Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations [1st ed.] 978-3-030-10818-2;978-3-030-10819-9

Table of contents :
Front Matter ....Pages i-x
Introduction (Johannes Sjöstrand)....Pages 1-5
Front Matter ....Pages 7-7
Spectrum and Pseudo-Spectrum (Johannes Sjöstrand)....Pages 9-28
Weyl Asymptotics and Random Perturbations in a One-Dimensional Semi-classical Case (Johannes Sjöstrand)....Pages 29-52
Quasi-Modes and Spectral Instability in One Dimension (Johannes Sjöstrand)....Pages 53-65
Spectral Asymptotics for More General Operators in One Dimension (Johannes Sjöstrand)....Pages 67-91
Resolvent Estimates Near the Boundary of the Range of the Symbol (Johannes Sjöstrand)....Pages 93-134
The Complex WKB Method (Johannes Sjöstrand)....Pages 135-155
Review of Classical Non-self-adjoint Spectral Theory (Johannes Sjöstrand)....Pages 157-181
Front Matter ....Pages 183-183
Quasi-Modes in Higher Dimension (Johannes Sjöstrand)....Pages 185-189
Resolvent Estimates Near the Boundary of the Range of the Symbol (Johannes Sjöstrand)....Pages 191-209
From Resolvent Estimates to Semigroup Bounds (Johannes Sjöstrand)....Pages 211-217
Counting Zeros of Holomorphic Functions (Johannes Sjöstrand)....Pages 219-244
Perturbations of Jordan Blocks (Johannes Sjöstrand)....Pages 245-293
Front Matter ....Pages 295-295
Weyl Asymptotics for the Damped Wave Equation (Johannes Sjöstrand)....Pages 297-311
Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small Random Perturbations, Results and Outline (Johannes Sjöstrand)....Pages 313-327
Proof I: Upper Bounds (Johannes Sjöstrand)....Pages 329-361
Proof II: Lower Bounds (Johannes Sjöstrand)....Pages 363-407
Distribution of Large Eigenvalues for Elliptic Operators (Johannes Sjöstrand)....Pages 409-426
Spectral Asymptotics for \(\mathcal {P}\mathcal {T}\) Symmetric Operators (Johannes Sjöstrand)....Pages 427-441
Numerical Illustrations (Johannes Sjöstrand)....Pages 443-486
Back Matter ....Pages 487-496

Citation preview

Pseudo-Differential Operators Theory and Applications 14

Johannes Sjöstrand

Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

Pseudo-Differential Operators Theory and Applications Vol. 14 Managing Editor M. W. Wong (York University, Canada)

Associate Editors Luigi G. Rodino (Università di Torino, Italy) Bert-Wolfgang Schulze (Universität Potsdam, Germany) Johannes Sjöstrand (Université de Bourgogne, Dijon, France) Sundaram Thangavelu (Indian Institute of Science at Bangalore, India) Maciej Zworski (University of California at Berkeley, USA)

Pseudo-Differential Operators: Theory and Applications is a series of moderately priced graduate-level textbooks and monographs appealing to students and experts alike. Pseudo-differential operators are understood in a very broad sense and include such topics as harmonic analysis, PDE, geometry, mathematical physics, microlocal analysis, time-frequency analysis, imaging and computations. Modern trends and novel applications in mathematics, natural sciences, medicine, scientific computing, and engineering are highlighted. More information about this series at http://www.springer.com/series/7390

Johannes Sjöstrand

Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

Johannes Sjöstrand Université de Bourgogne Franche-Comté Dijon, France

ISSN 2297-0355 ISSN 2297-0363 (electronic) Pseudo-Differential Operators ISBN 978-3-030-10818-2 ISBN 978-3-030-10819-9 (eBook) https://doi.org/10.1007/978-3-030-10819-9 Mathematics Subject Classification (2010): 30C15, 35P20, 81Q12 © 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, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms 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 specific 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, express 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 affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Abstract

The main purpose of this monograph is to describe some recent developments around the Weyl asymptotics for the eigenvalues of non-self-adjoint differential operators with random perturbations. It was natural to add some other results about non-self-adjoint operators, such as spectral stability and instability, growth bounds for semi-groups, and to review some of the classical theory.1

1 Chapter 20 has been prepared in collaboration with Agnès E. Sjöstrand.

v

Contents

1

Introduction .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

1

Part I Basic Notions, Differential Operators in One Dimension 2

Spectrum and Pseudo-Spectrum .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.1 Operators in Hilbert Spaces, a Quick Review .. . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.2 Pseudospectrum.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.3 Numerical Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.4 A Simple Example of a Large Matrix . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.5 The Non-self-adjoint Harmonic Oscillator . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

9 9 13 16 18 22

3

Weyl Asymptotics and Random Perturbations in a One-Dimensional Semi-classical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3.1 Preparations for the Unperturbed Operator .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3.2 Grushin (Schur, Feschbach, Bifurcation) Approach . . . . . . .. . . . . . . . . . . . . . . 3.3 d-Bar Equation for E−+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3.4 Adding the Random Perturbation .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

29 31 34 37 41

4

Quasi-Modes and Spectral Instability in One Dimension . . . . . .. . . . . . . . . . . . . . . 4.1 Asymptotic WKB Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 4.2 Quasimodes in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

53 53 60

5

Spectral Asymptotics for More General Operators in One Dimension.. . . . . 5.1 Preliminaries for the Unperturbed Operator .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.2 The Result . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.3 Preparations for the Unperturbed Operator .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.4 d-Bar Equation for E−+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.5 Adding the Random Perturbation .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.A Appendix: Estimates on Determinants of Gaussian Random Matrices . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

67 67 71 74 79 80 85

vii

viii

Contents

6

Resolvent Estimates Near the Boundary of the Range of the Symbol . . . . . . . 6.1 Introduction and Statement of the Main Result . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.2 Preparations and Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.3 The Factor hDx + G(x, ω; h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.4 Global Grushin Problem, End of the Proof .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.A Appendix: Quick Review of Almost Holomorphic Functions .. . . . . . . . . . .

93 93 104 108 120 133

7

The Complex WKB Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 135 7.1 Estimates on an Interval .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 135 7.2 The Schrödinger Equation in the Complex Domain . . . . . . .. . . . . . . . . . . . . . . 140

8

Review of Classical Non-self-adjoint Spectral Theory . . . . . . . . .. . . . . . . . . . . . . . . 8.1 Fredholm Theory Via Grushin Problems . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 8.2 Singular Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 8.3 Schatten–von Neumann Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 8.4 Traces and Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

157 157 164 171 173

Part II Some General Results 9

Quasi-Modes in Higher Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 185

10 Resolvent Estimates Near the Boundary of the Range of the Symbol . . . . . . . 10.1 Introduction and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 10.2 Evolution Equations on the Transform Side .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 10.3 Resolvent Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 10.4 Examples .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

191 191 196 204 208

11 From Resolvent Estimates to Semigroup Bounds . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 11.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 11.2 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 11.3 The Gearhardt–Prüss–Hwang–Greiner Theorem . . . . . . . . . .. . . . . . . . . . . . . . .

211 211 211 214

12 Counting Zeros of Holomorphic Functions .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 12.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 12.2 Thin Neighborhoods of the Boundary and Weighted Estimates .. . . . . . . . . 12.3 Distribution of Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 12.4 Application to Sums of Exponential Functions . . . . . . . . . . . .. . . . . . . . . . . . . . .

219 219 223 232 241

13 Perturbations of Jordan Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 13.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 13.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 13.3 Study in the Region |z| > R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 13.4 Study in the Region |z| < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 13.5 Upper Bounds on the Number of Eigenvalues in the Interior . . . . . . . . . . . . 13.6 Gaussian Elimination and Determinants.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 13.7 Random Perturbations and Determinants.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

245 245 247 249 250 253 257 259

Contents

13.8 13.9 Part III

ix

Eigenvalue Distribution for Small Random Perturbations .. . . . . . . . . . . . . . . 266 Eigenvalue Distribution for Larger Random Perturbations .. . . . . . . . . . . . . . 279 Spectral Asymptotics for Differential Operators in Higher Dimension

14 Weyl Asymptotics for the Damped Wave Equation . . . . . . . . . . . . .. . . . . . . . . . . . . . . 297 14.1 Eigenvalues of Perturbations of Self-adjoint Operators.. . .. . . . . . . . . . . . . . . 297 14.2 The Damped Wave Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 307 15 Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small Random Perturbations, Results and Outline. . . . . . .. . . . . . . . . . . . . . . 15.1 The Unperturbed Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 15.2 The Random Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 15.3 The Result . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 15.4 Comparison of Theorems 15.3.1 and 1.1 in [137] and [139] . . . . . . . . . . . . .

313 313 315 316 325

16 Proof I: Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 16.1 Review of Some Calculus for h-Pseudodifferential Operators . . . . . . . . . . . 16.2 Sobolev Spaces and Multiplication . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 16.3 Bounds on Small Singular Values and Determinants in the Unperturbed Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 16.4 Subharmonicity and Symplectic Volume .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 16.5 Extension of the Bounds to the Perturbed Case . . . . . . . . . . . .. . . . . . . . . . . . . . .

329 329 332

17 Proof II: Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.1 Singular Values and Determinants of Certain Matrices Associated to δ Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.2 Singular Values of Matrices Associated to Suitable Admissible Potentials . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.3 Lower Bounds on the Small Singular Values for Suitable Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.4 Estimating the Probability that det Pδ,z Is Small . . . . . . . . . . .. . . . . . . . . . . . . . . 17.5 End of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.A Appendix: Grushin Problems and Singular Values . . . . . . . .. . . . . . . . . . . . . . .

363

372 383 389 400

18 Distribution of Large Eigenvalues for Elliptic Operators .. . . . .. . . . . . . . . . . . . . . 18.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 18.2 Some Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 18.3 Volume Considerations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 18.4 Semiclassical Reduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 18.5 End of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

409 409 412 415 417 424

335 351 354

363 367

x

Contents

19 Spectral Asymptotics for PT Symmetric Operators.. . . . . . . . . .. . . . . . . . . . . . . . . 19.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 19.2 The Semi-Classical Case with Random Perturbations .. . . .. . . . . . . . . . . . . . . 19.3 Weyl Asymptotics for Large Eigenvalues . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 19.4 PT -Symmetric Potential Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 19.4.1 Simple Well in One Dimension . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 19.4.2 Double Wells in Arbitrary Dimension . . . . . . . . . . . .. . . . . . . . . . . . . . .

427 427 428 432 434 436 438

20 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.1 Jordan Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.2 Hager’s Operator.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.2.1 The Case When g(x) = i cos x . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.2.2 The Case g(x) = eix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.3 (hD)2 + i cos x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.4 Second-Order Differential Operators on S 1 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.4.1 A PT -Symmetric Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.4.2 A Non-PT -Symmetric Case . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.5 Finite Difference Operators on S 1 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.5.1 A PT -Symmetric Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.5.2 A Non-PT -Symmetric Case . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.6 PT -Symmetric Double Well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 20.7 The Figures .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

443 443 444 444 446 447 448 449 449 450 451 451 451 453

Bibliography . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 487

1

Introduction

Non-self-adjoint operators is an old, sophisticated and highly developed subject. See for instance Carleman [28] for an early result on Weyl type asymptotics for the real parts of the large eigenvalues of operators that are close to self-adjoint ones, with later results by Markus and Matseev [96, 97] in the same direction. (See the classical works Weyl [159], Avakumovi´c [8], Levitan [94], Hörmander [80] for the asymptotics of large eigenvalues of elliptic self-adjoint operators and Robert [119] and Dimassi and Sjöstrand [41] for corresponding results in the semi-classical case, not to mention numerous deep and sophisticated results by Ivrii [86] and others.) Abstract theory with the machinery of snumbers can be found in the book of Gohberg and Krein [48]. Other quite classical results concern upper bounds on the number of eigenvalues in various regions of the complex plane and questions about completeness of the set of all generalized eigenvectors. (Cf. Agmon [1, 2].) In my own work I have often encountered non-self-adjoint operators. Thus for instance problems about analytic regularity for PDE’s with pointwise degeneracy or in domains with conic singularities turned out to boil down to problems about non-self-adjoint operators; estimates on their resolvents and completeness of the set of generalized eigenvectors. Cf. [9, 10] with Baouendi. Non-self-adjoint quadratic operators turned up naturally in the study of certain classes of hypoelliptic operators with double characteristics, [24, 131] and in a number of recent works, this theme has been taken up again and is now quite an active field, see e.g., [34, 68–70, 75–77, 109, 112–114, 153, 154]. The study of resonances (scattering poles) of operators of Kramers–Fokker–Planck type with related problems of return to equilibrium are very rich domains where non-self-adjointness is an important ingredient; these, however, are beyond the scope of this book. A major difficulty in the non-self-adjoint theory is that the norm of the resolvent may be very large even when the spectral parameter is far from the spectrum. This causes the

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_1

1

2

1 Introduction

spectrum (and in this book we will only consider the discrete spectrum) to be unstable under small perturbations of the operator and this can be a source of numerical errors. Starting in the 90’ies the perspectives have changed thanks to works by numerical analysts, (see for instance the article Trefethen, [151], and the monograph [152] by Trefethen and Embree) who emphasized that the pseudospectrum—roughly the zone in the complex plane where the resolvent norm is large—can be of independent interest, for instance to understand the onset of turbulence from the (non-self-adjoint) linearizations of stationary flows. This spurred new interest among analysts like Davies [33, 34], Zworski and others [40, 161, 162]. Given the spectral instability it has been natural to study the effect on the eigenvalues of small random perturbations, and beautiful numerical results can be found for instance in [152]. In her thesis [53–55], M. Hager made a mathematical study of the spectrum of the sum of the simple operator hDx + g(x) on S 1 and a small random term. Here, Dx = (1/i)d/dx = (1/i)∂x and as quite often in this text, we work in the semi-classical limit h  0. The spectrum of the unperturbated operator is just an arithmetic progression  2π of eigenvalues on the line z = g = (2π)−1 0 g(x)dx and the effect of the random perturbation is to spread out the eigenvalues in the band {z ∈ C; inf g ≤ z ≤ sup g} = p(T ∗ S 1 ), where p(x, ξ ) = ξ +g(x) is the semi-classical principal symbol. The main result is that if  is a fixed bounded domain with smooth boundary in the interior of the band, then with probability very close to 1, the number of eigenvalues N() of the perturbed operator in  is given by N() = (2πh)−1 (volT ∗ S 1 (p−1 ()) + o(1)),

(1.0.1)

in the semi-classical limit, h → 0, with an explicit estimate on the remainder o(1).1 We here recognize the natural non-self-adjoint version of Weyl asymptotics in the semiclassical limit, well established for large eigenvalues of self-adjoint differential operators since more than a century and later in the semi-classical self-adjoint case. This came as a big surprise since in the cases known to me, one has to assume analyticity to get eigenvalue asymptotics via complex Bohr–Sommerfeld conditions. In one dimension such eigenvalues typically sit on curves which is incompatible with (1.0.1). (We here discuss genuinely non-self-adjoint operators with complex valued principal symbol.) Thus with hindsight one can say that the random perturbation will typically destroy (uniform) analyticity and hence destroy all possible asymptotic formulae in terms of complex phase space (like complex Bohr–Sommerfeld conditions). Among the possible remaining formulae in terms of real phase space, the Weyl asymptotics seems to be the only possible one.

