which (7.A.10) Y1 =  τ  ∈ R and the problem τ J(x ) − a(x  , D1 , ξ  ), ðb(x  , D1 , ξ  ) is microhyperbolic in no multidirection . Finally, let ◦

(7.A.11) Z0 be the set of the points (x, τ ) ∈ X × R such that for some ◦

ξ ∈ Tx∗ X the symbol τ J(x) − a(x, ξ) is microhyperbolic at (x, ξ) in no direction  with x = 0;

7.A. APPENDICES

735

and (7.A.12) Z1 be the set of points (x  , τ ) ∈ Y ×Rsuch that for some ξ  ∈ Tx∗ Y the problem τ J(x  )−a(x  , D1 , ξ  ), ðb(x  , D1 , ξ  ) is microhyperbolic at (x  , ξ  ) in no multidirection T = ( , ν1 , ... , νM ) with x = 0. We assume that (7.A.13) J(x) and a(x, ξ) are Hermitian (matrix-valued) symbols and J(x) is non-degenerate, (7.A.14) Either J(x) or a(x, ξ) is positive definite, (7.A.15) Operator a(x  , D1 , ξ  ) with the boundary condition ðb(x  , D1 , ξ  )v = 0 is self-adjoint on L2 (R+ , H), (7.A.16) Either J(x  ) or operator a(x  , D1 , ξ  ) (with the indicated boundary condition) is positive definite, and (7.A.17) The coefficient of D1m in a(x  , D1 , ξ  ) is non-degenerate. ◦

Here and in what follows X , Y and X¯ mean the the previous domain, its boundary and its closure respectively intersected with B(0, 1). Obviously, (7.A.18) Ξ0 and Ξ1 are closed sets. Let us recall that (7.A.19) If a symbol is microhyperbolic at some point in some direction then it is microhyperbolic in a neighborhood of this point in all the directions close to original one. Similarly (7.A.20) If the problem is microhyperbolic at some point in some multidirection then it is microhyperbolic in a neighborhood of this point in all multidirections close to original one (see Proposition 3.1.5).

CHAPTER 7. STANDARD THEORY NEAR THE BOUNDARY

736

Hence ◦

(7.A.21) Y0 , Y1 are closed subsets of R, Z0 is a closed subset of X × R, and Z0 ∩ (Y × R), Z1 are closed subsets of Y × R; obviously Z0 ∩ (Y × R) ⊂ Z1 . Our main goal is Theorem 7.A.1. Let conditions (7.A.13)–(7.A.17) be fulfilled and let K = K (d, m, D) in the smoothness conditions where D is the dimension of the matrices and m is an order of symbols with respect to ξ1 . Then (i) The Lebesgue measures of the sets Y0 and Y1 equal 0. ◦

(ii) For every x ∈ X , y ∈ Y the Lebesgue measure of the set {τ , (x, τ ) ∈ Z0 } equals 0. (iii) The Lebesgue measures of the sets Z0 and Z1 31) equal 0 (iv) The Lebesgue measure of the set Στ b (in T ∗ Y ) equals 0 provided τ ∈ / Y = Y0 ∪ Y 1 . Proof. We will prove Statement (i) here; Statement (ii) can be proven similarly. Obviously Z0 and Z1 are measurable sets and hence Statement (iii) follows from Statement (ii). In the proof of Statement (iv) one can assume that (∂x2 , ν1 , ... , νM ) is the multidirection of microhyperbolicity; then it follows from the proof below that for every x  = (x3 , ... , xd ) and ξ  the Lebesgue measure of the set {x2 , (x2 , x  , ξ  ) ∈ Sτb } equals 0; this fact and the obvious fact that Στ b is measurable imply Statement (iv). So let us prove that mes(Y1 ) = 0 (the equality mes(Y0 ) = 0 can be proven in the same way but without any preliminary analysis; this equality follows directly from Lemma 7.A.2). Let ε > 0 and let Ω be an open subset in T ∗ Y × R. Let us introduce set Y1 (Ω, ε) as we defined Y1 but with (z, τ ) ∈ Ω, z = (x  , ξ  ) and with condition (7.A.7) in the problem (7.A.5)–(7.A.7) replaced by (7.A.22)

u = O(e −εx1 ) as x1 → +∞;

then it is sufficient to prove that 31)

◦

In X × R and Y × R respectively.

7.A. APPENDICES

737

∗ ∗ Every (7.A.23)   point (z , τ ) has a neighborhood Ω = Ω(ε) such that mes Y1 (Ω, ε) = 0.

Without any loss of the generality one can assume that the polynomial g (z, ζ, τ ) has no root with the imaginary part equal to ε for (z, τ ) ∈ Ω. Let us reduce problem (7.A.5), (7.A.6) to a first-order problem and let us replace u by u new = Qu as we did in Subsection 3.1.2; then we obtain problem (p, ðb) where ⎛

(7.A.24)

⎞ pI 0 0 p = ⎝ 0 0 p II ⎠ , 0 p ∗II 0

b = (b I , b II , p III )

and all the roots of the polynomials det p I (z, ζ, τ ) and det p II (z, ζ, τ ) lie in the domains {| Im ζ| ≤ ε}, {| Im ζ| ≥ ε} of C respectively. Moreover, let b  = (b I , b II , b III ) be defined by the equality (7.A.25) p(z, D1 , τ )U, V + − U, p(z, D1 , τ )V + = b  (z, τ )U(0), b(z, τ )V (0) − b(z, τ )U(0), b  (z, τ )V (0); then (7.A.26) R = b ∗ II b II is a Hermitian matrix.   The microhyperbolicity of τ J(x  )−a(x  , D1 , ξ  ), ðb(x  , D1 , ξ  ) means exactly that (7.A.27) ( + νk ∂ζ )p I (z, ζ, τ )V , V  < 0 ∀V ∈ Ker p I (z, ζk , τ ) \ 0 ∀k = 1, ... , M and (7.A.28)

R(z, τ )V , V  < 0

∀V ∈ Ker b(z, τ ) \ 0.

Moreover, (7.A.29) Conditions (7.A.27) and (7.A.28) are fulfilled for  = ∂τ and ν1 = ... = νM = 0.

CHAPTER 7. STANDARD THEORY NEAR THE BOUNDARY

738

Let us introduce

⎞ 0 0 p I (z, ζ1 , τ ) ... ⎟ ⎜ .. .. .. ... ⎟ ⎜ . . . (7.A.30) S(z, ζ1 , ... , ζM , τ ) = ⎜ ⎟, ⎝ 0 ... p I (z, ζM , τ ) 0 ⎠ 0 ... 0 b II (z, τ ) ⎛ ⎞ I ... 0 0 ⎜ .. . . .. ⎟ .. ⎜ ⎟ . . . T (z, τ ) = ⎜ . (7.A.31) ⎟; ⎝0 ... I 0 ⎠ 0 ... 0 b ∗ II (z, τ )   then the equality mes Y1 (Ω(ε), ε) = 0 follows from Lemma 7.A.2 below. ⎛

Lemma 7.A.2. Let S(z, τ ) and T (z, τ ) be CK matrices of dimensions N × D and D × N respectively for (z, τ ) ∈ Ω where Ω ⊂ Ωx × R and Ωx are open domains and K = K (d, N, D). Let us assume that (7.A.32) ST is a Hermitian matrix and that for  = ∂τ the following condition holds (7.A.33)

(ST (z, τ )v , v  < 0

∀v ∈ Ker S(z, τ ) \ 0

∀(z, τ ) ∈ Ω.

Let (7.A.34) N = {(z, τ ) : there exists no vector field  = z ∈ Tx Ωx satisfying condition (7.A.33)}. Then (7.A.35)

mes



τ , ∃z : (z, τ ) ∈ N



= 0.

Proof. Let Xs = {(z, τ ) : rank S(z, τ ) ≤ s}. Then it is sufficient to prove that   (7.A.36) mes τ , ∃z : (z, τ ) ∈ N ∩ (Xs \ Xs−1 = 0. Moreover, it is sufficient to treat the case s = 0. In fact, in the proof of  S 0 (7.A.36) we can assume without any loss of the generality that S = 0 S 

7.A. APPENDICES

739

where S  is an invertible s × s matrix. This case can obviously be reduced to the case s = 0. Let t(z, τ ) = tr(TS)(z, τ ); then condition (7.A.33) with  = ∂τ yields that t(z, τ ) = 0, ∂τ t(z, τ ) > 0 for (z, τ ) ∈ X0 . Then without any loss of the generality one can assume that τ = λ(z) for (z, τ ) ∈ X0 where λ is a real-valued CK function. Let α1 (z), ... , αp (z) be elements of matrix R(z) = (TS)(z, λ(z)) arbitrarily enumerated. Then X0 = {(z, τ ) : τ = λ(z), α1 (z) = ... = αp (z) = 0}. Let us note that (7.A.37) Let (z, τ ) ∈ N ∩ X0 . Then for arbitrary dz, dλ is a linear combination of dα1 , ... , dαp . In fact, if that is not the case then there exists a direction  ∈ Tx Ωx such that λ < 0 and α1 = · · · = αp = 0. Then (TS)(z, τ ) = −∂τ (TS)(z, τ ) · λ(z) and hence (7.A.33) holds for this direction because it holds for ∂τ . Therefore N ∩ X0 is contained in the union of sets NK for K ⊂ {1, ... , p} where (7.A.38) NK = {τ = λ(z), αj (z) = 0 ∀j ∈ K, dαj (j ∈ K) are linearly independent and dλ is their linear combination}. Then Sard’s theorem yields that   mes τ , ∃z, (z, τ ) ∈ NK = 0 and this equality yields (7.A.36) with s = 0. The following assertion also seems to be of interest: Proposition 7.A.3. Let conditions (7.A.13)–(7.A.17) be fulfilled. Then (i) If τ ∗ ∈ ∂Ξ0 and z ∗ ∈ Στ ∗ then the symbol (τ ∗ J − a) is microhyperbolic in no direction at (z ∗ , τ ∗ ).

740

CHAPTER 7. STANDARD THEORY NEAR THE BOUNDARY

  (ii) If τ ∗ ∈ (∂Ξ1 )\Ξ0 and z ∗ ∈ Στ ∗ b then the problem τ ∗ J −a(x  , D1 , ξ  ), ðb is microhyperbolic in no multidirection at (z ∗ , τ ∗ ). Proof. In the proof of Statement (i) or (ii) let τ1 , ... , τM be continuous eigenvalues of the matrix J −1 (x)a(x, ξ) (or those of the problem (7.A.5)– (7.A.7)) depending on z ∈ T ∗ X or z ∈ T ∗ Y equal to τ ∗ for z = z ∗ . Let Π = Π(z) be the corresponding spectral projector in H or in the weighted space L2 (R+ , H, Jdx) respectively. Then both statements will be proven if we prove Statement (i) under the additional restriction τ ∗ J(z ∗ ) − a(z ∗ ) = 0. Conditions (7.A.5)–(7.A.13) yield that τj are one time differentiable at z ∗ in every direction (K. O. Friedrichs [1], e.g.) and hence it is sufficient to prove that if  is a microhyperbolicity direction then τj (z ∗ ) = 0 and therefore Ξ0 contains a neighborhood of τ ∗ . But the equalities   (τ1 + ... + τM )(z ∗ ) = tr (J −1 a)(z ∗ ) = − tr J −1 (τ ∗ J − a) (z ∗ ) yield that if the matrix ((τ ∗ J − a))(z ∗ ) is either positive or negative definite then (τ1 + ... + τM ) = 0 provided J is positive definite; if a is positive definite we can permute a and J and use the same arguments.

Comments The results of Section 7.1 are trivial enough, but the proofs given here were developed specially for this book. The construction of Section 7.2 (the method of successive approximations) in its present form was suggested first in V. Ivrii [8, 10] for differential and pseudodifferential operators (in fact in [8] the more complicated case of a domain with boundary was treated and namely the presence of the boundary was the reason that the traditional method did not work properly); the more sophisticated and general form which was used here was suggested in V. Ivrii [11]; for h-(pseudo)-differential operators this construction and its justification is more transparent. The results of Section 7.3 were well known in certain special cases: for scalar operators for which h = 1 and the large spectral parameter τ is considered (then this problem can be reduced to a semiclassical one with h = |τ |−1/m where m is the degree of the operator), asymptotics with the remainder estimate O(|τ |(d−1)/m ) are due to R. Seeley [1, 2]; later by other methods they were obtained by V. Ivrii [8, 11]; for scalar operators with m > 2 and for matrix operators with constant multiplicities of the eigenvalues of the principal symbol these asymptotics were first obtained by

7.A. APPENDICES

741

D. Vassiliev [1–3] and in the general case by V. Ivrii [11]; we do not discuss asymptotics with less precise remainder estimate because they are outside of the domain of the method of hyperbolic operators and we only mention that in the paper of V. M. Babich & B. M. Levitan [2] this estimate was obtained under the very restrictive condition that the domain is strictly concave in the corresponding metrics (which is fulfilled for no domain in Rd for the standard Laplacian) before it was obtained by R. Seeley in the general case. More precise asymptotics of Section 7.4 with the remainder estimate o(|τ |(d−1)/m ) were first obtained (under conditions on the Hamiltonian flow) by V. Ivrii [8, 11] and in all the papers mentioned above excluding those of R. Seeley such asymptotics were also obtained; this estimate was obtained by R. Melrose R. Melrose [8] in the case m = 2 simultaneously with V. Ivrii [8] but under the condition that the domain is strictly concave in the corresponding metric. Results of Section 7.5 were first published in V. Ivrii [27], I should mention less precise results of L. Zielinski [1–4]. One can find a lot of beautiful billiard pictures in many places, in particular on Wolfram MathWorld http://mathworld.wolfram.com/Billiards.html I am aware that these comments are very poor and inadequate. I send the reader to the excellent (but not very modern) survey of G. Rozenblioum, M. Z. Solomyak & M. Shubin [1] for more information. It is a real shame that nobody came with more modern exposition.

Chapter 8 Standard Local Semiclassical Spectral Asymptotics near the Boundary. Miscellaneous In this chapter we continue the treatment of local and microlocal semiclassical spectral asymptotics near the boundary. In Section 8.1 we consider asymptotics of e(x, x, τ ) near boundary and derive it with a remainder estimate O(h1−d ) and a boundary layer correction term h−d Υ(x  , h−1 x1 ). In Section 8.2 we consider rescaling technique near the boundary; here we have two scales: one comes from the principal symbol of operator and another is the distance to the boundary. Section 8.3 is devoted to operators with (partially) periodic Hamiltonian flow: for example we consider two manifolds with the common boundary and Hamiltonian flow is periodic only in one of them. Then asymptotics with the remainder estimate o(h1−d ) and a correction term h1−d Υ(τ h−1 ) appears not only due to “interference” when phase shifted differently along different trajectories, but also “dissipation” when at reflection some part of energy goes to manifold with non-periodic Hamiltonian flow and does not return to the original point. We consider also the case when Hamiltonian flows in both manifolds are periodic. Furthermore, in Section 8.4 we consider asymptotics when operator A is restricted to its invariant subspace, projector to which is either a pseudodifferential operator (away from the boundary), or sum of a pseudodifferential operator with the transmission property and a singular Green operator.

© Springer Nature Switzerland AG 2019 V. Ivrii, Microlocal Analysis, Sharp Spectral Asymptotics and Applications I, https://doi.org/10.1007/978-3-030-30557-4_8

742

8.1. POINTWISE SPECTRAL ASYMPTOTICS

8.1

743

Pointwise Spectral Asymptotics

In this section we consider boundary layer type term in e(x, x, τ ) appearing near boundary. Let us recall that due to rescaling technique   (8.1.1) e(x, x, τ ) = Nwx + O h1−d γ −1 (x) where γ(x) = dist(x, ∂X ) and this remainder estimate generated after integration O(h1−d | log h|) and basically the whole Chapter 7 we spent to eliminate logarithmic factor. Now we would like to consider e(x, x, τ ) without integration with respect to x.

8.1.1

Preliminary Analysis

Discussion Let us recall that in the previous chapter we derived asymptotics of   N := Γ Q1x e(., ., τ ) tQ2y (8.1.2) and   N := Γ ðx ðy Q1x e(., ., τ ) tQ2y (8.1.3) with the remainder O(h1−d ) provided problem (A, ðB) is microhyperbolic in multidirection T = ( , ν1 , ... , νM ); further, we derived asymptotics of   (8.1.4) Nx := Γx ðx ðy Q1x e(., ., τ ) tQ2y and   Nx := Γx Q1x e(., ., τ ) tQ2y (8.1.5) with the remainder O(h1−d ) provided  = (ξ , 0) and for (8.1.5) we also need to assume that ν1 = ... = νM in the microhyperbolicity condition; this asymptotics had a boundary layer type term h1−d Υ(x  , x1 h−1 ) with D α Υ(x  , r ) = O(r −∞ ) as r → +∞. Let us discuss this microhyperbolicity condition. Consider first the case of the Laplace operator; without any loss of the generality one can assume that a(x, ξ) = a (x, ξ  ) + ξ12 where a(x, ξ  ) is a positive definite quadratic form. Then microhyperbolicity means that  1 (8.1.6) ξ , ∇ξ a  + 2ν± ξ1 > 0 as ξ1 = ± τ − a(x, ξ  ) 2

744 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II and we can always take ξ = ξ  , ν+ = ν− = 0 as long as a (x, ξ  ) > 0 (i.e. ξ  = 0). So, with the exception of the case ξ  = 0 both microhyperbolicity conditions hold. In this exceptional case however ∇ξ a (x, ξ  ) = 0 and the first term in (8.1.6) is 0 no matter what ξ we pick up; so (8.1.6) boils up to  1 ±ν± τ − a(x, ξ  ) 2 > 0 and for microhyperbolicity we just take ν± = ±1 provided τ > 0; this is the standard ξ  -microhyperbolicity condition (we deliberately take  = (ξ , 0)). However it is not the case as we assume that ν+ = ν− ; then condition is impossible to satisfy. So, the previous section fails to find asymptotics of e(x, x, τ ) but not because of the rays tangent to the boundary but on the contrary, because of the rays orthogonal to it (in the corresponding metrics). This is a pleasant surprise because then we can find the solution of non-stationary problem in the form of the standard oscillatory integrals as we did in Subsubsection 5.3.1.4 Asymptotics without Spatial Mollification and Short Loops. The analogy does not stop here: the exceptional rays hit the boundary orthogonally, reflect and follow the same path as before forming the short loops of the length 2dist(x, ∂X ) where distance is measured in the corresponding Riemannian metrics. Remark 8.1.1. (i) However there is a difference: in the former case loop appeared because trajectory went “uphill”, lost velocity and rolled back “downhill’ repeating the path while now it reflects without losing velocity. (ii) Another difference is that in the former case the leading term in the asymptotics of e(x, x, τ ) at points where |V (x) − τ |  1 was smaller, of d/2 magnitude (τ − V (x))+ , than in the generic points and therefore the remainder estimate without correction term in higher dimensions (d ≥ 3) could be taken uniform. Now, however the leading terms is everywhere of the same magnitude and the correction term is needed in any dimension, however in the higher dimensions it matters in the smaller vicinities of the boundary (under appropriate assumptions) because the measure of “observable short loops” is smaller (see Remark 8.1.2).