1 At first, the formula appeared more complicated, depending on the method of proof, and the simpler form was pointed out to me by E. Amar-Servat.

1 Introduction

3

After this first result, there have been several works, by Hager [55], Bordeaux Montrieux [16–18] as well as [19, 56, 137, 139] that treated more general situations and obtained more precise results. During this process the methods were improved and the main purpose of this research monograph is to give a unified account,2 leaving out many other recent developments for non-self-adjoint operators. We also leave out some related results for resonances (see [144] and references given there). A related, very promising approach for studying directly the expected eigenvalue density and correlations can be found in work of Christiansen and Zworski [32] and Vogel [147, 155, 156]. Since this approach is currently less developed for spectral problems (especially in higher dimensions), we will not treat it here. Quite naturally, the classical theory of non-self-adjoint operators [48] is a fundamental ingredient of the methods. We will also use some results of complex analysis, in particular a result on counting the zeros of holomorphic functions with exponential growth. Microlocal analysis in its classical C ∞ version will be important also. Except for the complex WKB method in Chap. 7, we do not touch upon the subject of analytic operators and methods of analytic microlocal analysis, a subject that would require a much longer and deeper text. We will assume that the reader has some familiarity with standard microlocal analysis, roughly corresponding to the books [51] with A. Grigis and [41], but we think that even without formal prerequisites in that area, much of the book should remain accessible. Applications have not been our main motivation, but rather to present a coherent piece of mathematical analysis where Weyl asymptotic appeared first as a surprise. The text is split into three parts. • Part I is devoted to some functional analysis and to spectral theory in dimension 1. – In Chap. 2 we review some notions including spectrum and the pseudospectrum in the spirit of Davies [35] and Trefethen [151]. – In Chap. 3 we discuss the original result of Hager on Weyl asymptotics for random perturbations of the model operator hD + g, where many ideas appear in a nontechnical context. – In Chap. 4 we use a classical WKB-approach to construct quasimodes for differential operators, generalizing a result of Davies [33] for the non-self-adjoint Schrödinger operators. (The construction in any dimension is given in Chap. 9, and goes back to Hörmander [78, 79].) – Chapter 5 is devoted to Weyl asymptotics for more general differential operators in one dimension, in spirit close to [55]. – In Chap. 6 we establish a result of Bordeaux Montrieux [18] about the norm of the resolvent near the boundary of the range of the symbol. What is remarkable is that

2 Including several articles of the author, often with minor changes.

4

1 Introduction

we get not only a precise upper bound, but even the asymptotics of the norm. Later, in Chap. 10 we review upper bounds in any dimension. – In Chap. 7 we explain the complex WKB method for ordinary differential equations. – In Chap. 8 we review abstract theory of non-self-adjoint operators, following essentially [48]. • Part II deals with various general facts. – In Chap. 9 we generalize the construction of quasimodes to any dimension, cf. [40, 78, 79, 161]. – In Chap. 10 we give resolvent bounds near the boundary of the range of the symbol for semi-classical operators in any dimension. We follow closely [140]. – In Chap. 11 we discuss some abstract questions around the Gearhardt–Prüss– Hwang–Greiner theorem, getting estimates on semigroups from estimates on resolvents. This is mainly joint work with B. Helffer, originally published in [138] and later also in [57]. – In Chap. 12 we give a result about the number of zeros of a holomorphic function with exponential growth. A simpler version of this result was originally proved by Hager, see Proposition 3.4.6. The improvements here are essential for the more precise and general results in Chap. 13 and in Part III. – In Chap. 13, we study the distribution of eigenvalues of small random perturbations of large Jordan blocks. We show that with probability close to 1, the eigenvalues concentrate to a certain circle and have an approximately uniform angular distribution there. Originally I planned a last section, based on a joint work with Vogel [147] about the expected density of eigenvalues inside that circle, but we now refer to that work instead, as well as to related more recent works. • Part III. This part deals with spectral asymptotics for differential operators in arbitrary dimension. – In Chap. 14 we review a result of Markus and Matseev [97] (see also [96]) about Weyl distribution of the real parts of the eigenfrequencies of the damped wave equation. We choose to use Chap. 12. – In Chaps. 15–17 we present and prove a general result on Weyl asymptotics for semiclassical (pseudo-)differential operators on a compact manifold which basically improves the main result of [139]. The proof follows the strategy of [137, 139], with the difference that we can now use the improved results of Chap. 12. – Chapter 18 gives almost sure Weyl asymptotics for the large eigenvalues of differential operators, no longer in the semi-classical limit. We here follow [19]. – In Chap. 19 we apply the results to PT -symmetric operators. Such operators are generally non-self-adjoint, but with an additional symmetry that forces the spectrum to be symmetric around the real line. They have been proposed by physicists as building blocks in new versions of quantum mechanics and the reality of the spectrum is then important. We first show (following [143]) that most PT -symmetric operators have most of their eigenvalues away from R in the semi-classical case as well as

1 Introduction

5

in that of large eigenvalues. Then we describe without detailed proofs some results of [23] about the reality of eigenvalues for semi-classical PT -symmetric analytic Schrödinger operators with a simple potential well in dimension one and a result of [101] about the non-reality of the eigenvalues for semi-classical Schrödinger operators with a double-well potential. – In Chap. 20, prepared jointly with A. Sjöstrand, we give numerical illustrations to Weyl asymptotics and spectral instability, as well as double-well Schrödinger operator. The examples are one-dimensional, but often illustrate results that hold in all dimensions. As already mentioned, this text is a coherent description of some developments for nonself-adjoint operators, but far from all. Even within our special horizons, some aspects are missing and should be the subject of further research: • Give more general and intrinsic classes of random perturbations: a look at the results clearly indicates that this should be possible. • Treat non-elliptic operators like the Kramers–Fokker–Planck operator. • Study boundary value problems. • Discuss applications to resonances (scattering poles). Results for generic perturbations have been obtained by Christiansen et al. [29–31] and in probability in [144] • From the proofs for Weyl asymptotics for operators with random perturbations, it is quite clear that the resolvent bounds improve in the pseudospectral region (cf. [18]). We think that this is an important phenomenon, to be studied further. This book is a research monograph on topics in non-self-adjoint operator theory, and an attempt to give a systematic presentation of some recent developments. It is not selfcontained, and for the full understanding of some chapters, the reader is supposed to have some familiarity with microlocal analysis (or to revise that topic in parallel with the study of selected parts of this book). We will also use some complex analysis and some basic functional analysis, as well as some very basic probability theory, covered for instance by the first two chapters in the book [87] by Kallenberg. The following chapters are more elementary (not excluding various amounts of general mathematical analysis): Chapters 2–4, 7–9, assuming some notions of symplectic geometry, Chaps. 11–13, 15 (the proofs in Chaps. 16 and 17 do use some microlocal analysis), Chaps. 18–20. Acknowledgements Discussions with colleagues and coworkers have been a very important basis for this work. I am grateful to W. Bordeaux Montrieux, M. Hager, M. Hitrik, B. Helffer, K. PravdaStarov, J. Viola, M. Vogel, X. P. Wang, M. Zworski and many others. We also thank the two referees for a very careful work and most useful comments.

Part I Basic Notions, Differential Operators in One Dimension

2

Spectrum and Pseudo-Spectrum

2.1

Operators in Hilbert Spaces, a Quick Review

In this book all Hilbert spaces will be assumed to separable for simplicity. In this section we review some basic definitions and properties; we refer to [88, 116, 117] for more substantial presentations. Let H be a complex Hilbert space and denote by  · , (·|·) the norm and the scalar product, respectively.1 A unbounded (or rather not necessarily bounded) operator S : H → H is given by a linear subspace D(S) ⊂ H, the domain of S, and a linear mapping S : D(S) → H. We say that S is bounded if D(S) = H and Sx < ∞. 0 =x∈H x

S = sup

(2.1.1)

Let L(H, H) denote the corresponding normed space of bounded operators. These definitions and many of the facts below have straightforward extensions to the case of operators S : H1 → H2 , where Hj are two Hilbert spaces. If S : H → H is an unbounded operator, we introduce its graph graph (S) = {(x, Sx); x ∈ D(S)},

(2.1.2)

which is a linear subspace of H × H. We say that S is closed if graph (S) is closed. Every bounded operator S ∈ L(H, H) is closed; conversely, the closed graph theorem tells us

1 Linear in the first argument and anti-linear in the second.

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_2

9

10

2 Spectrum and Pseudo-Spectrum

that if S : H → H is closed and D(S) = H, then S is bounded, i.e., S ∈ L(H, H). In the next proposition, we introduce the adjoint S ∗ of a densely defined operator: Proposition 2.1.1 Let S : H → H be an unbounded operator whose domain D(S) is dense in H. Then there exists an unbounded operator S ∗ : H → H characterized by: D(S ∗ ) = {v ∈ H; ∃C(v) ∈ ]0, +∞ [, |(Su|v)| ≤ C(v)u, ∀u ∈ D(S)}, (Su|v) = (u|S ∗ v); ∀u ∈ D(S), v ∈ D(S ∗ ).

(2.1.3) (2.1.4)

Proof (2.1.3) is just a definition and the space D(S ∗ ) so defined is linear. For v ∈ D(S ∗ ), the map H  u → (Su|v) is a bounded linear functional, so there is a unique element w ∈ H such that (Su|v) = (u|w) for all u ∈ D(S). By definition, S ∗ v = w and it is   straightforward to see that S ∗ v depends linearly on v. We have the following easily verified properties: • S ∗ is closed. • Let S denote the closure of S, so that in general S is the relation H → H whose graph is the closure of the graph of S; graph (S) = graph(S). We say that S is closable if S is an operator. If D(S ∗ ) is dense, then S ∗∗ = S. In particular, S is closable. Definition 2.1.2 Let A, B : H → H be unbounded operators. We say that A ⊂ B if graph (A) ⊂ graph (B), or equivalently if D(A) ⊂ D(B) and Ax = Bx for all x ∈ D(A). If D(A) is dense and A ⊂ B, then B ∗ ⊂ A∗ . Definition 2.1.3 A densely defined operator A : H → H is symmetric if A ⊂ A∗ and self-adjoint if A = A∗ . Notice that a densely defined operator A : H → H is symmetric iff (Ax|y) = (x|Ay), ∀x, y ∈ D(A). An important general problem is to determine when a symmetric operator A : H → H has  : H → H (in the sense that A ⊂ A).  Notice that a self-adjoint a self-adjoint extension A  of a symmetric operator A operator is always closed, so every self-adjoint extension A has to contain the closure A. The “best” case is when A is essentially self-adjoint in the sense that A has a unique self-adjoint extension. It can be shown that when a symmetric operator A is essentially self-adjoint, then the unique self-adjoint extension of A is the closure A. Equivalently, a symmetric operator A is essentially self-adjoint precisely when

2.1 Operators in Hilbert Spaces, a Quick Review

11

A is self-adjoint. Recall that many of these statements are easy to understand if we make the observation that (J (graph (A))⊥ = graph (A∗ ), where the superscript ⊥ indicates the orthogonal space in H × H and 

 0 1 J = : H × H → H × H, −1 0 so that J ∗ J = 1 and J J ∗ = 1 on H × H, where 1 denotes the identity operator. We have the following theorem of von Neumann, see [88, p. 275]: Theorem 2.1.4 Let T : H → H be closed and densely defined. Then the operator T ∗ T with its natural domain D(T ∗ T ) = {x ∈ D(T ); T x ∈ D(T ∗ )} is self-adjoint. Moreover, D(T ∗ T ) is a core for T in the sense that {(u, T u); u ∈ D(T ∗ T )} is dense in D(T ). Definition 2.1.5 A closed densely defined operator T : H → H is normal if T ∗ T = T T ∗ (where the second operator T T ∗ is equipped with its natural domain). Every self-adjoint operator is normal. Many properties of self-adjoint operators extend to normal ones. Spectrum, Resolvent Let T : H → H be closed and densely defined. It is often practical 1 to equip D(T ) with then norm uD(T ) = (u2 + T u2 ) 2 := (u, T u)H×H . We say that z0 ∈ C belongs to the resolvent set ρ(T ) of T if T − z0 : D(T ) → H is bijective and the inverse (z0 − T )−1 : H → H is bounded. Here we write (T − z0 )u = T u − z0 u. (Notice that (T − z0 )−1 is bounded H → H precisely when it is bounded H → D(T ) and that the closed graph theorem tells us that it is indeed bounded, once it is well defined.) When z0 ∈ ρ(T ) we define the resolvent R(z0 ) = (z0 − T )−1 . For z ∈ C, we have (z − T )R(z0 ) = 1 + (z − z0 )R(z0 ),

(2.1.5)

whereas above “1” denotes the identity operator. Here (z − z0 )R(z0 ) ≤ |z − z0 |R(z0 ) is < 1 when z belongs to the open disc D(z0 , 1/R(z0 )) of center z0 and radius 1/R(z0 ), and the operator 1 + (z − z0 )R(z0 ) then has the bounded inverse given by the Neumann series (1 + (z − z0 )R(z0 ))−1 = 1 − (z − z0 )R(z0 ) + ((z − z0 )R(z0 ))2 − · · ·

12

2 Spectrum and Pseudo-Spectrum

We also see that (1 + (z − z0 )R(z0 ))−1  ≤

1 . 1 − |z − z0 |R(z0 )

(2.1.6)

(Here we use that the normed linear space L(H, H) is complete.) From (2.1.5) it now follows that z − T : D(T ) → H has the right inverse  = R(z0 )(1 + (z − z0 )R(z0 ))−1 R(z)

(2.1.7)

with norm  R(z) ≤

R(z0 ) . 1 − |z − z0 |R(z0 )

Since 1 + (z − z0 )R(z0 ) maps D(T ) into itself, we see that the inverse has the same property. Using also that R(z0 )(z − T ) = 1 + (z − z0 )R(z0 ),

(2.1.8)

 we see that R(z), defined in (2.1.7), is also a left inverse, so z belongs to the resolvent set  and R(z) = R(z). Proposition 2.1.6 Let z0 ∈ ρ(T ), z ∈ D(z0 , 1/R(z0 )). Then z ∈ ρ(T ) and R(z) ≤

R(z0 ) . 1 − |z − z0 |R(z0 )

It follows in particular that the resolvent set ρ(T ) is open. Definition 2.1.7 Let T : H → H be a closed densely defined operator. The spectrum σ (T ) is the closed subset of C defined as σ (T ) = C \ ρ(T ).

(2.1.9)

From Proposition 2.1.6 we see that R(z) ≥

1 , z ∈ ρ(T ), dist (z, σ (T ))

(2.1.10)

where dist (z, σ (T )) = infw∈σ (T ) |z−w| denotes the distance from z to the set σ (T ). From the spectral resolution theorem for self-adjoint operators, we have

2.2 Pseudospectrum

13

Theorem 2.1.8 Let T : H → H be self-adjoint. Then σ (T ) ⊂ R and R(z) =

2.2

1 , z ∈ ρ(T ). dist (z, σ (T ))

(2.1.11)

Pseudospectrum

Let P : H → H be closed and densely defined. Definition 2.2.1 Let > 0. The -pseudospectrum of P is the set σ (P ) = σ (P ) ∪ {z ∈ ρ(P ); (z − P )−1  > 1/ }.

(2.2.1)

Some authors put “≥” rather than “>” in this definition. We follow [35, 151] and the book [152]. With this choice, σ (P ) becomes an open subset of C. From (2.1.10) it follows that σ (P ) + D(0, ) ⊂ σ (P ).

(2.2.2)

It is a standard fact for non-self-adjoint operators that the second set in (2.2.2) may be much larger than the first one. For self-adjoint operators we have equality by (2.1.11). Thus for general non-self-adjoint operators, the existence of quasimodes for P − z does not necessarily imply that z is close to the spectrum of P . The following proposition shows though that there is a close link between the existence of quasimodes and the pseudospectrum. (The notion of “quasimode” is implicitly defined in (2.2.3) below.) Proposition 2.2.2 Let > 0, z ∈ C. The following two statements are equivalent: 1) z ∈ σ (P ). 2) z ∈ σ (P ), or ∃u ∈ D(P ), such that u = 1 and (P − z)u < .

(2.2.3)

Proof It suffices to show that for z ∈ ρ(P ) the statement z ∈ σ (P )

(2.2.4)

is equivalent to (2.2.3). Now (2.2.3) is equivalent to: ∃0 = v ∈ H, such that v < , (z − P )−1 v = 1, which in turn is equivalent to (2.2.4) with the choice u = (z − P )−1 v.

 

14

2 Spectrum and Pseudo-Spectrum

In the above situation we call u a quasimode and z the corresponding quasi-eigenvalue. The -pseudospectrum is a set of spectral instability: a small perturbation of P may change the spectrum a lot. This is formalized in the following easy result: Theorem 2.2.3 For every > 0, σ (P ) =



σ (P + A).

(2.2.5)

A∈L(H,H) A<

Proof Denote the right-hand side of (2.2.5) by  σ (P ). Clearly  σ (P ) contains σ (P ), so σ (P ) \ σ (P ). we only have to identify σ (P ) \ σ (P ) and  Let z ∈ σ (P ) \ σ (P ) so that (z − P )−1 exists and is of norm > 1/ . Then, by Proposition 2.2.2, there exist a vector u as in (2.2.3). Let v = (P − z)u, so that v < . Now we can find A ∈ L(H, H) with A < , Au = v. For instance, we can define A by Ax = (x|u)v. Then (P + A − z)u = 0, so z ∈ σ (P + A), and hence z ∈  σ (P ) \ σ (P ). Next, let z ∈ C \ σ (P ), so that (P − z)−1  ≤ 1/ . Let A ∈ L(H, H), A < . Then (P + A − z)(P − z)−1 = 1 + A(P − z)−1 ,

A(P − z)−1  < 1.

Thus 1 + A(P − z)−1 : H → H has a bounded inverse and we see that P + A − z : D(P ) → H has the bounded right inverse (P − z)−1 (1 + A(P − z)−1 )−1 . Similarly, (P − z)−1 (P + A − z) = 1 + (P − z)−1 A is bijective H → H and D(P ) → D(P ) and P + A − z has the bounded left inverse (1 + (P − z)−1 A)−1 (P − z)−1 . We   conclude that z ∈ σ (P + A) and varying A it follows that z ∈  σ (P ). Subharmonicity This property allows us to use the maximum principle in order to establish some general properties of pseudospectra. We start by recalling some general properties, see [84]. Definition 2.2.4 Let  ⊂ C be open and let u :  → [−∞, +∞ [ . We say that u is subharmonic if (a) u is upper semi-continuous, i.e., u−1 ([−∞, s [ ) is open for every s ∈ R, (b) If K ⊂  is compact, h ∈ C(K; R) is harmonic on the interior of K (in the sense that h = 0 there) and h ≥ u everywhere on ∂K, then h ≥ u in K. In this definition we can restrict K to the set of closed discs contained in O. (Theorem 1.6.3 in [84].) Another important property (Theorem 1.6.2 in [84]) is Theorem 2.2.5 Let uα , α ∈ A, be family of subharmonic functions such that u := supα∈A uα is pointwise < +∞ and upper semi-continuous. Then u is subharmonic.

2.2 Pseudospectrum

15

As the name indicates, every harmonic function is subharmonic. We recall the characterization of subharmonic functions as those for which u ≥ 0 in the sense of distributions [84, Theorem 1.6.9–1.6.11]. Theorem 2.2.6 Let  be open and connected and let L(dz) denote the Lebesgue measure on C. (a) If u is a subharmonic function on , not identically −∞, then u ∈ L1loc () and u is a positive distribution:  uvL(dz) ≥ 0, for all 0 ≤ v ∈ C02 ().

(2.2.6)

(b) Conversely, if u ∈ L1loc () and (2.2.6) holds, then there exists a unique subharmonic function U on  that is equal to u almost everywhere. We now return to our general closed and densely defined operator P : H → H. Proposition 2.2.7 The function z → (z − P )−1  is subharmonic on ρ(P ). Proof We use the variational formula u = sup (u|v), u ∈ H, v∈H v=1

to deduce that for z ∈ ρ(P ), (z − P )−1  =

sup ((z − P )−1 u|v). u,v∈H u,v≤1

Here ρ(P )  z → ((z−P )−1 u|v) is harmonic for every fixed (u, v) ∈ H×H. Moreover, ρ(P )  z → (z − P )−1  is continuous and pointwise finite, so Theorem 2.2.5 gives the desired conclusion.   Applying the maximum principle, we get Theorem 2.2.8 Every bounded connected component of σ (P ) contains some point of σ (P ). Proof We first remark that if z0 is a point in the spectrum σ (P ) of P but not in the interior of σ (P ), then (z − P )−1  → +∞, ρ(P )  z → z0 .

16

2 Spectrum and Pseudo-Spectrum

Let V ⊂ C be a bounded connected component of σ (P ). We notice that (z−P )−1  = 1/ everywhere on the boundary of V . If V does not intersect the spectrum of P , then the function f (z) := (z − P )−1  is continuous and subharmonic in a small open neighborhood  of V . Thus the maximum principle for subharmonic functions (apply Proposition 2.2.4 with K = V , h = 1/ ) implies that f (z) ≤ 1/ in V , which is in contradiction with the fact that (z − P )−1  > 1/ in V .  

2.3

Numerical Range

Another set which has interesting connections with the spectrum and the pseudospectrum is the numerical range. As above, let P : H → H be a closed densely defined operator. Definition 2.3.1 We define the numerical range W (P ) ⊂ C of P by  W (P ) =

(P u|u) ; 0 =

u ∈ D(P ) . u2

(2.3.1)

Notice that we get the same set if we restrict u to the set of all u ∈ D(P ) with u = 1. The following theorem is due to Hausdorff and Toeplitz (see [85]). Theorem 2.3.2 W (P ) is convex. Proof Let z0 , z1 ∈ W (P ) be two distinct points, so that zj = (P ej |ej ), ej ∈ D(P ), ej  = 1. Then e0 , e1 are linearly independent and it suffices to show that {(P u|u)/u2 ; u ∈ } is convex, where := Ce0 ⊕ Ce1 . For u ∈ , we have (P u|u) = ( P u|u), where  : H → is the orthogonal projection onto . Thus we may replace P by  P : → and thus reduce the proof of the theorem to the case when H is a two-dimensional Hilbert space. After choosing some orthonormal basis in H we can identify P with a square matrix and H with C2 . Write P = P + iP , where P = (P + P ∗ )/2 and P = (P − P ∗ )/(2i) are Hermitian matrices. After conjugation with a unitary matrix, we can assume that 



λ1 0 P = 0 λ2 where λ1 and λ2 are real. Thus for u = (u1 , u2 ) ∈ C2 \ 0, we have λ1 |u1 |2 + λ2 |u2 |2 (P u|u) (P u|u) = +i . 2 u |u1 |2 + |u2 |2 |u1 |2 + |u2 |2

(2.3.2)

2.3 Numerical Range

17

Now the Hermitian matrix P takes the form   μ1 a , μ1 , μ2 ∈ R, P = a μ2 so finally (P u|u) (λ1 + iμ1 )|u1 |2 + (λ2 + iμ2 )|u2 |2 2(au1 u2 ) = +i . 2 2 2 u |u1 | + |u2 | |u1 |2 + |u2 |2

(2.3.3)

When investigating the set of these values, we may assume that u2 = 1, so that |u1 |2 = 1 − t, |u2 |2 = t,

0 ≤ t ≤ 1.

(2.3.4)

Then

|2(au1 u2 )| ≤ 2|a| t (1 − t). √ √ More precisely, for every θ ∈ [−2|a| t (1 − t), 2|a| t (1 − t)] there exists a u satisfying (2.3.4) such that 2(au1 u2 ) = θ , and we conclude that the set of values in (2.3.2) is equal to

{(1 − t)(λ1 + iμ1 ) + t (λ2 + iμ2 ) + iθ ; |θ | ≤ 2|a| t (1 − t), 0 ≤ t ≤ 1}. The function ellipsoid.

(2.3.5)

√ t (1 − t) is concave, so the set (2.3.5) is convex. More precisely, it is an  

If λ ∈ C is an eigenvalue of P , so that P u = λu for some 0 = u ∈ D(P ), then it is immediate that λ ∈ W (P ). We have Theorem 2.3.3 Let  ⊂ be a connected component of the open set C \ W (P ), so that  is open. If  contains a point z0 which is not in σ (P ), then  ∩ σ (P ) = ∅, and moreover (z − P )−1  ≤

1 , z ∈ . dist (z, W (P ))

(2.3.6)

For the proof we will use the following proposition of independent interest: Proposition 2.3.4 For every z ∈ C, we have dist (z, W (P ))u ≤ (P − z)u, u ∈ D(P ).