8.1. POINTWISE SPECTRAL ASYMPTOTICS

745

Toy-Model Consider X = R+ × Rd−1 , a(ξ) = |ξ|2 and either Dirichlet or Neumann boundary conditions. Then propagator is defined by U(x, y , t) = U 0 (x, y , t) + U 1 (x, y , t)

(8.1.7)

with free space solution (8.1.8)

0

U (x, y , t) = (2πh)

−d

and reflected wave (8.1.9)

U 1 (x, y , t) = ς(2πh)−d

 e ih

−1 (|ξ|2 t+x−y ,ξ )

 e ih

dξ

−1 (|ξ|2 t+x−˜ y ,ξ )

dξ

where y˜ = (−y1 , y2 , ... , yd ) and ς = ∓1 for Dirichlet or Neumann boundary conditions respectively. Since U 1 is responsible to the difference e 1 (x, y , τ ) between e(x, y , τ ) and its free space counterpart e 0 (x, y , τ ) we manipulate only with it.   Taking −1 y = x we get x − x˜, ξ = 2x1 ξ1 while Ft→h−1 τ replaces e ih τ by hδ τ − |ξ|2 ; so    −1 (8.1.10) Ft→h−1 τ Γx U 1 = ς(2πh)1−d δ τ − |ξ|2 e 2ih x1 ξ1 dξ = ⎧  π   1 ⎨ cos 2h−1 τ 2 x1 cos φ sind−2 φ dφ d ≥ 2, 1−d d2 −1 · const · ςh τ 0 ⎩  −1 1  d =1 cos 2h τ 2 x1 where we introduced spherical coordinates (φ, ζ sin φ) on Sd−1 , φ ∈ (0, π), ζ ∈ Sd−2 . It leads to  1   (8.1.11) e 1 (x, x, 1) = h−1 Ft→h−1 τ Γx U 1 dτ = h−d Υς (h−1 x1 ) −∞

with (8.1.12) Υς (r ) = const · ς

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

z

0



0



1

π

d−1

  cos 2zr cos φ sind−2 φ dφdz

d ≥ 2,

0

1

  cos 2zr

d =1

746 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II where we plugged τ = z 2 . Here we took τ = 1 since any other value could be reduced to this one 1 by rescaling h → hτ − 2 . One can find the constant so that for Dirichlet/Neumann problem e 1 (x, x, τ ) = ςe 0 (x, x, τ ) as x1 = 0. Stationary phase method implies that (8.1.13) Υ(r ) = const · r −(d+1)/2 cos(2r ) + O(r −(d+3)/2 )

as r → +∞.

Therefore (8.1.14) With an error not exceeding (8.1.15)

Ch−d (x1 h−1 + 1)− 2 (d+1) + Ch1−d 1

the standard Weyl expression for e(x, x, τ ) holds;  −1 this error is better than O h−d (x1 h−1 + 1 ) which was due to rescaling but it is only because critical points of a(ξ  ) = |ξ  |2 are non-degenerate. x

x1

Figure 8.1: Reflection and short loops. Remark 8.1.2. We can interpret (8.1.13) in the following way: as |ξ  |  ρ the point after reflection deviates from the original point (on the same distance from the boundary) by |x  |  ρx1 (see Figure 8.1) and therefore this difference is quantum observable only if ρ2 x1  h; then “short loops” are formed by −1/2 −(d−1)/2 (d−1)/2 {ξ  : |ξ  |  x1 h1/2 } and their measure is  x1 h and their −(d−1)/2 (d−1)/2 1−d −1 contribution to the asymptotics is of magnitude h x1 · x1 h which is exactly the first term in (8.1.13). For second order operators this works automatically but in the general case it requires condition (8.1.49)r with r = d − 1.

8.1. POINTWISE SPECTRAL ASYMPTOTICS

8.1.2

747

Schr¨ odinger Operator

Let us consider Schr¨odinger operator with the symbol g jk (ξj − Vj )(ξk − Vk ) + V (8.1.16) a(x, ξ) = j,k

assuming that τ − V  1 and positive. We want to study trajectories on the energy level τ only and we can assume without any loss of the generality that (8.1.17)

τ = 0,

V (x) ≤ −,

V (x) = −1

where the last assumption is achieved by division a(x, ξ) by −V (x) which does not affect trajectories on energy level 0. General Theory We can assume without any loss of the generality that V1 = 0. Let us introduce coordinates x1 = dist(x, ∂X ) in the metrics (g jk ) 1) and x  which are constant along trajectories of a(x, θ) on level 0 which are orthogonal to the boundary, i.e. such that g j1 = δj1 .

(8.1.18)

We are interested in the construction of parametrix in (8.1.19)

U = {|ξj − Vj | ≤ 

j = 2, ... , d}.

It is be much simpler to consider this problem with the “time” x1 and with t as one of spatial variables. Then as l = 0  −1 0 l e ih (φς (x,y ,η)+tη0 ) fςnl (x, y , η)hn−d dη, (8.1.20) u (x, y , t) ≡ ς=±

n≥0

where η = (η  , η0 ) = (η2 , ... , ηd , η0 ) and phase functions φ0ς are defined from (8.1.21)

∂x1 φlς = λς (x, ∇x  φς , η0 ),

(8.1.22)

φ0ς (x, y , η, η0 )|x1 =y1 = x  − y  , η  ,  1 λς (x, ξ  , η0 ) = ς η0 − a(x, ξ  ) 2 .

(8.1.23) 1)

I.e. in the final run in (g jk (−V (x))−1 ).

748 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II Here amplitudes fςn0 (x, y , η) are smooth, satisfy transport equations and some initial conditions as x1 = y1 but we do not need to calculate them. Then we can calculate u 0 , hD1 u 0 as x1 = 0 and define u 1 (x, y , t) in the same way (8.1.20) but with φ1ς satisfying (8.1.21) and φ1ς = −φ0−ς

(8.1.24)

as x1 = 0.

Here amplitudes fςn1 (x, y , η) are smooth, satisfy transport equations and some initial conditions as x1 = 0 but we do not need to calculate them. Obviously (8.1.25) (8.1.26)

φ1ς (y , y , η) = y1 ψς (y , η)

as x1 = y1 ,

ψς (y , η) = λς (0, y  , η) − λ−ς (0, y  , η) + O(y1 ).

Therefore (as |t| ≤ T )   (8.1.27) Γy Q1x u 1 (x, y , t) t Q2y ≡ 

e ih

−1 (y

1 ψς (y ,η)+tη0 )

ς=±



fςn2 (y , η)hn−d dη,

n≥0

where Qj = Q(x, hD  , hDt ) are cut-offs with respect to ξ  , τ and finally    (8.1.28) Ft→h−1 τ χ¯T (t) Γy Q1x u 1 (x, y , t) t Q2y ≡  −1  e ih y1 ψς (y ,η ,τ ) fςn3 (y , η  , τ )h1+n−d dη  . ς=±

n≥0

Obviously, representation (8.1.28) implies that (8.1.29) For small constant T expression (8.1.28) does not exceed Ch1−d . Since (8.1.29) is true for u 0 under ξ-microhyperbolicity condition (8.1.30)

|τ − V (y )| ≥ 0

due to free space results, this statement is true also for u and therefore due to Tauberian arguments

8.1. POINTWISE SPECTRAL ASYMPTOTICS

749

 Q2(x, hD  ) Proposition 8.1.3. Let (8.1.30) be fulfilled and Q  1 = Q1 1 (x, hD ), t be operators with symbols supported in U . Then Γy Q1x e (x, y , τ ) Q2y equals modulo O(h1−d ) Tauberian expression  τ      Ft→h−1 τ  χ¯T (t) Γy Q1x u 1 (x, y , t) t Q2y dτ . (8.1.31) h−1 −∞

Due to results of Subsection 7.3.1 we need to calculate only (8.1.31) with an extra factor χ(τ − τ  ) with χ supported in the small vicinity of 0:  τ  −1    e ih y1 ψς (y ,η ,τ ) χ(τ − τ ) fςn3 (y , η  , τ  )hn−d dη  dτ  . (8.1.32) −∞

ς=±

n≥0

Obviously modulo O(h1−d ) we can skip in the right-hand expression of (8.1.32) all the terms with n ≥ 1; so we get h−d J with obviously defined J:   τ −1   χ(τ − τ  ) e i ψς (y ,η ,τ ) fς03 (y , η  , τ  ) dη  dτ  (8.1.33) J := ς=±

−∞

with  = hy1−1 . Further, for y1 ≥ h−1 one can consider J as an oscillatory integral with a semiclassical parameter . One can see easily that due to (8.1.25)– (8.1.26) in this oscillatory integral the phase function ψς (y , η, τ  ) satisfies |∂τ  ψς (y , η  , τ  )| ≥ 1 and has non-degenerate critical points with respect to η  ; we denote these points by ης (y , τ  ). Therefore −1 1 (8.1.34) J ∼ e i ϕς (y ,τ ) fς04 (y , τ ) 2 (1+d)+k as   1 ς=± k≥0

where (8.1.35)

ϕς (y , τ ) = ψς (y , η  (y , τ ), τ ).

Then (8.1.36) J = J0 +O(h) where J0 is also given by (8.1.33) but with y replaced by (0, y  ). Also the fact that f0l are defined through transport equations along trajectories implies then Proposition 8.1.4. Modulo O(h1−d ) term (8.1.31) coincides with the same ¯ ðB) ¯ = (a(0, y  , hDx ), ðb(0, y  , hDx )). term but calculated for (A,

750 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II Calculations and Main Theorem Now in virtue of Propositions 8.1.3, 8.1.4 we need only to recalculate Υ(r ) in the case of more general boundary conditions than Dirichlet and Neumann, and then make change of variables. Repeating arguments of Subsubsection 8.1.1.2 Toy-Model one can prove easily that Proposition 8.1.5. Consider Schr¨odinger operator with g jk = δjk , V = −1, τ = 0 and the boundary condition   (8.1.37) B0 (x, hD  )hD1 + iB1 (x, hD  ) u|x1 =0 = 0 with |b0 (x  , ξ  ) + |b1 (x  , ξ  )| ≥ .

(8.1.38)

Assume that supp(qj ) ⊂ U defined by (8.1.19). Then (i) The boundary layer term is h−d Υ(h−1 x1 ) with (8.1.39) Υ(r ) =   λ+ (ξ ,τ )   −1  −d ξ1 + iβ(ξ  ) ξ1 − iβ(ξ  ) e 2ir ξ1 dξ1 q1 (ξ  )q2 (ξ  ) dξ  (2π) λ− (ξ  ,τ )

where bj are principal symbols of Bj and due to self-adjointness assumption β = −ib1 : b0 is real-valued. (ii) Here function Υ(r ) admits decomposition 1 e 2ir ς κςn r − 2 (d+1)−n (8.1.40) Υ(r ) ∼ ς=±

as r → +∞.

n≥0

Therefore changing variables and using asymptotics in the zone where Theorems 7.3.10 and 7.3.11 work we arrive to Theorem 8.1.6. Let us consider Schr¨odinger operator with self-adjoint boundary condition (8.1.37) and let condition V (x) ≤ −, be fulfilled. Further, let ellipticity condition (8.1.41)

 0 |b0 (x  , ξ  )λ(x  , ξ  ) + b1 (x  , ξ  )| =

on supp(q1 ) ∪ supp(q2 )

8.1. POINTWISE SPECTRAL ASYMPTOTICS

751

be fulfilled where λ(x  , ξ  ) is a root of a(x, ξ1 , ξ  ) = 0 with Im λ > 0 2) . Then   √ (8.1.42) Γx Q1x e(x, y , 0) t Q2y − κ0 (x)h−d − h−d Υ(x  , h−1 x1 ) g | ≤ Ch1−d with Weyl coefficient κ0 and with g = det(g jk )−1 where x1 is the distance from x to ∂X in the metrics g jk V −1 and Υ(r ) is defined by (8.1.38) and satisfies (8.1.40). Remark 8.1.7. (i) Due to Theorems 8.1.6 and 8.1.8 below the above theorem but with Υ(r ) replaced by Υ(r ) + Υb (r ) remains true if ellipticity condition (8.1.41) is replaced by microhyperbolicity condition (8.1.43) |b0 (x  , ξ  )λ(x  , ξ  ) + b1 (x  , ξ  )|+    0 |∇ξ b0 (x  , ξ  )λ(x  , ξ  ) + b1 (x  , ξ  ) | =

on supp(q1 ) ∪ supp(q2 )

where Υb (r ) ∈ S(R+ ). (ii) Standard Weyl asymptotics holds with the remainder estimate O(h1−d ) for e(x, x, τ ) as x1 ≥ h(d−1)/(d+1) .

8.1.3

Generalizations

Now we need to consider the generalizations to the systems. We consider only Qj with symbols supported in U . The same approach leads to the following Theorem 8.1.8. Let (A, ðB) be self-adjoint and let U ⊂ T ∗ ∂X . Assume that (8.1.44) As (x, ξ  , τ ) ∈ [0, ] × U × [−, ] all the roots ξ1 of equation   (8.1.45) det τ − a(x, ξ) = 0  ± are λ± which are real-valued, smooth and of j (x, ξ , τ ) with j = 1, ... , M constant multiplicities and

(8.1.46)

± ∂ τ λ± j > 0.

2) So, this is the condition only in the complement of πΣ|∂X ; recall that Σ = {(x, ξ) ∈ T ∗ X , a(x, ξ) = 0}.

752 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II Then (i) Reflected solution has decomposition    (8.1.47) Ft→h−1 τ χ¯T (t) Γx Q1x u 1 (x, y , t) t Q2y ≡  −1  e ih x1 ψjk (x,η ,τ ) fjkn (x, η  , τ )h1+n−d dη  . 1≤j≤M + ,1≤k≤M −

n≥0

with real phase functions ψjk and does not exceed Ch1−d (1 + h−1 y1 )− 2 (r −1) . 1

(ii) Asymptotics (8.1.42) holds with h−d Υ(x, h−1 x1 ) constructed from (8.1.47) in the usual Tauberian way; thus (8.1.48) Υ(x, h−1 x1 ) ≡



1≤j≤M + ,1≤k≤M −

τ

 e ih

−1 x

1 ψjk (y ,η

 ,τ  )



fjkn (y , η  , τ  ) dη  dτ  .

n≥0

(iii) Under assumption   −   (8.1.49)r ∇ξ λ+ j (x, ξ , τ ) − λk (x, ξ , τ ) = 0   −   =⇒ rank Hessξ λ+ j (x, ξ , τ ) − λk (x, ξ , τ ) ≥ r expressions (8.1.47) and (8.1.48) do not exceed Ch1−d (1 + h−1 x1 )− 2 (r −1) and 1 C (1 + h−1 x1 )− 2 (r +1) respectively. 1

(iv) Under condition (8.1.49)3 one can replace x by x  in the phase functions  −1 and in (8.1.48);  amplitudes  then Υ(y , h y1 ) is constructed for the problem   a(0, y , hDx ), ðb(y , hDx ) . (v) As r = d − 1 h−d Υ(h−1 x1 ) admits stationary phase decompositions 1 −1   e ix1 h μjk (x ,τ ) fjkn (x  , τ )(x1 h−1 )− 2 (d+1)−n mod O(h1−d ). (8.1.50) j,k

n

Remark 8.1.9. (i) An easy proof is left to the reader. (ii) In contrast to the Schr¨odinger operator we do not define x1 as the distance to the boundary; eliminating x1 from the phases ψjk without deteriorating estimate is possible as r ≥ 3 while eliminating x1 from amplitudes is possible as r ≥ 1.

8.2. SCALAR OPERATORS AND RESCALING

8.2

753

Scalar Operators and Rescaling

Now our purpose is to consider scalar operators and apply to them rescaling technique. The bad news is that we do not have the all freedom of transformations we enjoyed in the smooth case since we need to preserve a boundary and also we need to look how boundary operator scales. The good news is that this restriction applies only in the boundary partition elements while in the inner partition elements we can apply the results of Chapter 5. Further, let us recall that in Section 5.4 we studied operators with irregular coefficients and did not have a freedom of transformations by means of Fourier integral operators at all. The important question is what we study:   (a) Γ Q1x e(., ., τ ) tQ2y ,     (b) Γ Q1x e(., ., τ ) tQ2y as τ < inf V or Γ Q1x e(., ., τ ) tQ2y ,   (c) Γx Q1x e(., ., τ ) tQ2y . In the first case R. Seeley’s method of Section 7.5 is our very powerful instrument to tackle balls close to the boundary. In two latter cases this method does not work very well. We consider only Schr¨odinger operator.

8.2.1

Schr¨ odinger Operator. I

General theory As usual let us start from the Schr¨odinger operator with the principal symbol (8.2.1)

a=



g jk (ξj − Vj )(ξk − Vk ) + V

j,k

(or any  matrix operator  with the principal symbol aI ) looking for asymptotics t of Γ Q1x e(., ., τ ) Q2y (where Qj are h-differential operators, usually just cut-off functions). We consider boundary problem   (8.2.2) hb j (hDk − Vk ) + b0 u|∂X = 0; j

754 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II we assume that it is either Dirichlet boundary problem (i. e. b 1 = ... = b d = 0, b0 = 1) or |b j | ≥ . (8.2.3) 1≤j≤d

Without any loss of the generality one can assume3) that X = {x1 > 0} locally and g jk = δjk

(8.2.4)

in which case self-adjointness of the problem implies that (modulo nonvanishing factor) (8.2.5) ib 1 , b 2 ,. . . ,b d and b0 are real-valued functions and the standard ellipticity of the problem means exactly that  jk 1 j (8.2.6) |ib 1 g ξj ξk 2 + b ξj |  |ξ  | ∀ξ  ∈ Rd . 2≤j,k≤d

2≤d

We know that with appropriate boundary conditions remainder estimate O(h1−d ) is achieved under ξ-microhyperbolicity condition (8.2.7)

|V | ≥ 0 .