(2.3.7)

18

2 Spectrum and Pseudo-Spectrum

Proof For every non-vanishing u ∈ D(P ), we have     |((z − P )u|u)| =  zu2 − (P u|u) = |z − w|u2 , where w = (P u|u)/u2 belongs to W (P ). Thus, dist (z, W )u2 ≤ |z − w|u2 ≤ |((P − z)u|u)| ≤ (P − z)uu. Dividing by the norm of u, we get (2.3.7) for non-zero u. When u = 0, (2.3.7) holds trivially.   Proof of Theorem 2.3.3 Let z0 ∈  \ σ (P ) and let z1 ∈ . Knowing that the resolvent (z0 − P )−1 exists, we can apply (2.3.7) to see that (z0 − P )−1  ≤

1 , dist (z0 , W (P ))

and hence that D(z0 , dist (z0 , W (P ))) ⊂ ρ(P ). Now let γ be a C 1 curve in  that connects z0 to z1 . Then we can find finitely many points w0 , w1 , . . . , wN with w0 = z0 , wN = z1 in (the image of) γ such that wk+1 ∈ D(wk , dist (wk , W (P ))). Iteratively, we see that D(wk , dist (wk , W (P ))) ⊂ ρ(P ) for k = 1, 2, . . . , N and in particular for k = N we get z1 ∈ ρ(P ) and again by (2.3.7) that (2.3.6) holds when z = z1 . Now z1 is an arbitrary point in  and the theorem follows.  

2.4

A Simple Example of a Large Matrix

Consider a Jordan block A0 : CN → CN that we identify with its matrix ⎛ 0 ⎜ ⎜0 ⎜ ⎜0 A0 = ⎜ ⎜. ⎜ ⎜ ⎝0 0

10 01 00 . . 00 00

⎞ 0 ... 0 ⎟ 0 . . . 0⎟ ⎟ 1 . . . 0⎟ ⎟. . . . . .⎟ ⎟ ⎟ 0 . . . 1⎠ 0 ... 0

(2.4.1)

It was observed by Zworski [162] that the open unit disc is a region of spectral instability for A0 , so we expect the eigenvalues to move in a vicinity of that disc when we add a small

2.4 A Simple Example of a Large Matrix

19

perturbation to A0 . Here we shall just look at the simple case of ⎛ 0 ⎜ ⎜0 ⎜ ⎜0 0 Aδ = ⎜ ⎜. ⎜ ⎜ ⎝0 δ

10 01 00 . . 00 00

⎞ 0 ... 0 ⎟ 0 . . . 0⎟ ⎟ 1 . . . 0⎟ ⎟ , δ > 0, . . . . .⎟ ⎟ ⎟ 0 . . . 1⎠ 0 ... 0

(2.4.2)

leaving the study of more general random perturbations for Chap. 13. If we identify CN with 2 ({1, 2, . . . , N}) in the natural way, then A0δ u(j ) = u(j + 1), j = 1, . . . , N − 1, A0δ u(N) = δu(1). Since A0 is a Jordan block, we already know that the spectrum of A0 is reduced to the eigenvalue λ = 0 which has the algebraic multiplicity N. We look for the eigenvalues of A0δ , δ > 0. Now λ ∈ C is such an eigenvalue iff there exists 0 = u ∈ 2 ({1, . . . , N}) such that u(j + 1) = λu(j ), 1 ≤ j ≤ N − 1, δu(1) = λu(N). The spectrum of A0δ consists of the N distinct simple eigenvalues 1

λk = δ N e

2π ik N

, k = 0, . . . , N − 1,

which are equidistributed on the circle of radius δ 1/N . If δ > 0 is fixed and N → +∞, then “the spectrum converges to the circle S 1 ”. We next look at the -pseudospectrum of A0 . Using that AN 0 = 0, we find for z = 0: (z − A0 )−1 =

1 1 1 1 1 (1 − A0 )−1 = (1 + A0 + · · · + N−1 AN−1 ), 0 z z z z z

which has the matrix ⎛

1 1 1 2 3 ⎜ z z1 z1 ⎜0 ⎜ z z2 ⎜ 1 ⎜0 0 z ⎜

1 z4 1 z3 1 z2

... ... ...

⎜. . . . ... ⎜ ⎜0 0 0 0 . . . ⎝ 0 0 0 0 ...

1 zN 1

⎞ ⎟ ⎟

zN−1 ⎟ 1 ⎟ zN−2 ⎟ ⎟

. ⎟ ⎟ ⎟ ⎠

1 z2 1 z

20

2 Spectrum and Pseudo-Spectrum

Applying this to the vector u with u(j ) = 0 for 1 ≤ j ≤ N − 1, u(N) = 1, we see that (z − A0 )−1  ≥

1 . |z|N

(2.4.3)

Moreover, (z − A0 )−1  ≤

 AN−1 A0  1 0 + + · · · + 2 N |z| |z| |z|

(2.4.4)

1 1 1 + = + ···+ N . |z| |z|2 |z| Inequality (2.4.3) shows that the resolvent is large inside D(0, 1) when N is large, while (2.4.4) implies (z − A0 )−1  ≤

1 , |z| > 1. |z| − 1

(2.4.5)

Combining (2.4.3) and (2.4.5) we see that, for 0 < < 1, 1

D(0, N ) ⊂ σ (A0 ) ⊂ D(0, 1 + ).

(2.4.6)

Let us next study the numerical range of A0δ . Notice that A01 describes translation by −1 on 2 (Z/NZ) with the natural identification Z/NZ ≡ {1, . . . , N}, so A01 is unitary and hence normal. It has the eigenvalues λk = e2πik/N , 0 ≤ k ≤ N − 1 (already computed) and the eigenvectors ek given by ek (j ) = N −1/2 e2πij k/N . Since A01 is normal, we can use the spectral resolution theorem to see that W (A01 ) = ch {e

2π ik N

; 0 ≤ k ≤ N − 1}

(2.4.7)

which is contained in the unit disc and contains a disc D(0, 1 − O(1/N 2 )). Here ch B denotes the convex null of a set B ⊂ C, i.e., the smallest convex set containing B. We have similar estimates for W (A0δ ), 0 ≤ δ ≤ 1: 1. W (A0δ ) ⊂ D(0, A0δ ) = D(0, 1) (except in the trivial case N = 1). 2. (A0δ ek |ek ) = λk −((A01 −A0δ )ek |ek ) = λk +(1−δ) k,N , where | k,N | := |ek (1)ek (N)| ≤ 1/N, so W (A0δ ) contains the points λk + (1 − δ) k,N = λk + O(1/N). We conclude that    1 D 0, 1 − O ⊂ W (A0δ ) ⊂ D(0, 1). N

(2.4.8)

2.4 A Simple Example of a Large Matrix

21

We pause to fix some conventions. If B is a normed space, we write “a = O(b)” , if “there is a constant C such that the norm of a is ≤ C|b|” for a family of (a, b) ∈ B × C, implicitly defined from the context. We adopt the less standard convention that 1/O(b) with b > 0 refers to a positive quantity which is ≥ 1/(Cb) for some constant C > 0. We shall return to A0 and its perturbations in Chap. 13 and as a preparation we establish some slightly more refined bounds on the resolvents. When |z| > 1 we can estimate the sum in (2.4.4) by that of the corresponding infinite  j series. When |z| < 1 we can write it as |z|−N N−1 j =0 |z| and estimate the sum by that of the corresponding infinite series. However, (which is of interest when |z| − 1 = O(1/N)) the finite sums can also be estimated by N and we get: ⎧ ⎨ 1 min(N, R ), R ≥ 1, R−1 (z − A0 )−1  ≤ F (|z|), F (R) := R ⎩ 1 min(N, 1 ), R ≤ 1. RN

(2.4.9)

1−R

A straightforward calculation gives the more explicit expression ⎧ 1 ⎪ ⎪ , R ≤ N−1 ⎪ N , N ⎪ R (1 − R) ⎪ ⎪ ⎨ N N−1 N ≤ R ≤ 1, F (R) = N/R , ⎪ N ⎪ ⎪ N/R, 1 ≤ R ≤ N−1 , ⎪ ⎪ ⎪ ⎩1/(R − 1), N R ≥ N−1 .

(2.4.10)

We see that this is a continuous strictly decreasing function on ]0, +∞ [ with range ]0, +∞ [. Consider the perturbation Aδ = A0 + δQ,

(2.4.11)

where Q is a general complex N × N-matrix, so that z − Aδ = (z − A0 )(1 − (z − A0 )−1 δQ). If F (|z|)δQ < 1,

(2.4.12)

(having in mind also the case |z| < 1 below), we can expand the last factor in a Neumann series and get (z − Aδ )−1  ≤ F (|z|)

1 . 1 − F (|z|)δQ

(2.4.13)

22

2 Spectrum and Pseudo-Spectrum

When |z| > 1, we have F (|z|) ≤ 1/(|z| − 1) and (2.4.13) gives (z − Aδ )−1  ≤

1 1 1 · , = |z| − 1 1 − δQ |z| − 1 − δQ |z|−1

(2.4.14)

provided that we replace (2.4.12) by the stronger assumption δQ < 1. |z| − 1

(2.4.15)

The spectrum of Aδ is confined to the disc D(0, R), if δQF (|z|) < 1 for every z outside that disc. Since F is strictly decreasing, we conclude that σ (Aδ ) ⊂ D(0, R) if F (R) =

1 . δQ

(2.4.16)

In view of (2.4.10), this relation splits into four different cases,   1 + O( N1 1 R N (1 − R) = δQ, if R ≤ 1 − , implying δQ ≤ , N eN 1 + O( N1 ) RN 1 1 = δQ, if 1 − ≤ R ≤ 1, implying ≤ δQ ≤ , N N eN N N 1 1 R = NδQ, if 1 ≤ R ≤ , implying ≤ δQ ≤ , N −1 N N −1 1 N , implying ≤ δQ. R = 1 + δQ, if R ≥ N −1 N −1

(2.4.17)

In these four cases we get, respectively, (δQ)1/N ≤ R ≤ (δQ)1/N N 1/N , R = (δQ)1/N N 1/N , R = NδQ,

(2.4.18)

R = 1 + δQ.

2.5

The Non-self-adjoint Harmonic Oscillator

Let c ∈ C\ ] − ∞, 0 ]. We consider the non-self-adjoint harmonic oscillator on R: Pc = Dx2 + cx 2 : L2 (R) → L2 (R).

(2.5.1)

2.5 The Non-self-adjoint Harmonic Oscillator

23

Pc is a closed operator with the dense domain B 2 ⊂ L2 (R), where we define, for general k ∈ N := {0, 1, 2, . . .}, B k = {u ∈ L2 (R); x ν D μ u ∈ L2 , ν + μ ≤ k}.

(2.5.2)

(For 0 > k ∈ Z, we define B k ⊂ S  (R) to be the dual space of B −k .) Here we use the d notation D = Dx = 1i dx . We will use the following facts, see for example [131]: 1. B k are Hilbert spaces when equipped with the natural norms and we have the compact inclusion maps: B k → B j when k > j . 2. Pc − z : B 2 → L2 are Fredholm operators of index 0 (cf. Sect. 8.1), depending holomorphically on z ∈ C. (Here the assumption that c ∈ / ] − ∞, 0 ] guarantees a basic ellipticity property.) 3. The spectrum of Pc : L2 → L2 is discrete: – It is a discrete subset σ (Pc ) of C. – Each element λ0 of σ (Pc ) is an eigenvalue of finite algebraic multiplicity, in the sense that the spectral projection λ0 =

1 2πi



(z − P )−1 dz

∂D(λ0 , )

is of finite rank. Here > 0 is small enough so that σ (Pc ) ∩ D(λ0 , ) = {λ0 }. 4. When c = 1 we get a self-adjoint operator with spectrum {λk = 2k + 1; k = 0, 1, 2, . . .}, and a corresponding orthonormal basis of eigenfunctions is given by   d k 1 2 e0 = √ e−x /2 , ek = Ck x − (e0 ) for k ≥ 1, dx 2π where x −

d dx

is the creation operator and Ck > 0 are normalization constants. Thus, ek =: pk (x)e−x

2 /2

(2.5.3)

are the Hermite functions and pk are the Hermite polynomials. Notice that pk is of order k of the form pk,k x k + pk,k−2 x k−2 + · · · , pk,k > 0 and that pk is even/odd when k is even/odd, respectively.

24

2 Spectrum and Pseudo-Spectrum

Calculation of the Spectrum of Pc When c > 0 this is easy by means of a dilation: Put x = c−1/4 y, so that Dx = c1/4Dy . Then we get Pc = c1/2 (Dy2 + y 2 ),

(2.5.4)

and we conclude that the eigenvalues and eigenfunctions of Pc are given by 1

1

λ(k, c) = c 2 λk =: c 2 (2k + 1),

1

1

ek,c (x) = c 8 ek (c 4 x).

(2.5.5)

Here c1/8 is the normalization constant, ensuring that {ek,c }k∈N is an orthonormal basis in L2 . A more formal proof of this is to introduce the unitary operator 1

1

U u(x) = c 8 u(c 4 x)

(2.5.6)

and to check that 1

Pc U = c 2 U P1 ,

Pc = c 2 U P1 U −1 . 1

(2.5.7)

In the general case when c is no longer real, we can no longer define U u as in (2.5.6) for general L2 functions u, but (2.5.5) still makes sense because ek is an entire function. When defining the fractional powers of c we here use the convention that arg c ∈ ] −π, π [, so that the argument of c1/4 is in [−π/4, π/4 [. We still have that ek,c ∈ B 2 and (Pc − λ(k, c))ek,c = 0,

(2.5.8)

so λ(k, c) are eigenvalues of Pc . We shall prove (cf. [24, 34, 131]) Proposition 2.5.1 σ (Pc ) = {λ(k, c); k ∈ N}. Proof It remains to show that there are no other eigenvalues. (The fact 3 above tells us that every element of σ (Pc ) is an eigenvalue.) For that, we shall use the (formal) adjoint of Pc which is given by Pc∗ = Pc and more precisely that (Pc u|v) = (u|Pc v), ∀u, v ∈ B 2 . We have Lemma 2.5.2 {ek,c ; k ∈ N} spans a dense subspace of L2 (R).