To get rid of this condition (as coefficients belong to C1 ) we applied scaling functions (8.2.8)

2

γ(x) = |V (x)| + h 3 ,

1

ρ(x) = γ(x) 2 .

Let us do the same; then as γ(x) ≤ (x), (8.2.9)

1 (x) = dist(x, ∂X ) 2

everything works as before in Section 5.4 and the total contribution of these balls to the remainder is O(h1−d ) as either d ≥ 3 or d = 2 and (8.2.10)

|V | + |∇V | ≥ 0 ;

3) Surely it may affect smoothness but we need this assumption only to write down conditions to B in more compact form

8.2. SCALAR OPERATORS AND RESCALING

755

as d = 1 under condition (8.2.10) we concluded that the total contribution of these balls to the remainder is O(| log h|); we needed to examine a propagation of singularities to fix this and we will do it later. On the other hand, as (x) ≤ γ(x) we have two choices: first, we can 1 2 snap γ(x) to (x) but leave ρ(x) = |V (x)| 2 (as it was before for γ(x) ≥ 2h 3 ): ρ(x) = |V (x)| 2 +h(x)−1 1

(8.2.11) γ(x) = (x),

as |V (x)| ≥ (x) ≥ h;

then the total contribution of such balls does not exceed  Ch1−d ρ(x)d−1 (x)−1 ≤ Ch1−d | log h| and we cannot generally improve this while contribution of the zone {x : (x) ≤ Ch} to the remainder is O(h1−d ). So we get remainder estimate O(h1−d | log h|) under condition (8.2.10) and d ≥ 1; then rescaling arguments of Chapter 5 continue to work and we get remainder estimate O(h1−d | log h|) as either d ≥ 2 or as d = 1 and non-degeneracy condition (5.2.100)m is fulfilled and the remainder estimate O(h−δ ) as d = 1 without non-degeneracy condition (in the latter case we should be happy as we got the same result as in the case without boundary). Alternatively we should not replace γ(x) by (x) but then we need to analyze boundary ball B(¯ x , γ(¯ x )) in which case we need to look how boundary condition scales. As before we can assume that Vj (¯ x ) = 0 as d ≥ 2 or V1 = 0 as d = 1. Obviously the Dirichlet boundary condition u|∂X = 0 scales well. Condition (8.2.2) also scales well as b0 = 0 identically otherwise assumptions (8.2.3), (8.2.6) may be in trouble. In fact, this condition scales to (8.2.12)



 b j (hDk − Vk ) + βb0 u|∂X = 0.

j

where β = 1, b0 = ρ−1 b0 as |b0 (x)| ≤ C ρ at some and thus in any point of partition element and β = ρ−1 b0 (¯ x ), b0 = b0 b0 (¯ x )−1 if |b0 (x)| ≥ C ρ at some and thus in any point of the partition element. So, we get a system with an additional large parameter β. However we use Seeley’s method of Section 7.5 and we need only estimate here (in the boundary strip {x1 ≤ } after rescaling). Therefore we can allow operators with a parameter as long

756 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II the boundary condition remains elliptic with a parameter i.e. (8.2.13) |ib0



g jk ξj ξk

 12

2≤j,k≤d

+



b j ξj + b0 β|  |ξ  | + β

2≤j≤d

∀ξ  ∈ Rd , ∀β ≥ 0. This condition scales well. So we assume that (8.2.14) One of the following condition is fulfilled: (a) b 1 = ... = b d = 0, b0 = 1, (b) b0 = 0 and (8.2.6) is fulfilled, (c) (8.2.13) is fulfilled. Thus, the contribution of ball B(¯ x , γ(¯ x )) to the remainder is O(1−d ) = 1−d d−1 d−1 O(h γ ρ ). Then the contribution of all such balls to the remainder does not exceed Ch1−d ρd−1 (x) dx  = O(h1−d ) without any extra assumption. Therefore we get the total remainder estimate O(h1−d ) as either d ≥ 3 or as d = 2 and condition (8.2.10) is fulfilled. Then using scaling with (8.2.15)

1

1

γ(x) = ρ(x) = (|V (x)| + |∇V (x)|2 ) 2 + h 2

and applying the same arguments we recover remainder estimate O(h−1 ) as d = 2 even without assumption (8.2.10). Thus we arrive to Theorem 8.2.1. Consider Schr¨odinger operator. Assume that the boundary problem is self-adjoint and condition (8.2.14) is fulfilled. Let either d ≥ 3 and coefficients belong to C1,1 or d = 2 and coefficients belong to C2 . Then (i) The remainder estimate is O(h1−d ). (ii) Moreover, the contribution of the zone {x : dist(x, ∂X ) ≤ ε} with ε ≥ h to the remainder does not exceed C εh1−d .

8.2. SCALAR OPERATORS AND RESCALING

757

Sharp Remainder Estimates We can combine Theorem 8.2.1(ii) with Theorem 5.3.3. Note that for Schr¨odinger operator the problem (A − τ , ðB) is automatically microhyperbolic at points where billiards hit the boundary. Details are left to the reader. Just to remind that as d = 2 the remainder estimate includes the term  Ch−1 |F12 | dx {|V −τ |+|∇V |2 ≤h1−δ }

where F12 = (∂x1 V2 − ∂x2 V1 ). Special Case d = 1 As d = 1 we still need to get rid of | log h|. Under assumption (8.2.10) it is easy. Without any loss of the generality one can assume that |V (0)| ≤  and then consider two cases: (a) ∂x1 V (0) ≥ 0 ; (b) −∂x1 V (0) ≤ −0 :

V

V

x1

x1

(a) V  > 0

(b) V  < 0

Figure 8.2: Case d = 1 under condition (8.2.10); gray covers classically forbidden zones. In the former case the slope goes to the boundary and after rescaling  1 3 2 x1 → x1 γ −1 , ξ1 → ξ1 γ − 2 , h →  = hγ − 2 with γ = 12 x1 ≥ γ¯ := |V (0)| + h 3

758 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II forbidden and its contribution to the the zone {x : x1new ≥ C } is classically 2  remainder is O(s ) = O hs γ − 3 s and the total contribution of the zone  2 {x : x1 ≥ C γ¯ } does not exceed C hs γ − 3 s γ −1 dγ  1 exactly as in the proof of Theorem 5.2.7. In the latter case the slope goes away from the boundary and therefore 1 launching trajectory in this direction we extend T  γ(x)ρ(x)−1  x12 to T  1. Then the contribution of interval x1  γ to the remainder improves 1 from O(1) to O(γ 2 ). Therefore the  total contribution of the 1zone {x : |x1 | ≥ 2 C (|V (0)| + h 3 )} improves from γ −1 dγ ≤ | log h| to γ 2 × γ −1 dγ  1 exactly as in the proof of Theorem 5.2.10. In both cases we can estimate contribution to the remainder of the zone {x : x1 ≤ γ¯ } by O(1). Therefore in both cases under condition (8.2.10) we reached the remainder estimate O(1).  1 1 Now we can use γ =  |V | + |∂x1 V |2 2 + h 2 and exactly as before we need to consider the case |V (0)| + |∂x1 V (0)| ≤  and then two cases: (a) ∂x21 V (0) > 0 ; (b) −∂x21 V (0) < −0 .

 Again in the former case Στ dμτ ≤ C leading us to the same remainder estimate O(1); so only case (b) needs to be addressed. Again, there are two subcases: V  (0) > 0 and V  (0) < 0. Let us introduce  1 (8.2.16) γ¯ = |V  (0)| + |V (0)| + h 2 . Then the zone {x : x1 < C γ¯ } is rescaled x1 → x1 γ¯ −1 , ξ1 → ξ1 γ¯ −1 , h →  = 1 h¯ γ −2 and either γ¯ ≥ 2h 2 in which case (8.2.10) is fulfilled after rescaling, or 1 γ¯  h 2 in which case   1; in both cases contribution of this zone to the remainder is O(1). Assume that function V (x) extended to negative x1 reaches its maximum on {x : |x1 | ≤ γ¯ } at x1∗ . Then in virtue of the above arguments we need to consider contribution of the zone {x : x1 ≥ C γ¯ }. Let us recall the proof of Theorem 5.2.10; adapting it to our case one can see easily that we considered there classical trajectories going “downhill” away from x1∗ unless x1 − x1∗ > hδ in which case we considered classical trajectories going “uphill” towards x1∗ . The boundary x1 = 0 does not affect our arguments in the former case

8.2. SCALAR OPERATORS AND RESCALING

V

x1

(a) V  (0) > 0

V

759

x1

(b) V  (0) < 0

Figure 8.3: Case d = 1 under condition (b); sign V (x1∗ ) does not really matter. and affects our arguments only in the latter case and only if x1∗ < 0 and V (0) < 0; sign V (x1∗ ) does not matter. However we can repeat the same arguments as in the proof of Theo1 rem 5.2.10 but with γ¯ defined by (8.2.16) rather than with γ¯ = (|V (x1∗ )|+h) 2 as we used there. Now we can go from condition (5.2.100)2 to (5.2.100)m by induction with respect to m. Note that in the proof of Theorem 5.2.14 in this case we always are moving “uphill” towards point x ∗ where γ(x) reaches its minimum and (8.2.17)

  T  max x1δ γ¯ −δ , 1

with γ¯ introduced there by (5.2.103) as γ reaches its minimum at x1∗ > 0 and with γ¯ replaced by γ(0) as γ reaches its minimum at x1∗ < 0. This leads us to Theorem 8.2.2. Consider Schr¨odinger operator in dimension d = 1 with sufficiently smooth coefficients. Assume that boundary problem is self-adjoint and satisfies condition (8.2.14) 4) . Then the remainder estimate is O(1) under condition (5.2.100)m and O(h−δ ) without it. 4)

Which as d = 1 boils down to |b0 | + |b 1 |  1

760 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II Generalizations Consider any scalar operator assuming that the classically principal part of it is (uniformly) elliptic and also classically principal part of boundary value problem is (uniformly) elliptic. Then in the smooth case we can apply arguments of Section 5.2 and in irregular case arguments of Section 5.4; in the latter case we either assume that the boundary is smooth or just satisfies regularity conditions of Section 7.5 and in the latter case we assume that there are Dirichlet boundary conditions. Then under these assumptions we recover in full results of Section 5.2 or Section 5.4 respectively.

8.2.2

Schr¨ odinger Operator. II

Preliminaries

  Now we consider asymptotics either of Γ Q1x e(., ., τ ) tQ2y as τ < inf V or   Γ Q1x e(., ., τ ) tQ2y . In this case we cannot get rid of the microhyperbolicity assumption of (A − τ , ðB) as easily as we did before. Also we assume that coefficients are smooth enough. Remark 8.2.3. (i) Ellipticity condition5) (8.2.6) has different implications for different dimensions: (i) As d ≥ 3 it is equivalent to gjk b j b k ≤ (1 − )|b 1 |2 (8.2.18) g  (b  ) := 2≤j,k≤d

where (gjk ) = (g jk )−1 ; (i) As d = 2 it is equivalent to either (8.2.7) or (8.2.19)

g22 |b 2 |2 ≥ (1 + )|b 1 |2 ;

(ii) Under condition (8.2.18) operator A with boundary condition ðBu = 0 is semi-bounded from below and under condition (8.2.19) it is not. (iii) Condition (8.2.13) is (8.2.6) in dimension dv rather than (d − 1) with ξ0 = β and g 0j = δj0 . 5)

As g jk = δjk .

8.2. SCALAR OPERATORS AND RESCALING

761

Microhyperbolicity Conditions Let us recall that the microhyperbolicity condition at the interior point (x, ξ) is |ξj − Vj (x)| + |∇x V (x)| ≥ 0 ; (8.2.20) |τ − V (x)| − 1≤j≤d

the microhyperbolicity condition for (A − τ ) at the boundary point (x, ξ  ) is (8.2.21) |τ − V (x)| − |ξj − Vj (x)| + |∇x  V (x)| ≥ 0 (x1 = 0); 2≤j≤d

moreover, ξ-microhyperbolicity in the interior and boundary point is |τ − V (x)| − (8.2.22) |ξj − Vj (x)| ≥ 0 1≤j≤d

and (8.2.23)

|τ − V (x)| −



|ξj − Vj (x)| ≥ 0

(x1 = 0);

2≤j≤d

respectively. Furthermore, if (A − τ ) is ξ-microhyperbolic in the boundary point, problem (τ − A, ðB) is ξ-microhyperbolic if and only if one of the following four conditions is fulfilled: (8.2.24) The point (x  , ξ  ) is either hyperbolic or “almost hyperbolic”, i.e. g jk (x)(ξj − Vj )(ξk − Vk ) + V −  τ≥ 2≤j,k≤d

with  =  (d, c, 0 ) > 0; (8.2.25)

ib1 b0 ≥ − ,

(8.2.26)

|τ − W (x)| ≥ 0

where (8.2.27)

W = V − β 2 (|b 1 |2 − g  (b  )−1

762 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II with g  (b  ) defined by (8.2.18), (8.2.28)

  ξj − Vj + (τ − V ) bj  ≥ 0 , b0 2≤j≤d

bj =



gjk b k ;

2≤k≤d

in cases (8.2.25)–(8.2.27) the operator B˜ with the principal symbol  jk 1 j b ξj g ξj ξk 2 + (8.2.29) ib 1 2≤j,k≤d

2≤d

and the problem (A − τ , ðB) are both either elliptic at this point or microhyperbolic in the multidirection (0, ξj − Vj )j=2,...d ); we assume that condition (8.2.24) is violated and hence operator A − τ is elliptic at this point. Finally, if all four conditions (8.2.24)–(8.2.27) are violated then the microhyperbolicity condition is fulfilled if and only if |∇x  W | ≥ 0 ;

(8.2.30) however x = 0 in this case.

Analysis in the “Elliptic” Zone Let us now consider the elliptic zone of the operator (A − τ ). Here    it is sufficient to consider only Γ Q1x e(., ., τ ) tQ2y as Γ Q1x e(., ., τ ) tQ2y in this zone can be reduced to it (with modified operators Q1x and Q2y and) with an extra factor h. In the zone in question case the ellipticity or microhyperbolicity of problem (A − τ , ðB) is equivalent to the same property of the symbol (8.2.31) where (8.2.32)

 1 b˜ = ib 1 (x  ) a (x  , ξ  ) − τ 2 + b  (x  , ξ  ) + b0 (x  ) a (x  , ξ  ) =



g jk (ξj − Vj )(ξk − Vk ) + V ,

j,k



b  = j b j ξj , we sum with respect to j, k = 2, ... , d and as before we assume that g 1k = δ1k ; let us recall that the elliptic zone is given by the inequality (8.2.33) with a constant  > 0.

a (x  , ξ  ) ≥ τ + 

8.2. SCALAR OPERATORS AND RESCALING

763

In turn symbol b˜ is elliptic in zone defined by   (8.2.34) ib 1 (x  )−1 b  (x  , ξ  ) + b0 (x  ) ≥ 0 and in zone (8.2.35)

  ib 1 (x  )−1 b  (x  , ξ  ) + b0 (x  ) < 0

the ellipticity or microhyperbolicity of b˜ is equivalent to those of 2    2  (8.2.36) p = − b 1 (x  ) a (x  , ξ  ) − τ + b  (x  , ξ  ) + b0 (x  ) , Due to assumption (8.2.18) or (8.2.19) quadratic form  2 jk   j 2 (8.2.37) − ib 1 (x  ) g (x )ξj ξk + b ξj j,k

j

is non-degenerate and thus we basically arrive to the Schr¨odinger operator on that ib 1 is disjoint from 0; otherwise  thej 2boundary; here we can assume 1 j |b | is disjoint from 0 and as ib is sufficiently small, the symbol p is ξ-microhyperbolic. For this Schr¨odinger operator we can invoke the full power of Subsection 5.3.1 and arrive to Theorem 8.2.4. Consider Schr¨odinger operator in dimension d ≥ 2. Assume that boundary problem is self-adjoint and satisfies condition (8.2.14). Let Qj = qj (x, hDx ) be operators with symbols supported in the zone defined by (8.2.33). Then (i) As d ≥ 3 the following estimates hold   (8.2.38) |Γ Q1x e(., ., τ ) tQ2y − κ0 h−d | ≤ Ch1−d and   |Γ Q1x e(., ., τ ) tQ2y − κ1 h1−d | ≤ Ch2−d . (8.2.39) (ii) As d ≥ 2 and nondegeneracy condition (5.2.100)m is fulfilled for symbol (8.2.40)

 2    2 W (x  ) = − ib 1 (x  ) V (x  ) − τ + b0 (x  ) ,

estimates (8.2.38) and (8.2.39) hold as well.

764 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II (iii) As d = 2 (8.2.41) and (8.2.42)

  |Γ Q1x e(., ., τ ) tQ2y − κ0 h−2 | ≤ Ch−1−δ   |Γ Q1x e(., ., τ ) tQ2y − κ1 h1−d | ≤ Ch−δ

hold with arbitrarily small exponent δ > 0. Remark 8.2.5. (i) Under some conditions on the Hamiltonian trajectories of p one can obtain two-term asymptotics; however one should keep in mind that these trajectories can leave the elliptic zone, so it is necessary either to forbid this or to use the complete analysis of propagation of singularities in the hyperbolic zone. (ii) Surely one can also consider averaging with respect to spectral parameter. Asymptotics as V ≈ 0 To cover this zone we introduce (8.2.43)

2

γ = |V | + h 3 ,

1

ρ = γ2

and after rescaling V ≈ 1 or   1. We do not need to care then about microhyperbolicity condition in the elliptic zone (unless d = 2). Then we arrive immediately to Theorem 8.2.6. Consider Schr¨odinger operator in dimension d ≥ 2. Assume that boundary problem is self-adjoint and satisfies condition (8.2.14). Then (i) Asymptotics (8.2.38) holds as d ≥ 3. (ii) Asymptotics (8.2.44)

  |Γ Q1x e(., ., τ ) tQ2y − κ0 h−d | ≤ Ch−1



γ −1 dx 

holds as d = 2 and the boundary condition is either Dirichlet or Neumann or non-degeneracy condition (5.2.100)m is fulfilled for symbol W (x  ) defined by (8.2.40).

8.2. SCALAR OPERATORS AND RESCALING

765

(iii) Asymptotics (8.2.45)

  |Γ Q1x e(., ., τ ) tQ2y − κ0 h−d | ≤ Ch−1−δ 



γ −1+ 2 δ dx  3

holds as d = 2. Proof. Really, in the frameworks of Statements (i) or (ii) the rescaling leads to the remainder estimate Ch1−d ρd−1 γ −1 dx  which implies (8.2.38) and (8.2.44) respectively. In the framework of Statement (iii) we get the remainder estimate   −δ  dx Ch1−d ργ −1 |hγ −1 ρ−1 | which leads to (8.2.45). Corollary 8.2.7. Let d = 2. Then (i) Asymptotics with remainder estimate O(h−1−δ ) holds under microhyperbolicity condition (8.2.46)

|V − τ | + |∇x  V | + |∇2x  V | ≥ .

(ii) Moreover, for Dirichlet or Neumann boundary condition asymptotics with remainder estimate O(h−1 | log h|) holds under condition (8.2.46). (iii) Asymptotics with remainder estimate O(h− 3 ) in the general case. 4

Remark 8.2.8. We cannot control ∂x1 V so we can neither scale microhyperbolicity condition |V − τ | + |∇x  V | ≥  properly. This prevents us from getting better results as d = 2. Problem 8.2.9. (i) Derive uniform with respect to τ ≈ 0 remainder estimates (the best possible) for toy-model operators   h2 D12 + D22 ± x1 (8.2.47)± under Dirichlet or Neumann boundary condition. (ii) Generalize these estimates.

766 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II

8.2.3

Schr¨ odinger Operator. III

  Consider now asymptotics of Γx Q1x e(., ., τ ) tQ2y . Assume for simplicity that either Dirichlet or Neumann boundary condition is imposed. Then the same scaling as before brings the remainder estimate O(h1−d ) as d ≥ 3 1 and the remainder estimate O(h−1 γ − 2 ) as d = 2; in particular as d = 2 the 4 remainder estimate O(h− 3 ) holds. Consider case d = 2 in more details. As (x) = dist(x, ∂X ) ≥ γ(x) we consider a ball B(x, (x)) and in this ball we replace γ(x) and ρ(x) by (x) 1 and (x) 2 respectively; we can do it because we have “free space asymptotics” without assumption that V (x) is disjoint from 0. This asymptotics was derived in Subsubsection 5.3.1.4 Asymptotics without Spatial Mollification and Short Loops. Let us recall that this asymptotics had remainder estimate O(h−1 ) and contained correction term described in (5.3.36) and (5.3.62) and concentrated near V = 0. 1 Scaling this ball leads us to the remainder estimate O(h−1 ρ 2 γ −1 ) where 1 1 we plug γ =  and ρ =  2 and arrive to the remainder estimate O(h−1 (x)− 2 ). Therefore in both cases the remainder does not exceed (8.2.48)

Ch−1 γ1 (x)− 2 , 1

 2 γ1 (x) = max |V (x)|, (x), h 3

but the correction term is different depending on which of (x), |V (x) is larger: Theorem 8.2.10. Consider Schr¨odinger operator in dimension d = 2 under of  Dirichlet or Neumann boundary condition. Then  asymptotics 1 Γx Q1x e(., ., τ ) tQ2y holds with the remainder estimate O h−1 γ1 (x)− 2 with the Weyl principal part and correction term h−2 Υ(h−1 x1 ) with function Υ 2 constructed in Section 8.1 as |V (x)| ≥ max((x), h 3 ) and with the correction 2 term described by Theorem 5.3.16 as (x) ≥ max(|V (x)|, h 3 ). Problem 8.2.11. In dimension d = 2 assuming that |∇x  V (x)| ≥  derive −1

asymptotics with the remainder O(h−1 ) or at least which is o(h−1 γ1 3 ) as γ1  1. This correction term is expected to contain a “circular wave” term corresponding to the “vertex” {x ∈ ∂X , V (x) = 0}. Related is the following

8.3. OPERATORS WITH PERIODIC HAMILTONIAN FLOWS

767

Problem 8.2.12. (i) In dimension d ≥ 2 consider non-degenerate Schr¨odinger operator in domain with edges and derive asymptotics with the remainder estimate O(h1−d ) or at least with o(h1−d γ1−1 (x)) as γ1 (x)  1 where γ1 is the distance to the nearest edge. (ii) Derive similar results in domains with edges, vertices of any dimension. (iii) Derive similar results in domains with conical points. Those are difficult problems solution of which clearly would deserve publication. Remark 8.2.13. Surely we can restore results of Section 5.2 for spectral mollifications.

8.3

Operators with Periodic Hamiltonian Flows

In Section 6.2 we considered the case when there was no boundary and the Hamiltonian flow was periodic and derived very sharp spectral asymptotics. Now we want to achieve similar results when there is a boundary. We do not expect asymptotics to be that sharp, but better than O(h1−d ) and have non-Weyl correction term. However the presence of the boundary, more precisely, branching of the rays on the boundary brings the new possibilities. For simplicity we consider only Schr¨odinger operators.

8.3.1

Discussion and Plan

We start our analysis in Subsection 8.3.2 from the case when there is no branching and the Hamiltonian flow (with reflections) is periodic. Then if period is T0 , ΨT0 = I we have the same equality (8.3.1)

e ih

−1 T

0A

Q = e iT0 L Q

as before where supp(q) is disjoint from the boundary and only transversal to the boundary trajectories originate at supp(q) and L is h-pseudodifferential operator. Then   t Ft→h−1 τ χ( (8.3.2) ¯ − n)Γ Q1x u tQ2y = (2πh)1−d J(n) + O(h1−d+δ ) T0

768 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II with (8.3.3)

 e in q1 q2 dμτ

J(n) = Στ

as χ¯ is supported in (− 23 , 23 ) and equals 1 in (− 13 , 13 ) and n ∈ Z, |n| ≤ h−δ . Here as before Στ = {a(x, ξ) = τ } and μτ = dxdξ : da is the measure there,  is the principal symbol of L. If  is “variable enough” on Στ so the right-hand expression is decaying as n → ∞ we can improve remainder estimate O(h1−d ) (to what degree depends on the rate of the decay). Recall that in Section 6.2 essentially  was defined by integral of the subprincipal symbol along periodic trajectories. Now however  can pick up extra terms at the moment of the reflection: for example, for Schr¨odinger operator increment of  is 0 and π for Neumann and Dirichlet boundary condition respectively; however more general boundary condition brings increment which depends on the point (x  , ξ  ) ∈ T ∗ Y of the reflection. Then in Subsection 8.3.3 we consider a more complicated case when two operators are intertwined through the boundary conditions and all Hamiltonian trajectories of one of them are periodic but of another are not and, moreover, if the generic Hamiltonian billiard is periodic, it is in fact some Hamiltonian billiard of the first operator. Then (8.3.1) fails but (8.3.2)–(8.3.3) is replaced by (8.3.2), (8.3.4)   (8.3.4) J(n) = e in−|n| q1 q2 dμ1,τ Σ1,τ

where Σ1,τ and μ1,τ correspond to the first operator and  ≥ 0 depends on the portion of the energy which was whisked away along trajectory of the second operator at the reflection point. Then if   (8.3.5) μ1,τ {(x, ξ) ∈ Σ1,τ ,  = 0, ∇ = 0} = 0 we conclude that J(n) = o(1) as n → ∞ and we will be able to improve remainder estimate O(h1−d ) to o(h1−d ) and again there will be a non-Weyl correction term but it will depend also on both  and  . Finally, in Subsection 8.3.3 we consider even more complicated case when two operators are intertwined through boundary condition and the Hamiltonian trajectories of both of them are periodic and, moreover, after a while all trajectories issued from some point at T ∗ Y assemble there. Then we can construct a matrix symbol and its eigenvalues play a role formerly played by .

8.3. OPERATORS WITH PERIODIC HAMILTONIAN FLOWS

8.3.2

769

Simple Hamiltonian Flow

Inner Asymptotics Let U be an open connected subset in T ∗ X disjoint from ∂X . Consider Hamiltonian billiards issued from U , let Φt denote the generalized Hamiltonian flow. Assume that (8.3.6) (8.3.7) (8.3.8)

|∇x,ξ a(x, ξ)| ≥ 

∀t ∀(x, ξ) ∈ Φt (U ),

|φ(x)| + |{a, φ}(x, ξ)| ≥  Φt (x, ξ) = (x, ξ)

∀t ∀(x, ξ) ∈ Φt (U ),

with t = t(x, ξ) > 0 ∀(x, ξ) ∈ U ,

where φ ∈ C∞ (X¯ ), φ > 0 in X and φ = 0 on ∂X and (8.3.9) Along Φt (U ) reflections are simple without branching, which means that ι−1 ι(y , η) ∩ {a(y , η) = τ } consists of two (disjoint) points (y , η ± ) such that ±{a, φ} > 0 as (y , η) ∈ Φt (U ), y ∈ ∂X and τ = a(y , η). Let us recall that ι : T ∗ X |∂X → T ∗ ∂X is a natural map. Then exactly as in the case without boundary (8.3.10) One can select t(x, ξ) = T (a(x, ξ)) in U with T ∈ C∞ (so period depends on the energy level only6),7) and (8.3.11) As (x, ξ) ∈ U and 0 < t < T (x, ξ) Φt (x, ξ) meets ∂X exactly (N − 1) times with N = const; so as N ≥ 1 billiard consists of N segments with the ends on the boundary. We can consider instead of the operator A with the domain (8.3.12)

D(A) = {u ∈ H m (X ), ðBu = 0}

an operator f (A) with  (8.3.13) 6)

f (τ ) =

T −1 (τ ) dτ

See clarification in footnote 6) in Chapter 6. But there could be subperiodic trajectories with periods n−1 T (a(x, ξ)), n = 2, 3, .... The structure of subperiodic trajectories described in Subsection 6.3.1. 7)

770 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II −1

and introduce the corresponding propagator e ih tf (A) so that the corresponding Hamiltonian flow Ψt of f (A) is periodic with period 1. However now structure of f (A) near ∂X is a bit murky; but it is not really serious obstacle since we can consider problem  −1  (8.3.14) fˇ(hDt ) − A e ih tf (A) = 0, ðBe ih

(8.3.15)

−1 tf (A)

=0

where fˇ is an inverse function. Therefore its Schwartz kernel U(x, y , t) satisfies problem (8.3.14)–(8.3.15) with respect to x and transposed problem with respect to y . −1 Note that due to assumptions (8.3.6)–(8.3.7) e ih tf (A) Q is a Fourier integral operator with the symplectomorphism Ψt provided Q is h-pseudodifferential operator with the symbol supported in U . Further, due to (8.3.8), −1 (8.3.13) Ψ1 = I and therefore e ih f (A) Q is h-pseudodifferential operator. Therefore, we arrive to Proposition 8.3.1. Let A be a Schr¨ odinger operator with the scalar principal symbol a(x, ξ) satisfying (8.3.6)–(8.3.8) and (8.3.13). Then (8.3.16)

e ih

−1 f (A)−iκ

≡ e ih

−1 εL

in U

with ε = h and h-pseudodifferential operator L; here κ is Maslov’s constant and the principal symbol  of L is defined by   (8.3.17) e i = e ik e ik 1≤k≤N

where k corresponds to k-th segment (see Theorem 6.2.3) and k corresponds to the k-th reflection. Example 8.3.2. Consider Schr¨odinger operator. Assume that at the reflection point a(x, ξ) = ξ12 + a(x  , ξ  ), b = ξ1 + iβ(ξ  ). Then  −1   1 1  (8.3.18) e ik = − (τ − a ) 2 − iβ (τ − a ) 2 + ıβ with the right-hand expression calculated in the corresponding reflection point. Taking β = 0 or β = ∞ (formally) we get k = 0 and k = π for Neumann and Dirichlet boundary conditions respectively.

8.3. OPERATORS WITH PERIODIC HAMILTONIAN FLOWS

771

Then we can use all the local results of Section 6.2, assuming that (8.3.16) holds with hl ≤ ε ≤ h and derive local spectral asymptotics with the same precision (up to O(h2−d )) as there. Corollary 8.3.3. (i) Let conditions of Proposition 8.3.1 be fulfilled. Then asymptotics with the remainder o(h1−d ) holds provided   (8.3.19) μτ {(x, ξ) ∈ Στ , ∇Στ (x, ξ) = 0} = 0 where ∇Στ means a gradient along Στ . (ii) Asymptotics with the remainder O(h1−d+δ ) with small exponent δ > 0 holds provided    (8.3.20) μτ {(x, ξ) ∈ Στ , |∇ (x, ξ)| ≤ ε} = o(εδ ) with the small exponent δ  > 0. (iii) Furthermore asymptotics with the remainder o(h1−d ) holds as (8.3.6)– (8.3.7) are replaced by their non-uniform versions (8.3.7)

|φ(x)| + |{a, φ}(x, ξ)| > 0

∀t ∀(x, ξ) ∈ Φt (U ∩ Στ \ Λτ ),

where μτ (Λτ ) = 0. We leave to the reader to formulate statement similar to Statement (iii) but with the remainder estimate O(h1−d+δ ). Usually our purpose is the asymptotics with Q1x = ψ1 (x), Q2y = ψ2 (y ) where ψj are smooth functions but not vanishing near the ∂X . Corollary 8.3.4. Asymptotics with the remainder estimate o(h1−d ) holds provided conditions (8.3.6), (8.3.7) , (8.3.8) and (8.3.19) are fulfilled in U = T ∗ X , Qj = ψj ∈ C∞ (X¯ ) compactly supported. Proof. Due to corollary 8.3.3(iii) a contribution of the zone {x : x1 > ε} to the remainder is oε (h1−d ) while a contribution of the zone {x : x1 ≤ ε} to the remainder does not exceed O(εh1−d ) as ε ≥ h. These arguments could be improved to ε = hδ and the remainder estimate could be improved to O(h1−d+δ ). We leave both precise statements and precise arguments to the reader. Our goal is to derive even more sharp remainder estimate.

772 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II Asymptotics near Boundary Let us consider X = {x : x1 > 0} and U open subset in T ∗ Rd−1 x  × [0, ε]x1 × Rτ where ε > 0 is a very small constant. We are interested in  −1  (8.3.21) Ft→h−1 τ χ¯T (t) Tr e ih tf (A) Q with Q = Q(x, hDx , hDt ) with symbol supported in U ; let us rewrite it as   −1 −1 −1 Ft→h−1 τ χ¯T (t) Tr(e ih (t−t¯)f (A) Q  e ih t¯f (A) ) = Ft→h−1 τ χ¯T (t) Tr e ih tf (A) Q  −1

−1

with Q  = e ih t¯f (A) Qe −ih t¯f (A) . Note that if (8.3.6), (8.3.7) are fulfilled, ε > 0 and t¯ ≥ C ε are small enough then Q  = Q  (x, hDx , hDt ) is h-pseudodifferential operator with the symbol supported in { t¯ ≤ x1 ≤ C  t¯} and we can rewrite   Q  = Q  (x, hDx ) + Q  (x, hDx , hDt ) hDt − f (A) ; therefore we can rewrite (8.3.21) in the same form but with Q replaced by Q  . Now we can apply all the arguments of the previous Subsubsection 8.3.2.2 Asymptotics near Boundary and of Section 6.2 and derive asymptotics with the remainder estimate as sharp as O(h2−d ) provided conditions (8.3.6)– (8.3.8) are fulfilled in the full measure as well as corresponding conditions of Section 6.2 to . Here as usual we lift the points of (y  , η  , τ ) ∈ T ∗ ∂X to (y  , η ± ) ∈ Στ . So, we need to analyze only a near tangent zone {x : x1 ≤ ε, |{a, x1 }| ≤ ε}. Let us assume that conditions (8.3.6), (8.3.7) and (8.3.8) are fulfilled for all (x, ξ) ∈ T ∗ X : τ1 ≤ a(x, ξ) ≤ τ2 with τ1 < τ2 . In all examples we know (see Subsubsection 8.2.2.3 Analysis in the “Elliptic” Zone below ∂X is bicharacteristically flat (8.3.22)

x1 = 0, τ1 ≤ a (x, ξ  ) ≤ τ2 =⇒ ax 1 (x, ξ) = 0;

so we assume in this section that this is the case. Then (8.3.8) and (8.3.22) imply that (8.3.23) All solutions of equation (8.3.24)

z  + b(Ψt (x  , ξ  ))z = 0

1 with b(x  , ξ) = ax 1 x1 (0, x  , ξ  ) 2

are T0 -periodic for any (x  , ξ  ) ∈ T ∗ ∂X , τ1 ≤ a (x, ξ  ) ≤ τ2 .