(2.5.9)

2.5 The Non-self-adjoint Harmonic Oscillator

25

Proof Let L ⊂ L2 (R) be the space of all finite linear combinations of the ek,c , k ∈ N. Then u belongs to the orthogonal space L⊥ iff (u|ek,c ) = 0, ∀k ∈ N,

(2.5.10)

and it suffices to show that this fact implies that u = 0. Rewrite (2.5.10) as  0=

u(x)e−c

1/2 x 2 /2

pk,c (x)dx,

(2.5.11)

where we used that ek,c = ek,c =: pk,c (x)e−c

1/2 x 2 /2

,

pk,c (x) = c1/8 pk (c1/4x).

It is easy to see that the Fourier transform  v (ξ ) of the function v(x) = u(x)e−c can be extended to all of Cξ as an entire function and (2.5.11) means that pk,c (−Dξ ) v (0) = 0, ∀k.

1/2 x 2 /2

(2.5.12)

Now, every polynomial can be expressed as a finite linear combination of the pk,c , k ∈ N, so (2.5.12) means that v )(0) = 0, ∀α ∈ N. (Dξα

(2.5.13)

In other words, the power series expansion of the entire function  v (ξ ) vanishes identically, hence  v = 0. Therefore, v = 0 and finally u = 0.   Now we can finish the proof of the proposition. Let λ ∈ σ (Pc ) \ {λ(k, c); k ∈ N} and assume that (Pc − λ)u = 0,

(2.5.14)

for some 0 = u ∈ B 2 . Then, for every k ∈ N, (u|ek,c ) =

1 1 (u|(Pc − λ)ek,c ) = ((Pc − λ)u|ek,c ) = 0, λ(k, c) − λ λ(k, c) − λ

so Lemma 2.5.2 shows that u = 0.

 

Remark 2.5.3 The arguments in the proof can be pushed further to show that the generalized eigenspaces (i.e., the ranges of the spectral projections) corresponding to λ(k, c) are one-dimensional:

26

2 Spectrum and Pseudo-Spectrum

Let λ0 = λ(k0 , c) ∈ σ (Pc ) and let u be a corresponding eigenfunction, so that (Pc − λ0 )u = 0. As in the proof above, we see that (u|ek,c ) = 0, ∀k = k0 . By unique holomorphic extension with respect to c from ]0, +∞[, we have also that ⎧ ⎨0, if  = k, (ek,c |e,c ) = ⎩1, if  = k. Hence we can find d ∈ C such that (u − dek0 ,c |ek,c ) = 0 for all k ∈ N, and as above we conclude that u − dek0 ,c = 0. Furthermore, there can be no Jordan blocks: if (Pc − λ0 )2 u = 0 for some u ∈ B 2 with Pc u ∈ B 2 , then (u|ek,c ) =

1 (u|(Pc − λ0 )2 ek,c ) = 0, (λ(k, c) − λ0 )2

for k = k0 , and as above we see that u = dek,c . A different way to reach the same conclusion would be to choose a continuous deformation [0, 1]  t → ct ∈ C\ ] − ∞, 0 ] with c0 = 1, c1 = c and to notice that the corresponding spectral projections λ(k,ct ) vary continuously with t for every fixed k ∈ N. Then the rank of λ(k,ct ) is independent of t and equal to 1 for t = 0. Thus λ(k,c) = λ(k,c1 ) is also of rank 1. The Numerical Range We have the following result of Boulton [22]: Proposition 2.5.4 W (Pc ) = {t + sc; t, s > 0, ts ≥

1 }. 4

We have already recalled that the lowest eigenvalue of P1 is 1 and this implies (by the spectral resolution theorem) that (P1 u|u) ≥ u2 for all u ∈ B 2 . By integration by parts, we also have (P1 u|u) = Du2 + xu2 . We now recall an additional inequality, the uncertainty relation: Lemma 2.5.5 We have u2 ≤ 2xuDu ≤ Du2 + xu2 , ∀u ∈ B 1 .

(2.5.15)

Proof The second inequality follows from “2ab ≤ a 2 + b2 ” and since S is dense in B 1 it suffices to show the first inequality for every u in S. For such a function, we have by

2.5 The Non-self-adjoint Harmonic Oscillator

27

Cauchy-Schwarz, |(xDu|u)| = |(Du|xu)| ≤ xuDu |(Dxu|u)| = |(xu|Du)| ≤ xuDu. Thus, |(Dx − xD)u|u)| ≤ 2xuDu. But Dx − xD = [D, x] = 1/i, so |((Dx − xD)u|u)| = u2 , and the lemma follows.   Proof of the Proposition For u ∈ B 2 , u = 1, we have (Pc u|u) = Du2 + cxu2 = t + sc, with t = Du2 , s = xu2 , and (2.5.15) implies that 1 ≤ 4ts. It is then clear that W (Pc ) ⊂ {t + sc; t, s > 0, ts ≥

1 }. 4

To show the opposite inclusion, we first notice that if u = e0 , we have equalities in (2.5.15). Indeed, the first expression is equal to 1 since e0 is normalized in L2 , and the last expression is equal to ((D 2 + x 2 )e0 |e0 ) = (e0 |e0 ) = 1, so the inequalities have to be equalities for this choice of u. Since De0  = xe0 , we also have 1 De0  = xe0  = √ . 2

(2.5.16)

Now dilate and consider the functions 1

fλ (x) := λ 2 e0 (λx), λ > 0.

(2.5.17)

Using the same change of variables as in the calculation of the L2 norms, we see that 1 xfλ  = λ−1 xe0  = λ−1 √ , 2 1 Dfλ  = λDe0  = λ √ . 2 Hence we still have equality in the first part of (2.5.15), fλ 2 = 2xfλ Dfλ ,

(2.5.18)

28

2 Spectrum and Pseudo-Spectrum

and (Pc u|u) = Dfλ 2 + cxfλ 2 = t + sc, where now t = Dfλ 2 = λ2 /2 and s = xfλ 2 = 1/2λ2 can take arbitrary positive values with ts = 1/4. We conclude that {t + sc; t, s > 0, ts = 1/4} is contained in W (Pc ). The convex hull of (i.e., the smallest convex set containing) this set is precisely {t +sc; t, s > 0, ts ≥ 1/4}   and the latter set is therefore contained in W (Pc ). Notes Sect. 2.1 is a review of standard material in Hilbert space theory. See e.g., [88, 116, 117]. The discussion of pseudospectra in Sect. 2.2 has been inspired by the influential papers [35, 151] and the monograph [152]. The discussion of the numerical range in relation to pseudospectra in Sect. 2.3 follows [35] and [152]. The discussion of large Jordan blocks in Sect. 2.4 owes a lot to the work [162], where the pseudospectral aspects where highlighted. Some of the results in this section will be used in Chap. 13, devoted to random perturbations of a large Jordan block. A discussion of non-self-adjoint harmonic oscillators from the pseudospectral point of view was given in [34], while the computation of the spectrum is more classical [24, 131] appearing in the study of PDE with double characteristics. See also [22, 152, 162]. Quadratic non self-adjoint operators have been the subject of rather intense investigations in recent years, in connection with resolvent estimates and evolution equations; see [5, 6, 68, 70, 75–77, 109, 112–114, 153, 154].

3

Weyl Asymptotics and Random Perturbations in a One-Dimensional Semi-classical Case

We consider a simple model operator P in dimension 1 and show how random perturbations give rise to Weyl asymptotics in the interior of the range of P . We follow rather closely the work of Hager [55] with some input also from Bordeaux Montrieux [16] and Hager–Sjöstrand [56]. Some of the general ideas appear perhaps more clearly in this special situation. Let P = hDx + g(x), g ∈ C ∞ (S 1 ), S 1  R/2πZ, with symbol p(x, ξ ) = ξ + g(x), (x, ξ ) ∈ T ∗ S 1  S 1 × R, and assume that g has precisely two critical points, a unique maximum and a unique minimum. Here and at many other places of this book we work in the semi-classical limit, i.e., for h > 0 sufficiently small, even though we may sometimes omit the wording “then for h > 0 small enough”. We notice that P is a closed operator L2 (S 1 ) → L2 (S 1 ) with domain equal to the Sobolev space H 1 (S 1 ). The spectrum is discrete and confined to the line 1 z = g, g := 2π



2π

g(x)dx. 0

Indeed, a direct computation shows that the eigenvalues are simple and given by zk = g + kh, k ∈ Z Let   {z ∈ C; min g < z < max g} be open. Put Pδ = Pδ,ω = hDx + g(x) + δQω ,  Qω u(x) = αk, (ω)(u|ek )e (x),

(3.0.1)

C |k|,||≤ h1

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_3

29

30

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

where C1 > 0 is sufficiently large, ek (x) = (2π)−1/2 eikx , k ∈ Z, and αj,k ∼ NC (0, 1) are independent complex Gaussian random variables, centered with variance 1.1 In (3.4.1) we will review the definition. In this book we only use basic probability theory, see for instance the first two chapters in [87]. Qω is compact, so Pδ has discrete spectrum. Let    have smooth boundary. Theorem 3.0.6 Let κ > 5/2 and let 0 > 0 be sufficiently small. Let δ = δ(h) satisfy δ hκ and put = (h) = h ln(1/δ). Then there exists a constant C > 0 such e− 0 / h 2 that for h > 0 small enough, we have with probability ≥ 1 − O( √δ h5 ) that the number of eigenvalues of Pδ in  satisfies |#(σ (Pδ ) ∩ ) −

√

1 vol(p−1 ())| ≤ C . 2πh h

(3.0.2)

Here, we recall that p : T ∗ S 1 → C and we take the volume with respect to the natural symplectic volume element, dxdξ . The conclusion in the theorem is of interest when δ2 √ 5

h

1,

√



1,

which is satisfied when κ>

11 . 4

If instead we let  vary in a collection of subsets that satisfy the assumptions uniformly, δ2 then with probability ≥ 1 − O( h 5 ) we have (3.0.2) uniformly for all  in that collection. The remainder of the chapter is devoted to the proof of this result. Remark 3.0.7 The estimate on the probability in Theorem 3.0.6 is rather crude rough and can be improved by adapting the arguments in [56], as we will do in Sects. 5.5 and 13.7.

1 The choice of Gaussian random variables is convenient in order to formulate a less technical result. In Chap. 15 more general classes of perturbations will be treated, also in higher dimensions.

3.1 Preparations for the Unperturbed Operator

3.1

31

Preparations for the Unperturbed Operator

For z ∈ , let x+ (z), x− (z) ∈ S 1 be the solutions of the equation g(x) = z, with ±g  (x± ) < 0. We sometimes view x± as elements in R (unique mod 2πZ) chosen so that x− − 2π < x+ < x− . Define ξ± (z) by ξ± + g(x± ) = z. Putting ρ± = (x± , ξ± ), we have p(ρ± ) = z,

1 ± {p, p}(ρ± ) = ∓g  (x± ) > 0. i

Here, {a, b} = ∂ξ a ∂x b − ∂x a ∂ξ b denotes the Poisson bracket of two sufficiently smooth functions a(x, ξ ), b(x, ξ ). In the following we us the notation “neigh (A, B)” for “some neighborhood of A in B”, when A is an element or a subset of the topological space B. Let χ ∈ C0∞ (neigh (0, R)) be equal to 1 in a neighborhood of 0 and consider the function i

u(x) = u(x, z; h) = χ(x − x+ (z))e h φ+ (x), where  φ+ (x) = φ+ (x, z) =

x

(z − g(y))dy.

x+ (z)

Then φ+ (x) " |x − x+ (z)|2 , x ∈ neigh (x+ (z), R) and we choose the support of χ small enough, so that this holds on supp (χ(· − x+ (z)).2 Then   2 2 |u(x)| dx = |χ(x − x+ (z))|2 e− h φ+ (x) dx. Applying the Morse lemma (here in a simple one-dimensional situation) as explained for instance in Chapter 2 of [51] and Exercise 2.4 of that book, we see that 

1

|u(x)|2 dx = h 2 b(z; h),

2 We write a " b, when a, b are real numbers with the same sign, such that |a| = O(|b|) and |b| = O(|a|). This notion has a natural extension to the case when a, b are positive Radon measures on the same set in Rn .

32

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

where the symbol b satisfies3 b(z; h) ∼ b0 (z) + hb1 (z) + · · · in C ∞ (), and in particular β

∂zα ∂z b = Oα,β (1), for all α, β ∈ N. Here the subscripts α, β indicate that the estimate is uniform for each fixed (α, β), but not uniformly with respect to these parameters. Moreover, b0 (z) =

√ 2π 2∂x2φ+ (x+ (z))

> 0.

In particular, b(z; h) > 0 on  for h > 0 small enough (since the above holds in a relatively compact neighborhood of ) and we can form a(z; h) = (b(z; h))−1/2, which satisfies a(z; h) ∼ a0 (z) + ha1 (z) + · · · in C ∞ (), a0 (z) = b0 (z)−1/2 . Put ewkb (x) = h−1/4 a(z; h)χ(x − x+ (z))e h φ+ (x) . i

Then ewkb  = 1 where we take the L2 norm over ] x− (z) − 2π, x+ (z)[. Moreover, 1

(P − z)ewkb = O(e− Ch ), where the remainder to the right comes from the cutoff function χ(x − x+ ). Define z-dependent elliptic self-adjoint operators  = (P − z)(P − z)∗ : L2 (S 1 ) → L2 (S 1 ), Q = (P − z)∗ (P − z), Q  = H 2 (S 1 ) (the usual Sobolev space of order 2). They have with domain D(Q), D(Q) discrete spectrum ⊂ [0, +∞ [ and smooth eigenfunctions. (It is a standard fact that elliptic formally self-adjoint differential operators on a compact manifold M with smooth coefficients are essentially self-adjoint with domain H m (M), where m is the order of the operator. Furthermore the spectrum is discrete.) Using that P − z : H 1 → L2 is Fredholm

3 If F is a seminormed space, we write b(h) ∼ b + hb + · · · in F, if for every seminorm q and 0 1 every N ∈ N, we have q(b − (b0 + hb1 + · · · + bN hN )) = O(hN+1 ).

3.1 Preparations for the Unperturbed Operator

33

 ≤ 1. Here N (A) denotes of index zero (exercise!), we see that dim N (Q) = dim N (Q) the kernel of the linear operator A and we shall use R(A) to denote the range. By elliptic regularity we know that the kernel of P − z in H 1 agrees with that of Q in H 2 . If μ = 0 is an eigenvalue of Q, with the corresponding eigenfunction e ∈ C ∞ , then f := (P − z)e is  with the same eigenvalue μ. Pursuing this observation, we see that an eigenfunction for Q  = {t02 , t12 , . . .}, 0 ≤ tj # +∞. σ (Q) = σ (Q) Proposition 3.1.1 There exists a constant C > 0 such that t02 = O(e−1/(Ch)), t12 − t02 ≥ h/C for h > 0 small enough. Proof We have Qewkb = r, r = O(e−1/Ch ), and since Q is self-adjoint, we deduce that t02 is exponentially small. (Cf. (2.1.11).) If e0 denotes the corresponding normalized eigenfunction uniquely determined up to a factor of modulus 1, we see that (P − z)e0 =: v with v exponentially small. Considering this ODE on ] x− (z) − 2π, x− (z)[, we get e0 (x) = Ch− 4 a(h)e h φ+ (x) + F v(x), C = C(h), 1

F v(x) = where φ+ (x) = of integration.

x

x+ (z − g(y))dy.

i

i h



x

i

e h (φ+ (x)−φ+ (y))v(y)dy,

(3.1.1) (3.1.2)

x+

We observe that (φ+ (x) − φ+ (y)) ≥ 0 on the domain

Lemma 3.1.2 We have F L(L2 ,L2 ) = O(h−1/2 ). Proof Indeed,  F w(x) =

K(x, y)w(y)dy,

where   i i i 1{x+ ≤y≤x≤x−} (x, y) − 1{x− −2π≤x≤y≤x+} (x, y) e h (φ+ (x)−φ+ (y)). K(x, y) = h h By treating the three cases • x, y = x+ + O(h1/2 ), • x, y = x− + O(h1/2 ), • x, y = x− − 2π + O(h1/2 )

34

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

separately, we check that for some C > 0, 1

 (φ+ (x) − φ+ (y)) ≥

h2 |x − y| − Ch, on supp K. C

It follows that  sup x

|K(x, y)|dy = O(h− 2 ), sup 1

y



|K(x, y)|dx = O(h− 2 ) 1

and the Schur lemma tells us that F L2 is bounded by the geometric mean of these two quantities.   For our particular v, we see that F v decays exponentially decaying in L2 and hence, since e0 is normalized, that C in (3.1.1) satisfies |C| = 1 + O(e−1/O(h)). Replacing e0 by eiθ e0 for a suitable θ ∈ R and recalling the form of ewkb (x), we conclude that e0 − ewkb is exponentially small. To show that t12 − t02 ≥ h/C, it suffices to show that (Qu|u) ≥ Ch u2 when u ⊥ e0 , or, in other words, that  u ≤

C (P − z)u. h

(3.1.3)

If v := (P − z)u, we again have on ] x− (z) − 2π, x− (z)[ that 1

i

u = Ch− 4 a(h)e h φ+ (x) + F v for some constant C, and the orthogonality requirement on u implies that 0 = (1 + O(h∞ ))C + (F v|e0 ), where (F v|e0 ) = O(h− 2 )v, so C = O(h−1/2 )v and we get the desired estimate on   u. Here we write O(h∞ ) for a quantity which is O(hN ) for every N ∈ N. 1

3.2

Grushin (Schur, Feschbach, Bifurcation) Approach

Let {e0 , e1 , . . .} and {f0 , f1 , . . .} be orthonormal bases of eigenfunctions of Q = (P −  = (P − z)(P − z)∗ , respectively, so that ej , fj ∈ H 2 , Qej = t 2 ej , z)∗ (P − z) and Q j  j = t 2 fj . As observed prior to Proposition 3.1.1, we have Qf j (P − z)ej = αj fj , (P − z)∗ fj = βj ej ,

αj βj = tj2 ,

3.2 Grushin (Schur, Feschbach, Bifurcation) Approach

35

and combining this with ((P −z)ej |fj ) = (ej |(P −z)∗ fj ), we see that αj = β j . Replacing fj by eiθj fj for suitable real values of θj , we can arrange so that αj = βj = tj , which will somewhat simplify the notations. Proposition 3.2.1 Define R+ : H 1 (S 1 ) → C, R− : C → L2 (S 1 ) by R+ u = (u|e0 ),

R− u− = u− f0 .

Then  P(z) :=

P − z R− R+ 0

 : H 1 × C → L2 × C

is bijective with the bounded inverse 

 E(z) =

E E+ E− E−+

 1 O(h− 2 ) O(1) , = O(1) O(e−1/Ch ) 

where the estimates refer to the norms in L(L2 , H 1 ), L(C, H 1 ), L(L2 , C), L(C, C), respectively. (Here when B1 , B2 are normed spaces.) We let L(B1 , B2 ) denote the space of bounded operators B1 → B2 Moreover, E+ v+ = v+ e0 ,

E− v = (v|f0 ).

Here we use the semi-classical norm on H 1 : 1  2 uH 1 = u2 + hDu2 . h

It is a general feature of such auxiliary (Grushin) operators that z ∈ σ (P ) ⇐⇒ E−+ (z) = 0. Indeed, it is easy to show the formulae −1 (P − z)−1 = E − E+ E−+ E− ,

−1 E−+ = −R+ (P − z)−1 R− ,

under the assumptions that E−+ is bijective and that P − z is bijective, respectively.

36

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

Proof of the Proposition We shall show that for every (v, v+ ) ∈ L2 × C, there is a unique solution (u, u− ) ∈ H 1 × C of the system ⎧ ⎨(P − z)u + R u − − ⎩R+ u

= v,

(3.2.1)

= v+ ,

and give estimates and explicit formulae for the solution. Actually, it suffices to find a unique solution (u, u− ) ∈ L2 × C, since we can then deduce that hDu ∈ L2 from the first equation in (3.2.1). Express u and v in the bases introduced above: u=

∞ 

uj e j = u0 e 0 + u⊥ ,

v=

0

and recall that v2 = Then (3.2.1) becomes

∞ 

vj ej = v0 e0 + v ⊥ ,

0

∞ 0

|vj |2 = |v0 |2 + v ⊥ 2 , vj = (v|ej ), and similarly for u.

⎧ ⎨∞ t u f + u f = ∞ v f , − 0 0 j j j 0 j j ⎩u0 = v+ , i.e., t0 v+ + u− = v0 , tj uj = vj for j ≥ 1. Thus, we get the unique solution ⊥

u0 = v+ ,

u =

∞  vj 1

tj

ej , u− = v0 − t0 v+ ,

from which we deduce the expressions for E, E± , E−+ where v = Ev =

∞ 

(vj /tj )ej ,

∞ 0

vj fj and v+ ∈ C:

E+ v+ = v+ e0 ,

1

E− v = v0 , It then suffices to recall that tj ≥

√

E−+ v+ = −t0 v+ .

h/C for j ≥ 1.

 

3.3 d-Bar Equation for E−+

37

d-Bar Equation for E−+

3.3

We will use the following notations for the holomorphic and anti-holomorphic derivatives in the complex variable z:  ∂ 1 ∂ + , ∂z i ∂z   1 1 ∂ ∂ ∂ ∂z = = − . ∂z 2 ∂z i ∂z

1 ∂ ∂z = = ∂z 2



Also recall the expressions for the Laplace operator on C  R2 ,  =

∂ ∂z

2

 +

∂ ∂z

2 = 4∂z∂z.

Proposition 3.3.1 We have ∂z E−+ (z) + f (z)E−+ (z) = 0, z ∈ ,

(3.3.1)

where f (z) = f+ (z) + f− (z),

f+ (z) = (∂z R+ )E+ ,

f− (z) = E− ∂z R− .

(3.3.2)

Thus, ∂z (eF (z)E−+ (z)) = 0 if ∂z F (z) = f (z).

(3.3.3)

Moreover, 2 F (z) = 4∂z f = h



1 1 i {p, p}(ρ+ )

−

1 1 i {p, p}(ρ− )

 + O(1).

(3.3.4)

Proof Formulae (3.3.1)–(3.3.3) follow from the general formula for the differentiation of the inverse of an operator, here ∂z E + E(∂z P)E = 0.   Let (z) : L2 → Ce0 be the spectral projection of Q corresponding to the exponentially small eigenvalue t02 , and choose e0 to be the normalization of (z)ewkb .

38

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

It is easy to see that the various z and z derivatives of ewkb have at most temperate growth in 1/ h. The same fact holds for (z): Lemma 3.3.2 For every (α, β) ∈ N × N, there exists a constant Nα,β ≥ 0 such that ∂zα ∂z (z)L(L2,L2 ) = O(h−Nα,β ), z ∈ . β

Proof By the Cauchy-Riesz functional calculus, (z) =

1 2πi



(w − Q(z))−1 dw,

(3.3.5)

γ

where γ is the oriented boundary of D(0, h/C) for a fixed sufficiently large constant C > 0. For w ∈ γ , we have (w − Q(z))−1 L(L2 ,H 1 ) = O h

  1 , h

(3.3.6)

where we let H 1 = Hh1 be the Sobolev space of order 1 in the semi-classical (h dependent) variant, equipped with the semi-classical norm 1  2 uH 1 = u2 + hDu2 h

with  ·  denoting the L2 -norm on S 1 . (Relation (3.3.6) even holds with Hh1 replaced by Hh2 = D(Q), equipped with its natural semi-classical norm, but this will not be needed.) In fact, we first notice that the resolvent (w −Q(z))−1 is O(1/ h) as a bounded operator in L2 when w ∈ γ , so it suffices to show that hD ◦ (w − Q)−1 is O(1/ h) as a bounded operator in L2 . Now Q = (P − z)∗ (P − z) is of the form (hD)2 + a(x)hD + hD ◦ a(x) + b(x), where a, b are smooth and b is real-valued. It follows that for u ∈ H 1 (S 1 ) (Qu|u) ≥ hDu2 − O(1)uhDu − O(1)u2 ≥ (1/2)hDu2 − O(1)u2 . On the other hand, |(Qu|u)| ≤

 1 Qu2 + u2 . 2

Hence, hDu2 ≤ Qu2 + O(1)u2 ,

3.3 d-Bar Equation for E−+

39

and we get hDu ≤ O(1)(Qu+u) which in turn implies that hDu ≤ O(1)((Q− w)u + u) uniformly for w ∈ γ . Since the L(L2 , L2 )-norm of the resolvent is O(1/ h), we have u ≤ O(1/ h)(Q − w)u, so hDu ≤ O(1/ h)(Q − w)u, which gives the desired estimate on the L2 → L2 norm of hD(w − Q)−1 , and we have verified (3.3.6). β On the other hand, ∂zα ∂z Q(z) vanishes when max(|α|, |β|) ≥ 2, and for (α, β) ∈ {(1, 0), (0, 1), (1, 1)} we see that β

∂zα ∂z Q(z) = O(1) : Hh1 → L2 . For (α, β) = (0, 0), ∂zα ∂z (w − Q(z))−1 is a linear combination of terms, β

(w − Q(z))−1 (∂zα1 ∂z 1 Q(z))(w − Q(z))−1 · · · (∂zαN ∂z N Q(z))(w − Q(z))−1 , β

β

with (αj , βj ) = (0, 0), α1 + · · · + αN = α, β1 + · · · + βN = β. It now suffices to apply β ∂zα ∂z to (3.3.5) and use (3.3.6).   Since e0 is the normalization of (z)ewkb we see that the various (z, z)-derivatives of e0 , and hence also of e0 − ewkb , are of temperate growth. The last quantity is exponentially small in L2 , and by elementary interpolation estimates for the successive derivatives in z, z we get the same conclusion for the higher derivatives of e0 − ewkb . (Cf. [51, Proposition 1.6].) It follows that f+ (z) = (e0 (z)|∂z e0 (z)) = (ewkb (z)|∂z ewkb (z)) + O(e− Ch ), 1

(3.3.7)

and the various z, z-derivatives of the remainder are also exponentially decaying. Using the same simple variant of the method of stationary phase as in Sect. 3.1, we get (ewkb |∂z ewkb ) = O(1) −

i −1 h 2 |a(z; h)|2 h



|χ(x − x+ (z))|2 ∂z φ+ (x, z)e− h φ+ (x,z)dx 2

i = − (∂z φ+ )(x+ (z), z) + O(1), h

(3.3.8)

where the remainder has a complete asymptotic expansion in powers of h, namely, ∼ r0 (z)+hr1 (z)+· · · in the space of smooth functions and, in particular, it remains bounded after taking z, z derivatives. Using that φ+ (x+ (z), z) = 0, (φ+ )x (x+ (z), z) = ξ+ (z), we get after applying ∂z to the first of these relations, that (∂z φ+ )(x+ (z), z) = −ξ+ (z)∂z x+ (z).