8.3. OPERATORS WITH PERIODIC HAMILTONIAN FLOWS

773

Therefore (8.3.8) and (8.3.22) imply that (8.3.25) ρ(x, ξ) ◦ Ψt  ρ with ρ = x1 + |{a, x1 }| as τ1 ≤ a(x, ξ) ≤ τ2 . Consider zone Uε = {(x, ξ) : ρ(x, ξ)  ε}. Blowing up (x1 , ξ1 ) → ε−1 (x1 , ξ1 ) we can prove easily that after this   (8.3.26) Ψt (x, ξ) = Ψt,ε (x, ξ), Ψ(x  ,ξ ,t),ε (x1 , ξ1 ) , with Ψt,ε (x, ξ) ∼ Ψt,0 (x  , ξ  ) + (8.3.27) εn Ψt,(n) (x, ξ) n≥2

and (8.3.28)

Ψ(x  ,ξ ,t),ε (x1 , ξ1 ) ∼



εn Ψ(x  ,ξ ,t),(n) (x1 , ξ1 )

n≥1

Ψt,0

where is the Hamiltonian flow on T ∗ ∂X and Ψt,(1) is the linearized with respect to (x1 , ξ1 ) billiard flow (described by (8.3.24)). One can prove easily that Ψt,(n) and Ψ(x  ,ξ ,t),(n) are uniformly smooth. Then in the blown-up coordinates (8.3.16) holds provided ε ≥ h 2 −δ because  = ε−2 h and then we can apply all the previous arguments and derive asymptotics; the remainder estimate is as sharp as O(h2−d ) provided (in the original coordinates) 1

(8.3.29)

|∇(x,ξ) (x, ξ)| ≥ 0

as x1  ρ(x, ξ) ≤ 1 ;

x1  ρ(x, ξ) means exactly that x1 ≥ |ξ1 |. Therefore we arrive to Proposition 8.3.5. Let A be a Schr¨odinger operator with the principal symbol a(x, ξ) satisfying (8.3.6)–(8.3.8) and (8.3.22)–(8.3.23). Let the subprincipal symbol and the boundary condition be such that (8.3.29) is fulfilled. All these conditions are supposed to be fulfilled in {(x, ξ) : ρ(x, ξ) ≤ 0 }. 1 Then the contribution of the zone {(x, ξ) : h 2 −δ ≤ ρ(x, ξ) ≤ 2 } to the remainder is O(h2−d ). Since the contribution of the zone {(x, ξ) : ρ(x, ξ) ≤ ε} to the remainder 1 does not exceed C ε2 h1−d and we can take ε = h 2 −δ we conclude that the contribution of the zone {(x, ξ) : ρ(x, ξ) ≤ 2 } to the remainder is O(h2−d−δ ). One can also estimate this way the contribution of the subperiodic trajectories. Thus we arrive (details are left to the reader) to

774 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II Theorem 8.3.6. Let A be a Schr¨odinger operator with the principal symbol a(x, ξ) satisfying (8.3.6)–(8.3.8) and (8.3.22)–(8.3.23). Let the subprincipal symbol and the boundary condition be such that (8.3.29) is fulfilled. All these conditions are supposed to be fulfilled in the zone {(x, ξ) : ρ(x, ξ) ≤ 0 }. Then contribution of the zone {(x, ξ) : ρ(x, ξ) ≤ 2 } to the remainder is O(h2−d−δ ) while its contribution to the principal part of the asymptotics is given by the standard two term Weyl formula plus a correction term constructed according to Section 6.3 for symbol . Problem 8.3.7. Recover O(h2−d ) remainder estimate. This should not be extremely hard especially if there are no subperiodic trajectories but definitely worth of publishing. Example 8.3.8. Consider Laplacian h2 Δ on the standard hemisphere Sd+ := {x : |x| = 1, x1 > 0} in Rd+1 or harmonic oscillator h2 Δ+|x|2 on the standard half-space Rd+ = {x : x1 > 0}. (i) Consider a boundary operator of Example 8.3.2. Then N = 2 and (z, τ ) =  (z, τ ) +  ((z), τ ) where z = (x  , ξ  ), (z) is the antipodal point and  is defined by (8.3.18). Then one can express conditions to  as conditions to β; we leave exact statements to the reader. (ii) Consider Dirichlet or Neumann boundary conditions. Then if a perturbation is hx1 one can calculate easily that  = κρ + O(ρ2 ) where ρ is the incidence angle of trajectory and κ > 0. Then condition (8.3.29) is fulfilled. Discussion Unfortunately almost nothing is known about manifolds with the boundaries and all billiards closed; we are discussing geodesic billiards but one can ask the same questions about Hamiltonian billiards associated with the Schr¨odinger operator; in the last case the phase space S ∗ X is replaced by the portion of the phase space {τ1 ≤ a(x, ξ) ≤ τ2 }. We assume that the phase space is connected. Example 8.3.9. Consider the standard Laplacian h2 Δ on one of the following manifolds: (i) The standard hemisphere Sd+ = {x ∈ Sd |x| = 1, x1 > 0} is the most obvious example.

8.3. OPERATORS WITH PERIODIC HAMILTONIAN FLOWS

775

(ii) Starting from this let X be a connected closed manifold with all geodesics periodic and let G be a finite group of isometries of X . Then {x : j(x) = x} are surfaces in X for any j ∈ G (except j = I ); all this surfaces together break X into several cells. Let one of them be X . Unless #G = 2 ∂X can have edges and other singularities but dealing with them is easy; see Sections 11.2 and 11.3. (iii) In particular, consider Zoll-Tannery manifolds with the rotational symmetry X := {x = (x1 , x  ) ∈ [−L1 , L2 ] × Sd−1 } with g 11 = 1, g jk = f (x1 )2 δjk for j = 1, ... , d, k = 2, ... , d and function f such that all geodesics are periodic8) . Then we can use j(x) = (x1 , j  (x  )) where j  ∈ G  where G  is a finite group of isometries of Sd−1 . (iv) Further, in the framework of (iii) if f is an even function9) , we can also use j(x) = (−x1 , x  ) alone or in conjugation with G  . Example 8.3.10. Consider Schr¨odinger operator h2 Δ + V (|x|) where (i) We consider harmonic oscillator V (r ) = r 2 . (ii) We consider Coulomb Hamiltonian V (r ) = −r −1 on the negative energy levels. Let G be a finite group of the isometries of Sd−1 . These isometries are extended to rotational isometries of Rd and {x : j(x) = x} are surfaces in Rd for any j ∈ Γ (except j = I ); all this surfaces together break Rd into several cells. Let one of them be X . Unless #G = 2 ∂X can have edges and other singularities but dealing with them is easy; see Sections 11.2 and 11.3. Problem 8.3.11. Either prove that if all billiards are closed then ∂X must be bicharacteristically flat or to construct counter-example (i) For Laplacians only. 8) These manifolds are isometric (up to a factor) to a sphere Sd with the metrics (r + h(cos(s)))2 ds 2 + sin2 (s) dθ2 with s ∈ [0, π], r = p/q, p, q ∈ N, gcd(p, q) = 1 and odd function h : (−1, 1) → (−r , r ). Then the length of equator is 2π, the length of meridian 2πr and the length of general geodesics 2πp. See A. Besse [1], Theorem 4.13. 9) Which happens only as h = 0 in footnote 8) .

776 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II (ii) For general Schr¨odinger operators without magnetic field. (iii) For general Schr¨odinger operators with magnetic field.

8.3.3

Branching Hamiltonian Flow with “Scattering”

Let us consider two manifolds X1 and X2 glued along the connected component Y of the boundary. We consider Schr¨odinger operators Aj on Xj and these operators are intertwined through the boundary conditions. We assume that Hamiltonian flows have simple (no branching) reflections on each of them, so branching comes from “reflection-refraction”. Further we are interested in the case when Hamiltonian flow on X1 is periodic while on X2 majority of trajectories are not periodic and, moreover, when majority of the branching billiards in X1 ∪ X2 containing trajectories in X2 are not periodic. Analysis in X2 Since we want to consider examples in which we can indeed analyze the longterm behavior of billiards rather than just postulate it, we will make rather restrictive assumptions about billiards. First we assume that billiard flow on X1 satisfy all assumptions of the previous Subsection 8.3.2; in particular (8.3.30) Φ1,τ (z)N = I 10) . We also assume that Φ1,τ and Φ2,τ commute (8.3.31)

Φ1,τ ◦ Φ2,τ = Φ2,τ ◦ Φ1,τ .

Then for points of Σ2,τ periodicity properties of the branching Hamiltonian flow with reflections Ψt coincide with those of Ψ2,t ,11) . We define Φj,τ in the following way: first, we lift z ∈ T ∗ Y to z ∈ T ∗ Xj |Y ∩ Σj,τ with {aj , x1 }(z) > 0, then launch a trajectory forward until it hits the boundary at (y  , η) ∈ T ∗ Xj |Y and, finally, project it back to T ∗ Y . 11) Indeed, it is sufficient to analyze such properties of Φ2,τ and a corresponding branching map Φτ which is Φk2,τ Φl2,τ due to (8.3.30), (8.3.31). But if Φk2,τ Φl2,τ (z) = z rl then Φrl2,τ (z) = Φrk 2,τ Φ2,τ (z) = z. 10)

8.3. OPERATORS WITH PERIODIC HAMILTONIAN FLOWS

777

Now we need to impose condition to the boundary operator in addition to {A, B} being self-adjoint. Namely, consider a point z = (x  , ξ  ) ∈ T ∗ Y , and consider an auxiliary problem (8.3.32)

aj (x  , D1 , ξ  )uj = 0 

j = 1, 2,



ðb(x , D1 , ξ )u = 0

(8.3.33)

(where u = (u1 , u2 )) and consider solutions which are combinations of the exponents e iλj x1 where for each j either Im λj > 0 or Im λj = 0 and {aj , x1 }(x  , λj , ξ  ) > 0 12) . Denote by Λ the set of points for which such non-trivial solutions exist. We assume that mesT ∗ Y (Λ) = 0.

(8.3.34)

Then we arrive to the following Proposition 8.3.12. Let in the described setup conditions (8.3.30), (8.3.31) and (8.3.34) be fulfilled. Let Q1 , Q2 be h-pseudodifferential operators with the symbols supported in T ∗ X2 and (8.3.35)

μ2,τ (Π2,τ ∩ supp(q1 ) ∩ supp(q2 )) = 0

where Π2,τ is the set of point (y , η) ∈ Σ2,τ periodic with respect to Ψ2,t . Then for Γ(Q1x e(., ., τ ) tQ1y ) the standard Weyl two-term asymptotics holds with the remainder estimate o(h1−d ). Remark 8.3.13. (i) Note that μj,τ (Λj,τ ) = 0

(8.3.36)

where Λj,τ = Λj,τ ∪ Λj,τ , Λj,τ is the set of dead-end points of billiard (not generalized billiard) flow and Λj,τ is the set of points (y , η) ∈ Σj,τ such that ιj Ψj,t (y , η) ∈ Ξ3−j,τ where Ξk,τ = ιk {(x, ξ) ∈ Σk,τ , {ak , x1 } = 0}. (ii) Note that the grows properties of the branching Hamiltonian flow with reflections Ψt coincide with those of Ψ2,t as long as we ensure that along billiards we do not approach to glancing rays on X1 : (8.3.37) {a1 , x1 }(x, ξ) ≥ hδ1

∀(x, ξ) ∈ ι−1 1 ι2 Ψ2,t (y , η) ∩ Σ1,τ

∀(y , η) ∈ supp(q2 ) ∀t : ±t ∈ [0, T ] Ψ2,t (y , η) ∈ T ∗ X2 |Y 12) As we assume that x1 > 0 on both X1 and X2 and x1 = 0 on Y , it means that we consider solutions which either are of boundary-layer type or outgoing.

778 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II where T = T (h). Then in the framework of Section 7.4 one can recover similar results: namely that two-term Weyl asymptotics holds with the remainder estimate O(T (h)−1 h1−d ) which is usually O(h1−d | log h|−1 ) or O(h1−d+δ ) (depending on the growth and non-periodicity conditions to Ψ2,t . Exact statement and proof are left to the reader (which is a relatively easy problem). (iii) In Statement (ii) we need also to assume a more sharp version of (8.3.34): namely if we consider approximate solutions to (8.3.32)–(8.3.33) with precision ε we get set Λτ ,ε and we need to impose either condition mes(Λτ ,ε ) = o(εδ ) or condition mes(Λτ ,ε ) = o(| log ε|−1 ) as ε → +0. (iv) In Subsubsection 7.4.3.6 Non-Periodicity Condition for Branching Billiards we did not consider points z ∈ T ∗ Y where some of operators aj are elliptic on ι−1 z rather than hyperbolic. If we want to consider such points we need to assume condition (8.3.34) or more sharp version of it described in Statement (iii). (v) Property (8.3.37) is guaranteed if either a2 is cylindrically symmetric on T ∗ X2 or if there no complete internal reflection for billiards from X2 at all: (8.3.38) |{a1 , x1 }(x, ξ)| ≥ |{a2 , x1 }(x, η)|l ∀(x, ξ) ∈ T ∗ X1 |Y ∩ Σ1,τ ∀(x, η) ∈ T ∗ X2 |Y ∩ Σ2,τ : ι1 (x, ξ) = ι2 (x, η). Analysis in X1 However if operators Q1 and Q2 have symbols supported in X1 situation is very different: trajectories of a1 are periodic but the typical closed branching billiard is the one which does not contain segments of Hamiltonian trajectories of a2 . More precisely, due to conditions (8.3.39)

μ2,τ (Π2,τ ) = 0

and (8.3.36) we conclude that (8.3.40)

μ1,τ (Π1,τ ) = 0

where Π1,τ is the set of points z ∈ Σ1,τ such that there exists a billiard trajectory of Ψt starting and ending in z and which is not entirely billiard of Ψ1,t .

8.3. OPERATORS WITH PERIODIC HAMILTONIAN FLOWS

779

Then for arbitrarily large T > 0 and arbitrarily small ε > 0 there exists a set U = UT ,ε ⊂ T ∗ X such that (8.3.41)

μ1,τ (Σ1,τ \ U) ≤ ε,

(8.3.42)

dist(U , Π1,τ ,T ∪ Λ1,τ ,T ) ≥ γ,

(8.3.43)

dist(Y , πx U ) ≥ γ

with γ = γ(T , ε) > 0 were Π1,τ ,T s the set of points z ∈ Σ1,τ such that there exists a billiard trajectory of Ψt of the length not exceeding T , starting and ending in z and which is not entirely billiard of Ψ1,t and Λ1,τ ,T is the set of points z ∈ Σ1,τ such that billiard trajectory of a1 hits the boundary at point of Λ2,T ,τ . As usual, these sets increase as T increases, their unions (with respect to T ) coincide with Π1,τ and Λ1,τ respectively and Λ1,τ ,T and Λ1,τ ,T ∩ Π1,τ ,T are closed sets. Therefore there exists operator Q = QT ,ε such that supp(q) ⊂ U and   (8.3.44) |Γ Q1x e(., ., τ ) t((I − Q)Q2 )y − κ0 h−d − κ1 h1−d | ≤ C0 εh1−d with κj = κj,Q1 ,Q2 ,Q (τ ). This estimate holds for any fixed  τ satisfying (8.3.39). Therefore we need to consider Γ Q1x e(., ., τ ) tQ2y with supp(q2 ) ⊂ U . We can assume without any loss of the generality that (8.3.42), (8.3.43) are fulfilled for all τ . Then for  ≤ |t| ≤ T     (8.3.45) Γ Q1x U(., ., t) tQ2y ≡ Γ Q1x Z (., ., t) tQ2y where as ±t > 0 Z (x, y , t) is the Schwartz kernel of “the approximate propagation with the scattering semigroup” Z (t) constructed in the following way (for a sake of simplicity in notations we consider only t > 0: (i) We represent Q2 = Q2,1 + ... + Q2,N where symbols Q2,ν have small supports; (ii) For each ν we select 0 = t0,ν < ... < tM,ν = t in such a way that for any z ∈ supp(q2,ν ) Ψt  (z) is disjoint from Y as t  = tj,ν and Ψt  (z) hits Y no more than once as tj,ν ≤ t  ≤ tj+1,ν ; (iii) We set (8.3.46)

Z (t)Q2,ν =



 0≤j≤M−1

ψe ih

−1 (t

j+1,ν −tj,ν )f (A)



· Q2,ν

780 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II where ψ ∈ C∞ is supported in {x ∈ X1 , dist(x, Y ) ≥ γ  } and equals 1 in {x ∈ X1 , dist(x, Y ) ≥ 2γ  }. Then (8.3.47)

Z (t1 + t2 )Q2 ≡ Z (t2 )Z (t1 )Q2

as ± t1 > 0, ±t2 > 0

and (8.3.48)

Z (t)∗ ≡ Z (−t)

and due to periodicity of Ψ1,t with period T0 Z (±T0 ) are h-pseudodifferential operators. Remark 8.3.14. In contrast to what we had before they are not necessarily unitary because reflection coefficients are not unitary anymore. More precisely, / ι2 Σ2,τ 13) the reflection (i) If Ψ1,t hits T ∗ X |Y at points (x  , ξ) with (x  , ξ  ) ∈     coefficient κ11 (x , ξ , τ ) still satisfies |κ11 (x , ξ , τ )| = 1. (ii) However if Ψ1,t hits T ∗ X |Y at points (x , ξ) with (x  ,ξ  ) ∈ ι2 Σ2,τ then the reflection-refraction matrix κ(x  , ξ  , τ ) = κjk (x  , ξ  , τ ) j,k=1,2 is unitary but the reflection coefficient κ11 (x  , ξ  , τ ) satisfies only |κ11 (x  , ξ  , τ )| ≤ 1 and the equality means exactly that κ12 (x  , ξ  , τ ) = κ21 (x  , ξ  , τ ) = 0 (then |κ22 (x  , ξ  , τ )| = 1 as well). So the principal symbol of Z (T0 ) is the product of e i0 (x,ξ) where 0 (x, ξ) is given by the usual formula for manifolds without boundary (but with trajectories replaced by billiards) and also of reflection coefficients coefficients in the point of reflection. If the reflection coefficients do not vanish we can rewrite the answer in the form (8.3.49)

Z (T0 ) ≡ e iεh

−1 L

with h-pseudodifferential operator L, L+L∗ ≤ 0 and ε = h (but in calculations below we are not assuming this). We will use this formula even if the coefficients vanish: in this case we simply formally allow Im  = +∞. 13)

I.e. at the point of the complete internal reflection.

8.3. OPERATORS WITH PERIODIC HAMILTONIAN FLOWS

781

Then as χ¯ ∈ C0∞ [ − 1, 1 − ]   (8.3.50) Ft→h−1 τ χ¯T0 (t − nT0 )Γ Q1x Z (., ., t) tQ2y =

    .¯ τ + O h2−d , h1−d J(h, ε, n, τ ) · χ h

with  (8.3.51)

J(h, ε, n, τ ) = (2π)1−d

e ih

−1 ((ε Re −τ )n+ε Im |n|)

dμ1,τ .

Σ1,τ

We can see easily that (8.3.52) J(h, ε, n, τ ) = o(1) as n → ∞ provided   (8.3.53) μτ (x, ξ) ∈ Στ : Im (x, ξ) = ∇ Re (x, ξ) = 0 = 0. On the other hand, note that (8.3.54)

n∈Z\0

e ih

−1 (nα−|n|β)

=

2e −β cos α − 2e −2β 1 − 2e −β cos α + e −2β

as β > 0

and formally we can extend this for β = 0 as well. Then after summation with respect to n for β = 0 we get exactly ∂α Υ(α) with Υ defined by (6.2.68) and for β > 0 we get that the expression (8.3.54) −1  is equal to ∂β Υ ∗α (2π)−1 G (α, β) where G (α, β) = 2β α2 + β 2 and as usual ∗α means a convolution with respect to α. Note that (8.3.55) Function (8.3.56)

Υ(α, β) := Υ ∗α G (α, β)

is harmonic function as β > 0, coincides with Υ(α) as β = 0, is o(1) as β → +∞ and is periodic with respect to α. Therefore taking in account results of this subsection we arrive to the following

782 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II Theorem 8.3.15. Let in the described setup conditions (8.3.30), (8.3.31), (8.3.34), (8.3.39) and (8.3.53) be fulfilled. Then asymptotics (8.3.57) Γe(., ., τ ) = κ0 (τ )h−d +       Υ h−1 (ε Re  − τ ), −h−1 ε Im  dμτ dμτ + o(h1−d ) κ1 (τ ) + Στ

holds. Problem 8.3.16. Calculate all coefficients κjk for two Schr¨odinger operators A1 and A2 . We will solve this problem in the very special case: Example 8.3.17. Assume that (at the given point z ∈ T ∗ Y ) aj = cj2 |ξ|2 and the boundary condition is (8.3.58)

u2 = αu1 ,

(8.3.59)

D1 u2 = −βD1 u1

(recall that x1 > 0 in both X1 , X2 ; in more standard notations of x1 < 0 in X2 one needs to skip “−”). Then {A, B} to be self-adjoint requires αβ † = c12 c2−2 . We do not consider gliding points since their measure is 0. (i) Consider point ξ  with c12 |ξ  |2 < τ < c22 |ξ  |2 . One can prove easily that (8.3.60) with (8.3.61)

κ11 (ξ  , τ ) = e 2iϕ   1  − 1  ϕ(ξ  , τ ) = arctan |β|−2 c2−2 c12 c22 |ξ  |2 − τ 2 τ − c12 |ξ  | 2

and therefore assumption (8.3.53) is fulfilled provided c1 = c2 , cj , α, β are symmetric (i.e. cj  = cj ) and the subprincipal symbol is 0. If c1 > c2 this case does not appear. (ii) Consider point ξ  with cj2 |ξ  |2 < τ for both j = 1, 2. One can prove easily that   −1 (8.3.62) κ11 (ξ  , τ ) = ω − 1 ω + 1 with  1  − 1 ω(ξ  , τ ) = |β|−2 c2−2 c12 τ − c22 |ξ  |2 2 τ − c12 |ξ  | 2 (8.3.63) and |κ11 | < 1 so assumption (8.3.53) is fulfilled.

8.3. OPERATORS WITH PERIODIC HAMILTONIAN FLOWS

783

The following problems seem to be rather straightforward but not easy and worth publishing: Problem 8.3.18. Under stronger conditions to Ψ2,t and  (see (8.3.20)) prove remainder estimate O(h1−d+δ ). Problem 8.3.19. Prove the same results as Hamiltonian billiards of a1 are assumed to be periodic only on one energy level τ rather than on neighboring levels.

8.3.4

Two Periodic Flows

Now we consider the case of both flows Ψ1t and Ψ2t generated by f (a1 ) and f (a2 ) both satisfying (8.3.31) and (8.3.30) which becomes N

(8.3.64) Φj,τj = I for j = 1, 2 where N1 and N2 are not necessarily equal. Remark 8.3.20. (i) Here as in the previous Subsection 8.3.3 we can consider several manifolds attached to one another: Xj is a connected manifold, ∂Xj = Yj ∪ Yj with disjoint Yj ∪ Yj = ∅ and Yj and Yj+1 are glued together (j = 1, ... , m) while Y1 and Ym may be or may be not or may be only partially glued together. (ii) Further, we can consider cases when along (some part of) Y more than two manifolds are glued together etc We leave such generalization to the reader. Examples and Discussion Let us start from examples: Example 8.3.21. (i) In the framework of Example 8.3.9(ii) we again consider partition of X into cells and on each cell Xj we define operator as either Laplacian h2 cj2 Δ or more generally as a Schr¨odinger operator h2 cj2 Δ − Ej . (ii) In the framework of Example 8.3.9(iv) we consider X1 = {(x1 , x2 ) ∈ (−1, 0) × (0, 2π)} and X1 = {(x1 , x2 ) ∈ (0, 1) × (0, 2π)} and we glue them along x1 = 0 identifying (+0, x2 ) with (−0, x2 ). Observe that Nj = pj . (iii) In Example 8.3.9(iv) with p1 = 1 we have N1 = 1 and we can glue it with anything else as (8.3.31) is fulfilled automatically then.

784 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II Example 8.3.22. Consider two Schr¨odinger operators in Rd+ = R+ × Rd−1 (8.3.65)

 1  Aj = ωj2 h2 Δ + |x|2 − Ej 2

intertwined through boundary conditions. Our idea was to make all flows periodic with the common period. It works with Example 8.3.22 where Tj = 2πωj−2 . Indeed, we want to find asymptotics of N− h (0) which is the maximal dimension of the negative subspace of the 1 1 operator A which does not change if we replace A by J − 2 AJ − 2 with the positive self-adjoint operator J. Picking up J = J(x) equal ωj2 in Xj we arrive to the case of ωj = 1 and Tj = 2π. Therefore in this example one can make periods equal T1 = T2 . However this approach does not seem work with the Example 8.3.21 (unless some unnatural conditions to ωj , Ej are imposed). The main obstacle 1 here is that multiplying operator by J − 2 from both sides and calculating f (A) do not play well together. Reduction to the Boundary First of all observe that the contribution to the remainder of the domain {z, dist(z, T ∗ Y ) ≤ ε0 } does not exceed C0 20 h1−d and therefore one needs to consider the contribution of {z, dist(z, T ∗ Y ) ≥ ε0 } with arbitrarily small but fixed ε0 14) . In this zone one can consider U(x, y , t) as the solution of the Cauchy problem with respect to x1 (or y1 ) with the data at {x1 = 0} (or {y1 = 0}). Rewriting therefore U = Rx U tRy where R is an operator resolving this Cauchy problem is given by an oscillatory integral we can rewrite     (8.3.66) Γ U(., ., t) tQy ≡ Γ Q(x  , hDx , hDt , h)V with h-pseudodifferential operator R and V = ðx,m−1 ðy ,m−1 U the Cauchy data for U (assuming that m is an order of A). One can see easily that as Q is an operator with the symbol supported in T ∗ Xj , the principal symbol of Q at (x  , ξ, τ ) is defined as an averaging of Q along Ψjt (zj ) with t ∈ [0, tj (zj )] where zj = ι−1 (x  , ξ  ) ∩ Σjτ and tj (zj ) is the time of the next hit of the boundary. 14)

If our target is o(h1−d ) remainder estimate.

8.3. OPERATORS WITH PERIODIC HAMILTONIAN FLOWS

785 −1

Then instead of continuous family of Fourier integral operators e ih tA we can consider a discrete family F n defined by the following way: consider vj = ðm−1 uj = (vj− , vj+ ) with vj∓ corresponding to incoming and outgoing solutions. We have two types of trajectories: those in Σjτ which hit T ∗ X |Y at points elliptic for (a3−j − τ ) where j = 1, 2 and those which hit T ∗ X |Y at points hyperbolic for (ak − τ ) for both k = 1, 2. The analysis of the former is of no different from what we have seen in the previous subsection: we just consider Ψjt and construct the (real-valued) symbol  as long as we assume that (x  , ξ) ∈ / Λ where Λ is defined in the paragraph preceding (8.3.34). The analysis of the latter is more interesting. Here as before we have a matrix of reflection-refraction (κjk )j,k=1,2 which is a symbol defined on U × (τ − ε , τ + ε ) where U is a a zone described above. We need to construct a transformation along trajectories matrix L but it will be of the different type. Consider generic point z ∈ T ∗ Y , and collection of {Φj1,τ Φk2,τ }i=0,...,N1 −1;k=0,...N2 −1 . Not all points here are necessarily distinct; let us numerate distinct points by greek indices: zα with α ∈ {1, ... , N} 15) . Let φj : {1, ... , N} → {1, ... , N} be permutations: β = φj (α) ⇐⇒ zβ = Φj,τ (zα ). We consider 2N-vectors vjα with norms (8.3.67)

v  =





|vαj |2 · |{aj , x1 }(zα )|

 12

.

α=1,...,N j=1,2

Then we introduce a sparse matrix 2N × 2N-matrix L in the following way: (8.3.68)

Lβk,αj = |{aj , x1 }(zβ )|− 2 |{aj , x1 }(zα )| 2 e iαj 1

1

if k = j and β = φj (α) and 0 otherwise; here αj is an integral of the subprincipal symbol along trajectory of Ψjt from zα to zβ . Obviously matrix L is unitary16) . Finally let us consider a matrix M = κL ˜ where κ ˜ is a matrix κ raised to 2N × 2N-matrix: κ ˜ βk,αj = κjk (zα ) as β = α and 0 otherwise. Matrix M shows how data on Y is transformed after a single “step” (propagation followed by reflection-refraction). One can see easily that (8.3.69) Matrices κ ˜ and M are unitary16) . 15) 16)

Then N divides N1 N2 and is divisible by N1 and N2 . In the norm (8.3.67).

786 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II Analysis of the Evolution One can prove easily that modulo OT (h2−d )   (8.3.70) Ft→h−1 τ χT (t)Γ Q(x  , hDx , hDt , h)V ≡  (2πh)1−d χT (Tj1 + ... + TjK )



K =1 (j1 ,...,jK )∈{1,2}K (α1 ,...,αK )∈{1,...,N}K

Mα1 j1 ,α2 j2 Mα2 j2 ,α3 j3 · · · MαK −1 jK −1 ,αK jK e −ih

−1 τ (T

j1 +...+TjN )

QαK jK ,α1 j1 dx  dξ 

where χ ∈ C0∞ is supported in [ 12 , 1] and symbols Mjk and Q and Tj are calculated at (x  , ξ  , τ ) 17) and since we consider only trajectories which are periodic on the energy levels (close to) τ in this sum QαK jK ,α1 j1 = 0 unless αK = α 1 . One can rewrite terms with equal K as  (2πh)

1−d

χ(λ)e ˆ

iT −1 λ(Tj1 +...+TjK )







j=(j1 ,...,jK )∈{1,2}K (α1 ,...,αK )∈{1,...,N}K

Mα1 j1 ,α2 j2 Mα2 j2 ,α3 j3 · · · MαK −1 jK −1 ,αK jK e −ih

−1 τ (T

j1 +...+TjK )

QαK jK ,α1 j1 dx  dξ  dλ

which in turn equals  (2πh)

(8.3.71)

1−d

 χ(λ) ˆ

  tr S(λ, τ , x  , ξ  )K Q dx  dξ  dλ

where S is 2N × 2N unitary16) matrix   −1 −1 S(λT −1 , τ , x  , ξ  ) = e i(λT +τ h )Tj Mβk,αj βk,αj . (8.3.72) Consider eigenvalues of this matrix; as T → ∞ they tend to eigenvalues of the same matrix with λ = 0:   −1 (8.3.73) S(0, τ , x  , ξ  ) = e iτ h Tj Mjk j,k=0,...,L−1 . Therefore if (8.3.74) 17)

mes



   (x  , ξ  ) : ρ ∈ Spec S(0, 0, x  , ξ  ) = 0)

∀ρ ∈ R

Where Tj = Tj∗ (τ )/Nj is the time to run one leg and Tj∗ is a full period.

8.3. OPERATORS WITH PERIODIC HAMILTONIAN FLOWS

787

then this term (8.3.73) is o(h1−d ) and therefore we estimated expression (8.3.70) by o(Th1−d ) (as τ = o(1)) which in the end of the day returns remainder estimate o(h1−d ) with the Tauberian main part. Consider correction to the main part; so we integrate (8.3.70) with χ = 1 from τ = −∞ to τ = τ resulting in (after multiplication by h−1 and modulo o(h1−d )) after taking trace we get (8.3.75) Ω(τ ) := (2πh)1−d







K =1 j=(j1 ,...,jK )∈{1,2}K (α1 ,...,αK )∈{1,...,N}K −1

(Tj1 + ... + TjK )−1 e −ih τ (Tj1 +...+TjK )   × tr Mα1 j1 ,α2 j2 Mα2 j2 ,α3 j3 · · · MαK −1 jK −1 ,αK jK QαK jK ,α1 j1 dx  dξ  . Therefore we have proven Theorem 8.3.23. Consider two Schr¨odinger operators A1 and A2 with all periodic trajectories on levels close to τ , satisfying (8.3.30) and (8.3.31). Further, let (8.3.34) and (8.3.74) be fulfilled. Then asymptotics   (8.3.76) Nh (τ ) = κ0 (τ )h−d + κ1 (τ ) + Ω(τ ) h1−d + o(h1−d ) holds as h → +0 and τ = o(1) with the standard coefficients κ0 , κ1 and Ω(τ ) defined by (8.3.75). Example 8.3.24. Consider Example 8.3.17 in the new conditions to Hamiltonian flows. We can use results of that example immediately to treat billiards with complete internal reflections (thus non-branching). To treat branching billiards we need to calculate eigenvalues of the matrix M. If Φ1,τ = Φ2,τ and therefore N1 = N2 it is easy as M = C ⊗ κ where C is a cyclic matrix of order M and condition (8.3.74) for M is equivalent to the same condition for κ. It follows from calculations of Example 8.3.17 that κ22 = −κ11 ; thus we arrive to  (8.3.77) λ = e ±iϕ , ϕ = arccos κ11 = arccos (ω − 1)(ω + 1)−1 ; then obviously (8.3.74) is fulfilled and asymptotics (8.3.76) holds. Problem 8.3.25. (i) In the framework of Example 8.3.24 derive formula for Ω through function Υ defined by (6.2.68).

788 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II (ii) In the framework of Example 8.3.24 consider the case when Φ1,τ = Φ2,τ is not fulfilled. In this case calculation of eigenvalues of M could be really challenging. Remark 8.3.26. On the contrary, assume that condition (8.3.74) is not fulfilled. Then there will be eigenvalues of high multiplicity or clusters of eigenvalues located in o(h)-vicinities of solutions τ of equation det(M −1) = 0. Assuming that Φ1,τ = Φ2,τ one can rewrite this equation as     1 1 (8.3.78) cos −ϕ + τ h−1 (T1 + T2 ) = cos α · cos −ψ + τ h−1 (T1 − T2 ) 2 2   i(φ+ψ) e cos α e i(φ+χ) sin α as κ = which is the general form of the i(φ−χ) −e sin α e i(φ−ψ) cos α unitary matrix. The following problems seem to be a difficult one and worth of publication: Problem 8.3.27. Prove the same results as flows Ψj,t are assumed to be periodic only on one energy level τ rather than on neighboring levels. Problem 8.3.28. Consider two manifolds satisfying the same assumptions as before albeit with the measure 0 of periodic billiards on each of them. Assume that Φp1,τ Φq2,τ = I for some p, q ∈ N (recall that we assume that Φ1,τ and Φ2,τ commute. Results should be similar to those of Subsection 8.3.3. Example 8.3.29. Consider manifolds X1 and X2 with metrics ri2 ds 2 +sin2 (s) dθ2 and s ∈ [o, π/2] and s ∈ [π/2, pi] respectively albeit with r1 , r2 which are irrational but such that q1 r1 + q2 r2 = p with q1 , q2 , p ∈ N. On the contrary, the following problem seems to be relatively easy: Problem 8.3.30. (i) Prove the same results as before when there are more than two manifolds Xj j = 1, ... , m with the all billiards periodic. (ii) Incorporate in this scheme Problem 8.3.28. (iii) Extend these results to the case when there are also manifolds Xj j = m + 1, ... , m with almost all billiards non-periodic.

8.4. SPECTRAL ASYMPTOTICS ON SUBSPACES

8.4

789

Spectral Asymptotics on Subspaces

So far we have considered operators with domain which is dense in L2 (X , H). We need to consider also spectral problems when eigenfunctions are restricted to a certain subspace H ⊂ L2 (X , H) such that orthogonal projector Π to H is either a pseudodifferential operator (inside the domain), or a singular Green operator or sum of pseudodifferential operator with transmission property and singular Green operator.

8.4.1

Examples and Discussion

Let X be a connected Riemannian manifold (possibly) with the boundary (p) ∂X , Λx the space of p-forms at point x, Λ(p) = Λ(p) (X ) be the corresponding bundle over X . Let d : C∞ (X , Λ(p) ) → C∞ (X , Λ(p+1) ) be an operator of external differentiation, d∗ := ∗d∗ : C∞ (X , Λ(p) ) → C∞ (X , Λ(p−1) ) an operator formally adjoint to it. Indeed, due to the Riemannian structure we have an antilinear operator ∗ : C∞ (X , Λ(p) )  u → ∗u ∈ C∞ (X , Λ(d−p) ) which provides an inner product  1  (u, v ) = ς X u ∧ ∗v and a norm ||u|| = X u ∧ ∗u 2 as ∗∗ = 1 18) . Now we can consider operator Δ = dd∗ +d∗ d with the domain H 2 (X , Λ(p) ); at this moment we do not specify boundary conditions. Example 8.4.1. (i) Consider subspace (8.4.1)

H := {u ∈ L2 (X , Λ(p) ) : d∗ u = 0}

with p < d 19) . On this subspace consider quadratic form (8.4.2)

Q(u) = (du, du) + (d∗ u, d∗ u)

with domain (8.4.3) D(Q) = {u ∈ L2 (X , Λ(p) ) : du ∈ L2 (X , Λ(p+1) ), d∗ u ∈ L2 (X , Λ(p−1) )} 18) 19)

In contrast to most of textbooks where ∗∗ = (−1)p(d−p−1) . Then dim H = ∞; as d = p dim H = 1.

790 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II and an operator A = Δ generated by this form; it is given by a desired differential expression Δ = d∗ d = d∗ d + dd∗

(8.4.4) with the domain (8.4.5)

D(A) = {u ∈ H ∩ H 2 (X , Λ(p) ), ðn du = 0}

where ðn = ð − ðt , ðt is an operator of the restriction to ∂X ; so if w = w0 + w1 ∧ dx1 where uk are (p − k)-forms with respect to x  depending on x1 , then (8.4.6) (8.4.7)

ðw = w0 |x1 =0 + w1 |x1 =0 ∧ dx1 , ðt w = w0 |x1 =0 ,

ðn w = w1 |x1 =0 ∧ dx1 ;

We call ðt w and ðn w tangential and normal components of ðw respectively. However unless either p = 0 or ∂X = ∅ the quadratic form and operator have infinite-dimensional kernel (8.4.8)

K = {u ∈ L2 (X , Λ(p) ) : du = d∗ u = 0}

and we need to restrict both form and operator to (8.4.9)

H0 = H - K = {u ∈ L2 (X , Λ(p) ) : du = 0, ðn u = 0}.