(3.3.9)

40

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

On the other hand, if we apply ∂z and ∂z to the equation p(x+ (z), ξ+ (z)) = z, we get ⎧ ⎨p ∂ x + p ∂ ξ = 1, x z + ξ z + ⎩p ∂z x+ + p ∂z ξ+ = 0, x ξ

(3.3.10)

and using that x+ (z) and ξ+ (z) are real valued, ⎧ ⎨p ∂ x + p  ∂ ξ = 1, x z + ξ z + ⎩p ∂z x+ + p ∂z ξ+ = 0, x

(3.3.11)

ξ

which has the solution ∂z x+ =

pξ {p, p}

(ρ+ ),

∂z ξ+ =

−px (ρ+ ). {p, p}

(3.3.12)

Combining (3.3.7)–(3.3.9), we get f+ = O(1) + hi ξ+ ∂z x+ , where the O(1) is stable under differentiation. To this we apply ∂z , then take real parts and notice that ∂z ∂z x+ is real: i ∂z f+ = O(1) +  ∂z ξ+ ∂z x+ . h It now suffices to apply (3.3.12) to get the second (non-trivial) identity in (3.3.4) for the contribution from f+ . The one from f− can be treated similarly.   Using the expressions for the z-derivatives of x+ , ξ+ and the analogous ones for x− , ξ− , we have the following easy result relating (3.3.4) to the symplectic volume: Proposition 3.3.3 Writing z = x + iy, we have dξ+ (z) ∧ dx+ (z) =

2 1 i {p, p}(ρ+ )

−dξ− (z) ∧ dx− (z) = − 1

dy ∧ dx,

2

i {p, p}(ρ− )

dy ∧ dx,

so by (3.3.4), F (z)dy ∧ dx =

1 (dξ+ ∧ dx+ − dξ− ∧ dx− ) + O(1). h

(3.3.13)

3.4 Adding the Random Perturbation

3.4

41

Adding the Random Perturbation

Let X ∼ NC (0, σ 2 ) be a complex Gaussian random variable, meaning that X has the probability distribution X∗ (P(dω)) =

1 − |X|22 e σ d(X)d(X). πσ 2

(3.4.1)

Here σ > 0. For t < 1/σ 2 , we have the expectation value 1 . 1 − σ 2t

E(et |X| ) = 2

(3.4.2)

Bordeaux Montrieux [16] observed that we have the following possibly classical result (improving a similar statement in [56]). Proposition 3.4.1 There exists C0 > 0 such that the following holds: Let Xj ∼ NC (0, σj2 ), 1 ≤ j ≤ N < ∞ be independent complex Gaussian random variables. Put  2 s1 = max σj2 . Then for every x > 0, the probability that N 1 |Xj | ≥ x statistics

P(

N  1



N C0  2 x |Xj | ≥ x) ≤ exp σj − 2s1 2s1 2

 .

1

Proof For t ≤ 1/(2s1 ), we have P(



|Xj |2 ≥ x) ≤ E(et ( 

= exp

N 

ln

1



|Xj |2 −x)

1 − tx 1 − σj2 t



) = e−t x

N 

E(et |Xj | ) 2

1

   ≤ exp C0 σj2 t − tx .

It then suffices to take t = (2s1 )−1 .

 

Recall that Qω u(x) =

 |k|,|j |≤C1 / h

αj,k (ω)(u|ek )ej (x),

1 ek (x) = √ eikx . 2π

(3.4.3)

42

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

Since the Hilbert-Schmidt norm of Qω is given by Qω 2HS = the preceding proposition:



|αj,k (ω)|2 , we get from

Proposition 3.4.2 If C > 0 is large enough, then Qω HS ≤

C − 1 with probability ≥ 1 − e Ch2 . h

(3.4.4)

√ Actually, we get a sharper statement: If λ ≥ 2C0 (1 + 2C1 / h), then Qω HS ≤ λ 2 with probability ≥ 1 − e−λ /4 . Now, we work under the assumption that Qω HS ≤ C/ h and recall that Qω L(L2 ,L2 ) ≤ Qω HS . Assume that 0≤δ so that δQω 

h3/2 ,

(3.4.5)

h1/2 . Then, by simple perturbation theory, Proposition 3.2.1 shows that  Pδ (z) =

Pδ − z R− R+ 0

 : Hh1 × C → L2 × C

is bijective with the bounded inverse  Eδ =

δ E δ E+ δ Eδ E− −+

 ,

δ ) = O(h−1/2 ) in L(L2 , L2 ), h2 δ δ E+ = E+ + O( 3/2 ) = O(1) in L(C, L2 ), h δ δ E− = E− + O( 3/2 ) = O(1) in L(L2 , C), h E δ = E + O(

δ E−+ = E−+ − δE− QE+ + O(

δ2 ). h5/2

In fact, Pδ ◦ E = 1 + K,

  δQω E δQω E+ , K= 0 0

  (δQω E)n (δQω E)n−1 δQω E+ K = 0 0 n

(3.4.6)

3.4 Adding the Random Perturbation

43

and Proposition 3.2.1 implies that 1 + K : H 0 × C → H 0 × C is bijective with inverse 1 − K + K2 · · · . Hence Pδ has the right inverse  δ E δ E+ , = δ Eδ E− −+ 

Eδ = E(1 + K)

−1

where Eδ = E

∞ 

(−δQω E)n ,

0 δ E+ =

∞ 

(−EδQω )n E+ ,

0 δ E− = E−

∞ 

(−δQω E)n ,

0 δ E−+

= E−+ − E− δQω E+ + E− δQω EδQω E+ . . .

Similarly, Pδ has a left inverse, necessarily equal to Eδ . δ and we have the d-bar equation As before, the eigenvalues of Pδ are the zeros of E−+ δ δ ∂z E−+ + f δ (z)E−+ = 0, δ δ f δ (z) = ∂z R+ E+ + E− ∂z R− = f (z) + O(

1 δ ). h h3/2

δ

δ holomorphic) with We can solve ∂z F δ = f δ in  (making eF E−+

F δ = F + O(

δ δ 1 ) = F + O( 3/2 ) . h5/2 h h

(3.4.7)

In fact, recall that  is open and relatively compact in {z ∈ C; min g < z < max g} and that ∂z has the fundamental solution 1/(πz), so we can take  F δ (z) = 

1 f δ (w)L(dw), π(z − w)

δ (z) as the multiplicity of z as Remark 3.4.3 We define the multiplicity of a zero z0 of E−+ 0 δ δ (z). As we observed after Proposition 3.2.1, a zero of the holomorphic function eF (z) E−+ δ (z) agree; this follows from the set of eigenvalues of Pδ in  and the set of zeros of E−+

44

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

the formulae  δ δ (Pδ − z)−1 = E δ (z) − E+ (z) E−+ (z)

−1

δ E− (z),

δ E−+ (z) = −R+ (Pδ − z)−1 R− ,

(3.4.8) (3.4.9)

δ (z) and P − z are bijective (and implying the equivalence of valid respectively when E−+ δ these two properties). If z0 is an eigenvalue, its (algebraic) multiplicity is given by

m(z0 ) = tr (z0 ), where 

1 (z0 ) = 2πi

(z − Pδ )−1 dz

γ

and γ is the oriented boundary of a small disc centered at z0 . Choose γ = γr = ∂D(z0 , r) and let 0 < r → 0. (Here we follow an idea of Vogel [155], to exploit the fact that δ (z ) = 0.) By (3.4.8), ∂z E−+ 0  m(z0 ) = lim tr r→0

1 = lim r→0 2πi

1 2πi  γr

 γr

δ E+

 δ E−+

 δ tr E−+

−1

−1

 δ E− dz

(3.4.10) δ δ E− E+ dz.

For the last identity we used the fact that the integrand is a composition of trace class operators, to move the trace inside the integral and then apply the cyclicity of the trace. Differentiating the identity E δ P δ = 1, we get δ δ δ δ δ δ ∂z E−+ = E− E+ − ((∂z R+ )E+ + E− ∂z R− )E−+ .

If we insert this in (3.4.10), we see that the last term gives the contribution 0, so 1 r→0 2πi





m(z0 ) = lim

γr

δ E−+

−1

δ ∂z E−+ dz.

(3.4.11)

δ is given by On the other hand, the multiplicity m (z0 ) of z0 as a zero of E−+

1 r→0 2πi

m (z0 ) = lim

1 = lim r→0 2πi

  

γr

 γr

δ

δ eF E−+

−1

 δ  δ dz ∂z eF E−+

−1 δ δ E−+ ∂z E−+ dz +

1 lim r→0 2πi



(3.4.12) δ

∂z F dz, γr

3.4 Adding the Random Perturbation

45

and the last term in the last member vanishes, and hence m (z0 ) = m(z0 ). Proposition 3.4.4 Assume that 0 < t δt ( e

h3/2 ,

1, δ − C1 h 0

,

t(

δ , h5/2

(3.4.13)

where C0 ( 1 is fixed. Then for h > 0 small enough we have with probability ≥ 1−e that 1

δ |E−+ (z)| ≤ e− Ch +

Cδ , ∀z ∈ . h

For every z ∈ , we have with probability ≥ 1 − O(t 2 ) − e δ (z)| ≥ |E−+

−

−

1 Ch2

(3.4.14)

1 Ch2

, that

tδ . C

(3.4.15)

Proof By Proposition 3.2.1, E− Qω E+ = (Qω e0 |f0 )  = α,k (ω)(e0 |ek )(e |f0 ) |k|, ||≤

=

1 2π

C1 h



α,k (ω) e0 (k)f0 (),

C |k|, ||≤ h1

of e0 , f0 . This is a sum of independent Gauswhere e0 (k), f0 () are the Fourier  coefficients 2    1 sian random variables ∼ NC 0, 2π | e0 (k)| · |f0 ()| . Now, if X = M 1 Xj , where  2 Xj ∼ NC (0, σj2 ) are independent Gaussian random variables, then X ∼ NC (0, M 1 σj ). Thus, E− Qω E+ ∼ NC (0, σ 2 ), where σ2 =

1 (2π)2 ⎛

⎜ 1 =⎝ 2π

 |k|, ||≤

 |k|≤

C1 h

| e0 (k)|2 |f0 ()|2 C1 h

⎞⎛

⎟⎜ 1 | e0 (k)|2 ⎠ ⎝ 2π

⎞  ||≤

C1 h

⎟ |f0 ()|2 ⎠ .

(3.4.16)

46

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

If C1 is large enough, we get by repeated integration by parts that for every N ∈ N,  | ewkb (k)| ≤ CN

h k

N , |k| >

C1 , h

and similarly for fwkb . Since ewkb  = fwkb  = 1, we get by Parseval’s formula that 1  1  | ewkb (k)|2 = 1 − | ewkb(k)|2 = 1 − O(h∞ ), 2π 2π C C |k|≤

|k|>

1 h

1 h

and similarly for fwkb . On the other hand, we know that e0 − ewkb, f0 − fwkb  = O(h∞ ), so 1  | e0 (k)|2 = 1 − O(h∞ ), 2π C |k|≤

1 h

and similarly for f0 . Equation (3.4.16) now shows that σ = 1 − O(h∞ ).

(3.4.17)

To finish the proof, we combine this with the last relation in (3.4.6) and the fact that 2 |E−+ | ≤ e−1/Ch , to see that with probability ≥ 1 − e−1/(Ch ) , δ |E−+ (z)|

≤e

1 − Ch

 2  δ Cδ +O , z ∈ . + h h5/2

Since δ h3/2 , we get (3.4.14). Similarly by using again the last relation in (3.4.6), for 2 every z ∈ , we have with probability ≥ 1 − O(t 2 ) − e−1/(Ch ) that δ |E−+ (z)|

≥ tδ − e

−1/(Ch)

which together with (3.4.13) implies (3.4.15).



δ2 −O h5/2

 ,  

Proposition 3.4.5 Let κ > 5/2 and fix 0 ∈ ]0, 1 [ sufficiently small. Let δ = δ(h) satisfy 2 δ hκ , and put = (h) = h ln 1δ . Then with probability ≥ 1 − e−1/(Ch ) we e− 0 / h δ | ≤ 1 for all z ∈ . have |E−+

3.4 Adding the Random Perturbation

47

δ | ≥ e −C / h with probability Moreover, if C ( 1, then for any z ∈ , we have |E−+ ≥ 1 − O(δ 2 / h5 ).