(ii) We can add to (8.4.3) condition (8.4.10)

ðt u = 0

and get operator A = Δ with domain (8.4.11)

D(A) = {u ∈ H ∩ H 2 (X , Λ(p) ), ðt u = 0}

(iii) Using duality operator ∗ we can simultaneously permute d and d∗ , ðt and ðn , p = 0 and p = d. (iv) Let 0 < p < d. Then we can consider quadratic form (8.4.2) on H1 := L2 (X , Λ(p) ) - K and get an operator A = Δ with the domain (8.4.12)

D(A) = {u ∈ H ∩ H 2 (X , Λ(p) ), ðt du = ðn d ∗ u = 0}

8.4. SPECTRAL ASYMPTOTICS ON SUBSPACES

791

Remark 8.4.2. (i) Note that the boundary conditions for A are equivalent to equality (8.4.13)

(du, dv ) = (d∗ du, v )

∀v ∈ D(Q).

(ii) Also note that in Example 8.4.1(i),(ii) there is a “hidden” boundary condition ð t d∗ u = 0

(8.4.14)

due to d∗ u = 0; with the boundary conditions (8.4.5), (8.4.14) or (8.4.10), (8.4.14) operator A = Δ given by differential equation (8.4.4) is self-adjoint; (iii) One can see easily that (8.4.15) AB and Π commute20) . We leave to the reader Problem 8.4.3. Describe operators Π, Π0 and Π1 (projectors onto H, H0 and H1 in example 8.4.1;   u 2 (p) 2 (p−1) ) Example 8.4.4. Consider operator in L (X , Λ ) ⊕ L (X , Λ v   0 αdβ A= (8.4.16) β † d∗ α † 0 restricted to kernel H of operator (8.4.17)

R=



dα−1 d∗ β †−1



under boundary condition (8.4.18)

ðt βv = 0

or, alternatively, (8.4.19) 20)

ðn α † u = 0

Which means that spectral projectors of AB and Π commute.

792 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II (p)

(p)

(p−1)

(p−1)

where α : Λx → Λx , β : Λx → Λx are non-degenerate matrices smoothly depending on x. Then we can make the same conclusions as before with the corresponding subspace H. Maxwell equations in anisotropic media (crystals) fit the example above 1 1 as α = ε− 2 , β = μ− 2 where ε, μ are positive Hermitian matrices; for scalar such matrices we get Maxwell equations in isotropic media. Example 8.4.5. (i) Consider H = {u : Δu = 0} ⊂ L2 (X , Λ(p) ) and operator generated by the quadratic form Q(u) = ∇u2 . Expressing u = Rv 1 with v = ðu we get quadratic form Q (v ) = Q(Rv ) on H − 2 (Y , E) where (p) (p−1) E = Λ (Y ) ⊕ Λ (Y ), Y = ∂X . Then (8.4.20)

−1

−1

−1

Q (v ) = (BΔY 2 v , ΔY 2 v )Y ,

u2 = ΔY 2 2Y

with the pseudodifferential operator B on ∂X , acting on C∞ (Y , E) and self-adjoint in H = L2 (Y , E). Thus the problem is reduced to study of spectral properties of B. (ii) Consider H = {u : du = d∗ u = 0} ⊂ L2 (X , Λ(p) ) and operator generated by quadratic form Q(u). Expressing u = Rv with v = ðu and the same operator R as before, we get 1

(8.4.21)

1

H = {w ∈ L2 (Y , E) : dRΔY2 w = d∗ RΔY2 w = 0}.

Thus the problem is reduced to study of spectral properties of B  defined by a quadratic form Q (v ) on H . (iii) In (i) and (ii) we can also impose certain conditions to v thus further reducing the space H . We leave to the reader the following Problem 8.4.6. (i) Describe projector Π in L2 (X , Λ(p) ) which is a singular Green operator and operator B in Example 8.4.5(i). (ii) Describe projector Π in L2 (X , Λ(p) ) which is a singular Green operator, 1

1

projector Π = ΔY2 R ∗ ΠRΔY2 which is a pseudodifferential operator on Y and B  in Example 8.4.5(ii).

8.4. SPECTRAL ASYMPTOTICS ON SUBSPACES

8.4.2

793

Analysis away from the Boubdary

If X is a closed manifold or at least B(0, 1) ⊂ X there is an already developed theory: projector on H we are interested in is (locally) a pseudodifferential  operator and asymptotics of Γ Πx e(., ., τ ) we already defined. However there is a twist: Theorem 8.4.7. Let Π be an h-pseudodifferential projector commuting with A. Then all results of Chapters 4–6 remain true if we redefine microhyperbolicity according to Definition 8.4.8 below (with r = I − π) and for sharp results consider only trajectories on manifold (8.4.22)

Στ := {Ker a(x, ξ) ∩ Ker(1 − π(x, ξ)) = {0}}

where π is the principal symbol of Π. Definition 8.4.8. We say that operator A is microhyperbolic at point z in direction  at energy level τ under restriction Ru = 0 if (8.4.23)

(a(z))v , v  ≥ v 2 − C (a(z) − τ )v 2 − C r (z)v 2

∀v

with small enough constant  > 0; here R is a pseudodifferential operator (not necessary a projector) with the principal symbol r (x, ξ) such that Ker r (x, ξ) is invariant with respect to a(x, ξ). Proof of Theorem 8.4.7. The easiest way to recover all local results is to note that one can find pseudodifferential D × D -matrix operator E (where D = rank π and A is D × D-matrix operator) such that (I − Π)u ≡ 0 near z ∈ T ∗ X if and only if there exists v such that u ≡ Ev near z. Then the problem locally becomes D × D -matrix and microhyperbolic in the ordinary sense. Here D is constant on each connected component of modified Στ . The easy details are left to the reader. Sharper results propagation are also easy: we just remember that condition (I −Π)u = 0 makes some points of old Στ elliptic and thus removes them from propagation. Since so far we did not discuss in details propagation near boundary we simply announce that at this moment all propagation near boundary remains the same as without restriction (but at moment when the boundary is not considered Στ is “trimmed”). We will modify this in the next subsections.

794 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II Finally, results for operators with the periodic Hamiltonian flows and −1 −1 based on equality e ih A ≡ e iεh B are now based on e ih

(8.4.24) (8.4.25) (8.4.26)

(I − Π)e Πe

ih−1 A

−1 A

ih−1 A

Π ≡ e iεh

−1 B

Π ≡ (I − Π)e

(I − Π) ≡ Πe

iεh−1 B

Π,

iεh−1 B

Π ≡ 0,

(I − Π) ≡ 0.

Again, the easy details are left to the reader. Remark 8.4.9. (i) If Π does not commute with A defining quadratic form. situation here does not change drastically: this quadratic form defines operator ΠA = ΠAΠ on H = Ran Π and all we need is to replace in the definitions (8.4.22) of Στ and (8.4.23) of microhyperbolicity principal symbol a of A by the principal symbol a = πaπ of A where π is the principal symbol of Π. (ii) In particular, if a and π commute, (8.4.22) and (8.4.23) do not need to be modified. (iii) This would not be the case near the border since then ΠA would not be operator of the type we have studied.

8.4.3

Analysis near the Boundary. I

This theory is much more delicate partially because it involves not only pseudodifferential operators but also Boutet-de-Monvel operators and we do not have semi-classical theory here and development of this theory is rather tedious. Luckily we do not need a full theory but a limited one and it is rather obvious: Problem 8.4.10. Develop semi-classical theory parallel to one of Section 1.4. In particular prove that operators described there–pseudodifferential operators with transmission property (see Definition 1.4.3), boundary, trace and singular Green operators (see Definitions 1.4.10–1.4.12) and also hpseudodifferential operators on the boundary have the property that if symbols of two such operators R1 and R2 of different but “matching” types have disjoint supports then R1 R2 is a negligible operator.

8.4. SPECTRAL ASYMPTOTICS ON SUBSPACES

795

We assume that (8.4.15) holds (i.e. AB and Π commute). Then we need −1 to study UΠ (x, y , t) the Schwartz kernel of e ih tAB Π and thus we can apply the theory of Chapter 7 with the only difference that now Q2 = Π is the sum of pseudodifferential operator with the transmission property and a singular Green operator (with respect to (x, t)) while before it was of the type Qj,k (x, t, hDx , Dt )(hDx1 )k . (8.4.27) Qj (x, t, hDx , hDt ) = 0≤k≤l

Remark 8.4.11. One can rewrite operator Qj = Qj (x, t, hDx , hDt ) in the form (8.4.28)

Qj = Qj (x, t, hDx , hDt )(hDt − A) + Qj + hPðmin(l,m) + G

where ðl u = ð(u, hD1 u, ... , (hD1 )l−1 u), Qj is an operator of (8.4.27) type, P, G are Poisson and singular Green operators respectively. But (hDt − A) annihilates U, so one needs to consider only Qj = Qj + Pðmin(l,m) + G . Repeating arguments of Sections 3.1–3.3 we get immediately that Proposition 8.4.12. All statements of Theorem 7.2.17 remain true as Q1 , Q2 are sums of h-pseudodifferential operators with the transmission property and h-singular Green operators. Proof. The only not completely trivial part of the proof is the Proposition  3.3.4 reinvented:  then additionally appear terms of the same type  t Γ Q1x Ujk (., ., t) Q2y but in virtue of Remark 8.4.11 with at least one of operators Q1 and Q2 being the sum of Pðl and G .  As Q1 is operator of this type, it effectively confines x1 to {x1 ≤ h1−δ } and then there is no difference if we take νj x1 or νk x1 in the application of the propagation theorem, so we get terms of the type we deal in another parts of the proof. Details and the rest of arguments are left to the reader. Now we immediately arrive to Theorem 8.4.13. All statements of Theorems 7.3.1 and 7.3.2 remain true as Q1 , Q2 are sums of h-pseudodifferential operators with the transmission property and h-singular Green operators.

796 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II Remark 8.4.14. The leading term in the asymptotics is κ0 (τ , τ  )h−d . Consider Q2 = Q1∗ and τ > τ  . Then κ0 = 0 if and only if (8.4.29)

  q1 θ(a − τ  ) − θ(a − τ ) q2 = 0;

recall that I − θ(a − τ ) is a spectral projector of symbol a. As q2 = π is the projector-valued symbol the latter condition becomes (8.4.30)



 θ(a − τ  ) − θ(a − τ  ) Ran π = 0.

One can also prove easily the following Theorem 8.4.15. All statements of Theorem 7.3.7 remain true as Q1 , Q2 are sums of h-pseudodifferential operators with the transmission property and h-singular Green operators and principal symbol of Q1 is q2 = π, the projector-valued symbol satisfying (8.4.30). Problem 8.4.16. Investigate if theorems 7.3.10, 7.3.11 also remain true. Then one can prove rather easily Theorem 8.4.17. All statements of Theorems 7.4.1, 7.4.4, 7.4.6, 7.4.7, 7.4.8 and 7.4.21 remain true as Q1 , Q2 are sums of h-pseudodifferential operators with the transmission property and h-singular Green operators and principal symbol of Q1 is q2 = π, the projector-valued symbol satisfying (8.4.30). Problem 8.4.18. (i) Investigate if statements Theorems 8.1.6 and 8.1.8 also remain true. (ii) Investigate if statements of Theorems 8.2.1, 8.2.2, 8.2.4 and 8.2.6 also remain true. (iii) Investigate if statements of Theorems 8.3.6, 8.3.15, 8.3.23 and 8.2.6 also remain true. Much more difficult problem would be to weaken notion of “microhyperbolic in multidirection” and still recover the same results:

8.4. SPECTRAL ASYMPTOTICS ON SUBSPACES

797

Definition 8.4.19. Let conditions (3.1.1)–(3.1.6), (3.1.23), (3.1.9) be fulfilled. Let Π be a sum of the h-pseudodifferential operator with the transmission property and singular Green operator. Then the problem (P, B) is microhyperbolic at the point z ∈ Ω ⊂ T ∗ ∂X in the multidirection T = ( , ν1 , ... , νn ) under constrain (I − Π)u = 0 if the following two conditions are fulfilled: (i) For every k = 1, ... , n the operator P is microhyperbolic at the point zk = (z, ηk ) in the direction k = (, νk , 0) ∈ Tzk (T ∗ X |∂X ) ⊂ (Tzk T ∗ X )|∂X under restrain π0 where π0 is the principal symbol of the pseudodifferential part of Π; (ii) The following estimate holds (8.4.31)

− Re (T  p)(z, D1 )v , v + + Re i(T  b)(z)ðv , β(z)ðv b ≥

||v ||2+ − c1 (c1 ||b(z)ðv ||2 − T (z)ðv , ðv m )

∀v ∈ H (z) ∩ Ker(I − π(z))

where π(z) = π0 (z, hD1 ) + π (z) and π (z) is a principal symbol of singular Green part of Π; π(z) is a projector in S(R+ , CD ). We leave to the reader Problem 8.4.20. Correspondingly redefine Σ and Σb , modify definitions 3.1.1, 3.1.3, 3.1.4, 3.2.2, 3.2.3, 3.2.7. Here as we weaken notion of microhyperbolicity, we increase the cone of microhyperbolic multidirections and reduce the dual cones and the propagation “conoids”. The really challenging task would be to Problem 8.4.21. (i) Prove Theorems 3.1.2, 3.1.7 3.1.12 3.1.13, 3.2.4, 3.3.1, 3.3.2, 3.3.3, 3.3.5, 3.3.6 and 3.3.7 under new definition of microhyperbolicity as we consider only solutions satisfying (I − Π)u = 0 where Π and AB commute. (ii) Prove Theorems 8.4.13, 8.4.15 and 8.4.17 under new definition of microhyperbolicity as we consider only solutions satisfying (I − Π)u = 0 where Π and AB commute.

798 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II

8.4.4

Analysis near the Boundary. II. Seeley’s Method

Seeley’s approach works especially well (as long as (8.4.30) is violates). In this approach we analyze zone without actually considering boundary. Let us recall this approach in the simplified version as ∂X is smooth. (a) We consider first zone X  = {x : γ(x) := dist(x, ∂X ) ≥ h1−δ }, take  T (x) = γ(x)1−δ which confines propagation (in appropriate direction) to (8.4.32)

  h 1+δ  h   h 1−δ ≤ ≤ . Z (x) := y : γ(x) γ(y ) γ(x)

However in this zone A is just a regular h-pseudodifferential operator with the principal symbol a and all we need is its microhyperbolicity with the restriction (I − π)u = 0. Further, if π and a commute we need to check microhyperbolicity of a with the restriction (I − π)u = 0. So, the same arguments as before bring us that the contribution of  the strip {x : γ(x)  γ} to the remainder is O(h1−d γ δ ) and the total contribution of X  is O(h1−d ). (b) Then we consider zone X  = {x : h ≤ γ(x) ≤ h1−δ } and rescale it x → x/γ1 , h →  = h/γ1 , γ → γ  = γ/γ1 selecting γ1 = (γ/h)s h so that γ  = 1−δ ; then γ  = (h/γ)s−1 . Therefore in virtue of (a) contribution of the strip {x : γ(x)  γ}   to the remainder does not exceed C 1−d γ  δ γ11−d ≤ Ch1−d (h/γ)δ ) and the total contribution of this zone to the remainder is O(h1−d ). This part of arguments does not require any modification. (c) Furthermore, we consider zone x  = {x : γ(x) ≤ h} and we use the rough estimate similar to those of Section 7.1 we estimate contribution of this zone also by O(h1−d ). Easy details are left to the reader. (d) Further, these arguments prove that the contribution of the zone {x : μ−1 h ≤ γ(x) ≤ μ} to the remainder does not exceed ε(μ)h1−d with ε(μ) = o(1) as μ → 0. Improved analysis in zone {x : γ(x) ≤ μ−1 h} (see Section 7.5) proves that its contribution to the remainder is o(h1−d ) for any fixed μ > 0. Finally, propagation arguments prove that under nonperiodicity billiard conditions contribution of the zone {x : γ(x) ≤ μ} to the remainder is o(h1−d ). So we arrive to Theorem below:

8.4. SPECTRAL ASYMPTOTICS ON SUBSPACES

799

Theorem 8.4.22. Let ∂X be smooth. Then all statements of Theorems 7.5.10, 7.5.15, 7.5.19 and 7.5.20 remain true as Q1 , Q2 are sums of h-pseudodifferential operators with the transmission property and singular Green operators. Example 8.4.23. Consider H defined by (8.4.1) and operator A with boundary condition B defined by arbitrary elliptic m-th order quadratic form Q(u). Then (8.4.33)

A = A + G + hPð2m ,

with the domain (8.4.34) (8.4.35)

D(AB ) = {u ∈ H 2 (X , Λ(p) ), Bu = 0}, B = Bð2m + T

where ðm u = (ðu, ðhD1 u, ... , ð(hD1 )m−1 u), m is an order of Q and A is a pseudodifferential operator of order 2m with transmission property, B is a pseudodifferential operator on ∂X , G , P and T are singular Green, Poisson and trace operators respectively. We leave to the reader the following Problem 8.4.24. Describe operators A, B. Repeating arguments above21) we arrive to Theorem 8.4.25. Let ∂X be smooth. Then (i) Statements of Theorems 7.5.10 and 7.5.15 remain true as Q1 , Q2 are sums of h-pseudodifferential operators with the transmission property and singular Green operators and AB is of the type described above. (ii) As operator A and generalized boundary operator B differ from differential operator A and operator B by lower order terms only, statement of Theorems 7.5.19 and 7.5.20 also remain true. The following problem seems to be extremely challenging: Problem 8.4.26. To derive asymptotics with remainder estimate o(h1−d ) for general operators of (8.4.33)–(8.4.35)-type. Here treatment of both zones {x : γ(x) ≤ μ−1 h} and {x : γ(x) ≥ μ} seems to be difficult since propagation near boundary is not clear. 21)

In part (c) correct scaling of all terms of the operator is essential.

800 CHAPTER 8. STANDARD THEORY NEAR THE BOUNDARY. II

Comments The systematic theory outlined in Section 8.4 even without Problem 8.4.26 would be definitely worth publications.

Bibliography Agmon, S. [1] Lectures on Elliptic Boundary Value Problems. Princeton, N.J.: Van Nostrand Mathematical Studies, 1965. [2] On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems. Comm. Pure Appl. Math. 18, 627–663 (1965). [3] Asymptotic formulas with remainder estimates for eigenvalues of elliptic operators. Arch. Rat. Mech. Anal., 28:165–183 (1968). [4] Spectral properties of Schr¨odinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 4:151—218 (1975). [5] A perturbation theory of resonances. Comm. Pure Appl. Math. 51(1112):1255—1309 (1998). Agmon, S.; Douglis, A.; Nirenberg, L. [1] Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I.. Commun. Pure Appl. Math., 12(4):623–727 (1959). [1] Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II.. Commun. Pure Appl. Math., 17(1):35–92 (1964). Agmon, S.; H¨ ormander, L. [1] Asymptotic properties of solutions of differential equations with simple characteristics. J. Analyse Math. 30: 1—38 (1976).

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Presentations [1]

Sharp spectral asymptotics for irregular operators

[2]

Sharp spectral asymptotics for magnetic Schr¨odinger operator

[3]

25 years after

[4]

Spectral asymptotics for 2-dimensional Schr¨odinger operator with strong degenerating magnetic field

[5]

Magnetic Schr¨ odinger operator: geometry, classical and quantum dynamics and spectral asymptotics

[6]

Spectral asymptotics and dynamics

[7]

Magnetic Schr¨ odinger operator near boundary

[8]

2D- and 3D-magnetic Schr¨odinger operator: short loops and pointwise spectral asymptotics

[9]

100 years of Weyl’s law

[10]

Some open problems, related to spectral theory of PDOs

[11]

Large atoms and molecules with magnetic field, including selfgenerated magnetic field (results: old, new, in progress and in perspective)

[12]

Semiclassical theory with self-generated magnetic field

[13]

Eigenvalue asymptotics for Dirichlet-to-Neumann operator Available at http://weyl.math.toronto.edu/victor_ivrii/research/talks/

© Springer Nature Switzerland AG 2019 V. Ivrii, Microlocal Analysis, Sharp Spectral Asymptotics and Applications I, https://doi.org/10.1007/978-3-030-30557-4

873

PRESENTATIONS

874 [14]

Eigenvalue asymptotics for Fractional Laplacians

[15]

Asymptotics of the ground state energy for relativistic atoms and molecules

[16]

Etudes in spectral theory

[17]

Eigenvalue asymptotics for Steklov’s problem in the domain with edges

[18]

Complete semiclassical spectral asymptotics for periodic and almost periodic perturbations of constant operators

[19]

Complete Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operators and Bethe-Sommerfeld Conjecture in Semiclassical Settings

Index action, 540 adjoined boundary value problem, 117 adjoint operator pseudodifferential, 12 admissible box, 94 function, 11, 94 operator, 94 almost analytic function, 400 amplitude, 19, 69 approximaion error, 288 Tauberian, 287 approximation method of successive, 322 Weyl, 288 approximation error, 377 argument Tauberian, 300 asymptotics large eigenvalue elliptical domain, 692 polyhedron, 692 spectral local, 446 microlocal, 446 two-term, XXVII

averaging, 287, 618 averaging function, 312 bicharacteristic boundary, 236 generalized, 151 semi-, 155 bicharacteristic billiard generalized, 228 bicharacteristic loop, 359 bicharacteristically concave, 237 bicharacteristically convex, 236 bicharacteristically flat, 772 bicharacteristics, XXXIII generalized, XXXIII limiting, 160 periodic, 359 billiard generalized bicharacteristic, 228 Bohr-Sommerfeld quantization condition, 76 boundary characteristic, 197 boundary bicharacteristic, 236 boundary wave front set, 200 Boutet de Monvel algebra, 110 box admissible, 94 bracket, 376

© Springer Nature Switzerland AG 2019 V. Ivrii, Microlocal Analysis, Sharp Spectral Asymptotics and Applications I, https://doi.org/10.1007/978-3-030-30557-4

875

876 canonical graph, 80 Cauchy problem multi-time, 143 causality, 102 characteristic boundary, 197 point of a problem, 200 point of an operator, 200 polynomial, 201 symbol, 149 pseudodifferential operator, 53 cluster spectral, 542 coherent states, 78 commutator pseudodifferential operators, 13 commuting operators spectral asymptotics, 522 concave bicharacteristically, 237 condition no critical point, 412 configuration space, 63 constant, 93 continuity modulus, 376 continuous with continuity modulus, 376 convex bicharacteristically, 236 manifold, 689 creeping rays, 237 Dirac matrix, 516 operator, 463 Dirac correction term, 595 Dirac operator generalized, 464

INDEX massless, 463 Dirichlet-Neumann bracketing, XXV dissipative problem, 198 distribution Lagrangian, 69 domain bicharacteristically concave, 237 bicharacteristically convex, 236 elliptical, 692 polyhedral, 692 domination, 314 drift, 543 of singularities, 553 dual operator, 139 dynamics classical, 63 Egorov’s theorem, 82 eigenvalue counting function, XXIV eikonal, 63 equation, 63 elliptic point of a problem, 200 point of an operator, 200 zone, 235 elliptic boundary value problem, 114 elliptic component, 233 elliptic Green operator, 111 elliptic operator, 40 elliptical annulus, 692 domain, 692 error approximation, 493 combined, 493 essentially supported function, 95

INDEX

877 r -almost analytic, 400 roughness, 490 scaling, 412 tempered, 94

operator, 95 exponent, 93 families of commuting operators spectral asymptotics, 522 family of commuting operators propagator, 524 spectral projector, 523 field Hamiltonian, 87 finite speed of propagation of singularities, 135, 240 fixed object, 93 flow Hamiltonian, 87, 179 generated by a Hamiltonian field, 87 form symplectic skew-symmetric, 88 Fourier integral operator, 80 Fourier transform, 5 inverse, 5 funcion averaging, 287 function admissible, 94 almost analytic, 400 averaging, 312 essentially supported in a box, 95 H¨ormander, 301 negligible, 94 in a box, 94 of a self-adjoint pseudodifferential operator, 45 phase non-degenerate, 69

gap spectral semiclassical, 294, 581 Ga ˚rding inequality, 30 gauge transformation, 34, 448 generalized bicharacteristic, 151 bicharacteristic billiard, 228 ρ-bicharacteristic, 151 generalized metaplectic operator, 90 germ of an operator, 92 glancing zone, 235 gliding rays, 237 grazing rays, 237 Green operators, 110 h-Fourier transform partial, 72, 81 h-pseudodifferential operator, XXVII H¨ormander function, 301 multidimensional, 526 Hamilton-Jacobi equation, 63 Hamiltonian, 63, 179 dynamics, 63 field, 63, 87 flow, 63, 87, 179 generated by a Hamiltonian field, 87 trajectory, XXVIII Hamiltonian dynamics with the reflections, 271

878 Hamiltonian dynamics with the reflections and branching, 271 Hamiltonian flow generalized multitime, 533 heat equation method, XXV Heisenberg representation, 177 Hessian skew, 88 hyperbolic zone, 235 hyperbolic operator method, XXV incoming component, 232 incoming point, 232 index Maslov, 75 inequality Ga ˚rding, 30 Ivrii’s conjecture, 698 K # (p, z), 147 K ± (p, z, Ω), 151 # Kρε (p, z), 147 ± Kρε (p, z, Ω), 151 Kρ (p, z), 146 (l, r )-germ of an operator, 92 Lagrangian distribution, 69 principal symbol of, 74 Lagrangian manifold, 70 generated by a phase function, 70 Lagrangian submanifold, 65 large eigenvalue asymptotics elliptical annulus, 692 domain, 692 polyhedron, 692

INDEX elliptic operator, 40 parametrix, 40 left parametrix of boundary value problem, 115 Legendre transformation, 66 local spectral asymptotics, 446 uncertainty principle, 49 manifold convex, 689 Lagrangian, 70 generated by a phase function, 70 Maslov canonical operator, 75 constant, 541 index, 75, 541 Maslov canonical operator, 65 matrix Dirac, 516 Pauli, 516, 518 matrix signature, 66 mean spectral smooth, 342 Melin inequality, 39 metaplectic operator, 90 generalized, 90 method approximation successive, 287 heat equation, XXV hyperbolic operator, XXV of successive approximations, 322 partition-rescaling, XXXVIII rescaling, XXXVIII resolvent, XXV

INDEX

Tauberian, XXV variational, XXV ζ-function, XXV microhyperbolic problem, 201 under restriction, 793 microhyperbolic symbol, 127 microhyperbolic under constrain problem, 797 microhyperbolicity under restriction, 793 microhyperbolicity cones cones, 147 microlocal spectral asymptotics, 446 microlocalization, 92 anisotropic, 92 mollification with respect to non-spectral parameters, 614 multi-time Cauchy problem, 143 multidirection, 201 multiplicator, 6 negligible double symbol, 11 function, 11, 94 operator, 11, 94 negligible function, 94 negligible operator, 94 no critical point condition, 412 non-spectral parameter mollification with respect to, 614 non-strictly microhyperbolic, 131 normal singularity property, 141, 245 operator

879 admissible, 94 commuting families spectral asymptotics, 522 Dirac, 463 essentially supported in a box, 95 Fourier integral, 80 h-pseudodifferential, XXVII Maslov canonical, 75 metaplectic, 90 generalized, 90 negligible, 94 in a box, 94 scalar with periodic Hamiltonian flow, 538 Schr¨odinger, 447 tempered, 94 pseudodifferential operator, 53 ordinary differential inclusions, 154 outgoing component, 232 outgoing point, 232 parameter, 93 averaging, 287 non-spectral mollification with respect to, 614 roughness, 490 parameterized family of objects, 93 parametrix, 39 parametrix of boundary value problem, 116 parametrix of operator in domain with boundary, 116 wave front set, 135 partial h-Fourier transform, 72, 81 partition-rescaling method, XXXVIII

INDEX

880 Pauli matrix, 516 Pauli matrix, 518 periodic classical dynamics., 540 phase function, 19 non-degenerate, 69 phase space, 63 point absolutely periodic, 589 antipodal, 774 characteristic of a problem, 200 of an operator, 200 elliptic of a problem, 200 of an operator, 200 regular, 71 points diffractive, 260 elliptic, 259 glancing, 259 degenerate, 260 diffractive, 260 gliding, 260 grazing, 260 gliding, 260 grazing, 260 hyperbolic, 259 subperiodic, 539 Poisson relation, XXVIII Poisson brackets, 13 Poisson operator, 106 polarization wave front set, 194 polyhedron, 692 polynomial characteristic, 201

positive Lagrangian ideal, 78 principal symbol of a Lagrangian distribution, 74 problem dissipative, 198 microhyperbolic, 201 microhyperbolic under constrain, 797 multi-time Cauchy, 143 strictly dissipative, 198 symmetric, 243 procedure sandwich, 328 product of operators pseudodifferential, 13 propagation cones cones, 147 propagation of singularities, XXVIII finite speed of, 135, 240 propagator, 177 property normal singularity, 141, 245 pseudodifferential operator, 7, 15 adjoint, 12 applied to exponent, 21 classical, 53 function of, 42 Hilbert-Schmidt norm, 16 order, 53 product of, 13 Schwartz kernel, 8 self-adjoint function of, 45 trace of, 16 quantization, 4 pq-quantization, 4, 7

INDEX

qp-quantization, 4, 7 abstract, 15 symmetric quantization, 7 Weyl quantization, 7 r -almost analytic function, 400 rays creeping, 237 gliding, 237 grazing, 237 regular point, 71 relation Poisson, XXVIII representation, 6 coordinate, 6 momentum, 6 Rescaling, 29 rescaling, 142, 408, 413 method, XXXVIII with variable scale parameters, 408 resolvent method, XXV (ρ, ε)-bicharacteristic generalized, 151 elliptic operator, 40 parametrix, 40 right parametrix of boundary value problem, 116 sandwich procedure, 328 scalar operator with periodic Hamiltonian flow, 538 scaling functions, 409 parameters, 408 scaling function, 35, 412

881 admissible, 412 Schr¨odinger operator, 447 self-adjoint boubdary value problem, 118 self-adjoint pseudodifferential operator function of, 45 semibicharacteristic, 155 semiclassical parameter, 93 semiclassical spectral gaps, 294, 581 senior symbol, 636 set wave front, 95 boundary, 200 set-valued function, 146 continuous, 146 lower semicontinuous, 146 upper semicontinuous, 146 singular Green operator of order operator, 107 singularities propagation of, XXVIII singularity drift of, 553 normal, 141, 245 normal type, 405 propagation of finite speed of, 135, 240 skew-Hessian, 38, 88 smooth spectral mean, 342 spectral asymptotics commuting operators, 522 local, 446 microlocal, 446 bands, 558 gap semiclassical, 294, 581

882 gaps, 541, 558 mean smooth, 342 spectral asymptotics, XXXV local, XXXV microlocal, XXXV pointwise, XXXVIII standard set of assumptions first, 296 second, 296 standard Weyl expression, 611 stationary phase method, 20 strictly dissipative problem, 198 strip boundary, 708 successive approximation method of, 322 successive approximation method, 287 symbol pq-symbol, 11 qp-symbol, 11 characteristic, 149 double, 7 microhyperbolic, 127 principal, 12 of a Lagrangian distribution, 74 senior, 636 simple, 7 Weyl symbol, 11 symmetric in the principal, 197 symmetric problem, 243 symmetric product, 91 symplectic, 37 symplectic form, 65 symplectic form skew-symmetric, 88

INDEX symplectomorphism, 63 system symmetric in the principal, 197 Tauberian arguments, 300 Tauberian approximaion, 287 Tauberian error, 287 Tauberian formula alternative, 312 main, 311 mollified, 312 Tauberian method, XXV Tauberian technique simplified, 299 technique Tauberian, 300 tempered function, 94 operator, 94 term correction, 454 theorem Egorov’s, 82 trace operator, 107 trajectory Hamiltonian, XXVIII transform partial h-Fourier, 81 transformation gauge, 448 transmission property, 102, 103 transport equation, 64, 178 two-term asymptotics, XXVII uncertainty principle, XXXII, 30 logarithmic, XXXII, 30, 49 microlocal, XXXII, 30

INDEX

standard, XXXII variational method, XXV wave front set, 95 boundary, 200 Weyl approximation, 288 Weyl error, 288 quantization, 4 Weyl’s law, XXIV local, XXIV pointwise, XXIV WFs , 95 WFs , 135 ζ-function method, XXV zone inner, 708

883

Content of All Volumes Volume I. Semiclassical Microlocal Analysis and Local and Microlocal Semiclassical Asymptotics Preface

V

Introduction

I

Semiclassical Microlocal Analysis

1 Introduction to Microlocal Analysis

XXII

1 2

2 Propagation of Singularities in the Interior of the Domain

126

3 Propagation of Singularities near the Boundary

196

II

Local and Microlocal Semiclassical Spectral Asymptotics in the Interior of the Domain

4 General Theory in the Interior of the Domain

285 286

5 Scalar Operators in the Interior of the Domain. Rescaling Technique 6 Operators in the Interior of Domain. Esoteric Theory

407 521

III

Local and Microlocal Semiclassical Spectral Asymptotics near the Boundary

622

7 Standard Local Semiclassical Spectral Asymptotics near the Boundary

623

8 Standard Local Semiclassical Spectral Asymptotics near the Boundary. Miscellaneous

742

Bibliography

801

Presentations

873

Index

875

Volume II. Functional Methods and Eigenvalue Asymptotics Preface

V

Introduction

IV

Estimates of the Spectrum

9 Estimates of the Negative Spectrum

XII

1 2

10 Estimates of the Spectrum in the Interval

45

V

94

Asymptotics of Spectra

11 Weyl Asymptotics of Spectra

95

12 Miscellaneous Asymptotics of Spectra

262

Bibliography

440

Presentations

512

Index

514

Volume III. Magnetic Schr¨ odinger Operator. 1 Preface

V

Introduction

XVII

Smooth theory in dimensions 2 and 3

VI

13 Standard Theory

1 2

14 2D-Schr¨odinger Operator with Strong Degenerating Magnetic Field

182

15 2D-Schr¨odinger Operator with Strong Magnetic Field near Boundary

317

Smooth theory in dimensions 2 and 3

VII

(continued)

414

16 Magnetic Schro¨dinger Operator: Short Loops, Pointwise Spectral Asymptotics and Asymptotics of Dirac Energy

415

17 Dirac Operator with the Strong Magnetic Field

564

Bibliography

647

Presentations

719

Index

721

Volume IV. Magnetic Schr¨ odinger Operator. 2 Preface Introduction

I XX

VIII Non-smooth theory and higher dimensions

1

18 2D- and 3D-magnetic Schr¨odinger operator with irregular coefficients

2

19 Multidimensional Magnetic Schr¨odinger Operator. Full-Rank Case

104

20 Multidimensional Magnetic Schr¨odinger Operator.

IX

Non-Full-Rank Case

222

Magnetic Schr¨ odinger Operator in Dimension 4

324

21 4D-Schr¨odinger Operator with a Degenerating Magnetic Field

325

22 4D-Schr¨odinger Operator with the Strong Magnetic Field

433

X

Eigenvalue Asymptotics for Schr¨ odinger and Dirac Operators with the Strong Magnetic Field

497

23 Eigenvalue asymptotics. 2D case

498

24 Eigenvalue asymptotics. 3D case

569

Bibliography

632

Presentations

704

Index

706

Volume V. Applications to Quantum Theory and Miscellaneous Problems Preface

V

Introduction

XI

Application to Multiparticle Quantum Theory

25 No Magnetic Field Case 26 The Case of External Magnetic Field

XX

1 2 68

27 The Case of Self-Generated Magnetic Field

208

28 The Case of Combined Magnetic Field

284

Bibliography

395

XII

467

Articles

Spectral Asymptotics for the Semiclassical Dirichlet to Neumann Operator Spectral Asymptotics for Fractional Laplacians

468 495

Spectral Asymptotics for Dirichlet to Neumann Operator in the Domains with Edges

513

Asymptotics of the Ground State Energy in the Relativistic Settings

540

Asymptotics of the Ground State Energy in the Relativistic Settings and with Self-Generated Magnetic Field

559

Complete Semiclassical Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operator

583

Complete Differentiable Semiclassical Spectral Asymptotics

607

Bethe-Sommerfeld Conjecture in Semiclassical Settings

619

100 years of Weyl’s Law

641

Presentations

730

Index

732