This follows from Proposition 3.4.4 by choosing t such that 

1 − C1 h δ e 0 , 5/2 , Cδ C−1 max δ h

t ≤ O(

δ ), h5/2

which is possible because 1 − C1 h e 0 , Cδ C−1 δ

δ . h5/2

Under the same assumptions, we also have with probability ≥ 1 − e−1/(Ch ) , 2

|Fδ − F | ≤ O(

δ h3/2

)

1 1 ≤ O( ) . h h

δ (z): Thus for the holomorphic function u(z) = eFδ (z) E−+

• With probability ≥ 1 − e−1/(Ch ) we have |u(z)| ≤ exp(F (z) + C / h) for all z ∈ . • For every z ∈ , we have |u(z)| ≥ exp(F (z) − C / h) with probability ≥ 1 − O(δ 2 / h5 ). 2

To conclude the proof of Theorem 3.0.6, we will use the following result of Hager [55, Proposition 6.1] with φ = hF . Proposition 3.4.6 Let   C have smooth boundary and let φ be a real valued C 2 function defined in a fixed neighborhood of . Let z → u(z; h) be a family of holomorphic 1. Assume functions defined in a fixed neighborhood of , and let 0 < = (h) • |u(z; h)| ≤ exp( h1 (φ(z) + )) for all z in a fixed neighborhood of ∂. • There exist z1 , . . . , zN depending on h, with N = N(h) " −1/2 such that ∂ ⊂ !N √ 1 1 D(zk , ) and such that |u(zk ; h)| ≥ exp( h (φ(zk ) − )), 1 ≤ k ≤ N(h). Then, the number of zeros of u in  satisfies   √    −1

 #(u (0) ∩ ) − 1 . φ(z)dxdy  ≤ C  2πh  h End of the Proof of Theorem 3.0.6 Consider the holomorphic function δ u(z) = eFδ (z) E−+ (z)

48

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

and put φ(z) = hF. Then with probability ≥ 1 − e−1/(Ch ) , we have 2

1

|u(z)| ≤ e h (φ(z)+C ) for all z ∈ , and for each z ∈ , we have 1

|u(z)| ≥ e h (φ(z)−C ) with probability ≥ 1 − O(δ 2 / h5 ). √ ! D(zk , C ), N = O( −1/2 ). Then with Choose z1 , z2 , . . . , zN ∈ ∂ such that ∂ ⊂ N 1 √ probability ≥ 1 − NO(δ 2 / h5 ) ≥ 1 − O(δ 2 /( h5 )), we have 1

|u(zj )| ≥ e h (φ(zj )−C ) , j = 1, 2, . . . , N.

(3.4.18)

Having already identified the eigenvalues of Pδ with the zeros of u and their multiplicities in Remark 3.4.3, we conclude from Proposition 3.4.6 that with probability as in the theorem,   √    C

#(σ (Pδ ) ∩ ) − 1  . (F )(z)L(dz) ≤  2π  h

(3.4.19)

Returning to (3.3.13), we notice that the maps z → (x+ (z), ξ+ (z)) and z → (x− (z), ξ− (z)) have Jacobians of different sign, so when passing to densities, this relation shows that the direct image under p of 1/ h times the symplectic volume density is equal to F L(dz) + O(1). Consequently,  (F )L(dz) =

O(1) + 

1 vol p−1 (), h  

and the theorem follows. Proof of Proposition 3.4.6 Define φj (z) by iφj (z) = φ(zj ) + 2∂z φ(zj )(z − zj ). Then φ(z) = (iφj (z)) + Rj (z), φj (z) =

Rj (z) = O((z − zj )2 )

2 ∂z φ(z) + O((z − zj )). i

3.4 Adding the Random Perturbation

49

Consider the holomorphic function i

vj (z; h) = u(z; h)e− h φj (z) . 1

1

C

h

Then |vj (z; h)| ≤ e h (φ(z)−iφj (z)) = e h Rj ≤ e

√ , when z − zj = O( ), while

|vj (zj ; h)| ≥ e−

C

h

.

√ In a -neighborhood of zj we put v = vj and make the change of variables w = (z − √ zj )/ ,  v (w) = v(z), so that | v (w)| ≤ eC / h on D(0, 2),

| v (0)| ≥ e−C / h.

(3.4.20)

Using Jensen’s formula (which we review at the end of Sect. 8.4, see also Sect. 13.5), v in D(0, 3/2) (repeated according to their we see that the number of zeros w1 , . . . , wN of  multiplicity) is O( / h). Factorize  v (w) = eg(w)

N  (w − wk ).

(3.4.21)

1

In order to estimate g and g  , we need to find circles on which we have good lower bounds on the product above. This is a classical argument in complex analysis and we follow [135, Section 5].   Lemma 3.4.7 Let x1 , x2 , . . . , xN ∈ R and let I ⊂ R be an interval of length |I | ∈ ]0, +∞ [. Then there exists x ∈ I such that N 

|x − xj | ≥ e

 −N 1+ln

2 |I |



.

j =1

Proof Consider the function N 

F (x) =

ln

j =1

1 . |x − xj |

We have  ln I

1 dx ≤ 2 |x − xj |



|I |/2

ln 0

  2 1 = |I | 1 + ln , t |I |

50

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

since the first integral takes its largest possible value when xj is the midpoint of I . It follows that    2 1 F (x)dx ≤ N 1 + ln . |I | I |I | We can therefore find x ∈ I such that F (x) ≤ N(1 + ln(2/|I |)), so N 

|x − xj | = e

−F (x)

≥e

 −N 1+ln

2 |I |



.

1

  The lemma shows that there exists r ∈ [4/3, 3/2] such that, for every w ∈ ∂D(0, r), N    N    −N 1+ln 43   |r − |wk || ≥ e ≥ e−O( )/ h .  (w − wk ) ≥   1

1

Consequently, for |w| = r,    g(w)  v (w)| ≤ eO( )/ h , e  ≤ eO( )/ h | and by the maximum principle, this estimate extends to D(0, r): g(w) ≤

O( ) , |w| ≤ r. h

If C > 0 is large enough, C

h − g(w) is a non-negative harmonic function on D(0, r) that in view of (3.4.20), (3.4.21) satisfies O( ) C

− g(0) ≤ . h h We can then apply Harnack’s inequality (see e.g., Ahlfors [4, Ch. 6, Section 6.3]) to conclude that C

O( ) 5 − g(w) ≤ , |w| ≤ , h h 4 i.e., g(w) ≥ −

O( ) 5 , |w| ≤ , h 4

3.4 Adding the Random Perturbation

51

and hence, |g(w)| ≤

O( ) 5 , |w| ≤ . h 4

(3.4.22)

Representing g(w) by means of the Poisson kernel for D(0, 5/4):  g(w) =

K(w, ω)g(ω)|dω|, ∂D(0,5/4)

and using the smoothness of the kernel K(w, ω) for w in the interior of the disc, we see that ∇g =

O( ) , |w| ≤ 6/5, h

and hence by the Cauchy-Riemann equations, g =

O( ) , |w| ≤ 6/5. h

(3.4.23)

We now return to (3.4.21). If γ : [0, 1]  t → γ (t) ∈ D(0, 6/5) with γ˙ and 1/γ˙ uniformly bounded and which avoids the zeros wk , then on one hand, 1 1 var argγ  v= 2π 2π

 d(ln v) =  γ

1 2πi

 d ln v= γ

1 2πi

 γ

 v dw,  v

and on the other hand, 1 1 var argγ  v= 2π 2πi

 gdw + γ

N 

var argγ (w − wk ) =

1

O( ) . h

√ We notice that these relations are invariant under the substitution z = zj + w. Now √ √ cover ∂ by " 1/ discs D(zj , ) with zj ∈ ∂. Then there are at most O( / h) zeros of u in each such disc and we assume for simplicity that none of them is situated on ∂. We equip ∂ with the positive orientation and partition it into segments  γj so that √ γj ⊂ D(zj , ). Then  1  2πi



vj γj

vj

dz = O

  h

.

52

3 Weyl Asymptotics and Random Perturbations in a One-Dimensional. . .

Writing u = vj eiφj / h along  γj , we get 1  2πi

 γj 

1 u dz =  u 2πh

1 = 2πh =

1 2πh

 γj 

   2 ∂z φ(z) + O(z − zj ) dz + O i h   2 ∂z φ(z)dz + O . i h

  

γj 

γj 

 v 1 j 2πi  γj vj " #$ % =O ( h )

φj dz + 

Summing over j , we get the number of zeros of u in : 

1 2πi

 ∂

√  1 u (z) 2

dz =  ∂z φ(z)dz + O( ) u(z) 2πh ∂ i h √ 

1 ). = φ(x)dxdy + O( 2πh h

Here, we used Stokes’s formula for the last equality: 2 2 d ∂z φ(z)dz = ∂z ∂z φ(z)dz ∧ dz = 4∂z ∂z φ(z)dx ∧ dy = φ(z)dx ∧ dy. i i   Notes Hager [54] has studied generic perturbations of hD + g(x) and random perturbations of more general operators in dimension 1 in [55]. In this chapter we have restricted our attention to random perturbations of hD + g, following rather closely the approach of [54, 55]. We have added some features (like the use of singular values) from later works, in particular [16] and [56]. Philosophically, one can say that the random perturbation generates Weyl asymptotics by creating an interaction between the points ρ+ (z) and ρ− (z). For the present class of operators this cannot be achieved with multiplicative random perturbations.

4

Quasi-Modes and Spectral Instability in One Dimension

4.1

Asymptotic WKB Solutions

In this section we describe the general WKB construction of approximate “asymptotic” solutions to the ordinary differential equation P (x, hDx )u =

m 

bk (x)(hDx )k u = 0,

(4.1.1)

k=0

on an interval α < x < β, where we assume that the coefficients bk ∈ C ∞ ( ]α, β[ ). Here h ∈ ]0, h0 ] is a small parameter and we wish to solve (4.1.1) up to any power of h. We look for u in the form u(x; h) = a(x; h)eiφ(x)/ h,

(4.1.2)

where φ ∈ C ∞ ( ]α, β[ ) is independent of h. The exponential factor describes the oscillations of u, and when φ is complex valued it also describes the exponential growth or decay; a(x; h) is the amplitude and should be of the form a(x; h) ∼

∞ 

aν (x)hν in C ∞ ( ]α, β[ ).

(4.1.3)

ν=0

Here the sum is in general not convergent and (4.1.3) says that a is the asymptotic sum in the following sense:

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_4

53

54

4 Quasi-Modes and Spectral Instability in One Dimension

For all N, M ∈ N and every interval K  ]α, β[, there is a constant C = CK,N,M such that    N  d M     (a(x; h) − aν (x)hν ) ≤ ChN+1 , ∀x ∈ K.   dx 

(4.1.4)

ν=0

For any sequence aν ∈ C ∞ ( ]α, β[ ) there exists an asymptotic sum by virtue of the following Borel lemma (see e.g., [41, Ch. 2]): Lemma 4.1.1 Let a0 , a1 , a2 , . . . ∈ C ∞ ( ]α, β[ ). Then there exists a(x; h) C ∞ ( ]α, β[ ), 0 < h ≤ h0 , such that (4.1.3) holds as defined in (4.1.4).

∈

Thus, a(x; h) is smooth in x for every h ∈ ]0, h0 ] and that is in general enough in practice. A closer look at the proof shows that we can choose a smooth also in h and even in both variables simultaneously; a ∈ C ∞ ( ]α, β[×[0, h0 ]). Let us make a remark about the uniqueness of a: If  a (x; h) is a second function with ∞ ν  a ∼ 0 aν h , then ∀N, M ∈ N, K  ]α, β[, ∃CK,N,M such that     d N    (a(x; h) −  a (x; h)) ≤ CK,N,M hN+1 , x ∈ K.   dx 

(4.1.5)

We will write this more briefly as  a = a + O(h∞ ) locally uniformly on ]α, β[ and similarly for all the derivatives, or simply  a = a + O(h∞ ) in C ∞ ( ]α, β[ ). To P we associate its semi-classical principal symbol: p(x, ξ ) =

m  k=0

bk (x)ξ k ∈ C ∞ ( ]α, β[ ×R)

(4.1.6)

4.1 Asymptotic WKB Solutions

55

which is a polynomial of degree m in ξ with coefficients that depend smoothly on x. More generally, we can let bk depend on h and consider the semi-classical differential operator P = P (x, hD; h) =

m 

bk (x; h)(hD)k ,

k=0

bk (x; h) ∼

∞ 

bk,ν (x)hν ,

(4.1.7) bk,ν ∈ C ∞ ( ]α, β[ )

ν=0

and its semi-classical principal symbol p(x, ξ ) =

m 

bk,0 (x)ξ k ,

(4.1.8)

k=0

as well as its full symbol P (x, ξ ; h) =

m 

bk (x; h)ξ k ∼ p(x, ξ ) + hp1 (x, ξ ) + · · · ,

(4.1.9)

k=0

m k where pν (x, ξ ) = k=0 bk,ν (x)ξ , and we will sometimes write p = p0 . The last asymptotic sum takes place in the Fréchet space of smooth functions on ]α, β[ ×R that are polynomials of order m in ξ . If we consider a second semi-classical differential operator Q = Q(x, hD; h) =

m  

ck (x; h)(hD)k , where ck ∼

k=0

∞ 

ck,ν (x)hν in C ∞ ,

(4.1.10)

ν=0

then by using Leibnitz’ formula we shall prove that the composition R := P ◦ Q is a semi-classical differential operator of order m + m  with full symbol R(x, ξ ; h) =

∞  hμ μ (∂ P (x, ξ ; h))Dxμ Q(x, ξ ; h), μ! ξ

(4.1.11)

μ=0

where the sum is finite. (When considering pseudodifferential operators, one encounters infinite sums of the same type and they can be defined as asymptotic sums.) Here Dx =  i −1 ∂x denotes the partial derivative with respect to x. Writing Q ∼ k0 hk qk (x, ξ ), r ∼ k k 0 h rk (x, ξ ) similarly to (4.1.9), we get from the composition formula above, that

r1 (x, ξ ) =

r0 (x, ξ ) = p0 (x, ξ )r0 (x, ξ ),

(4.1.12)

1 ∂ξ p0 (x, ξ )∂x q0 (x, ξ ) + p0 (x, ξ )q1 (x, ξ ) + p1 (x, ξ )q0 (x, ξ ). i

(4.1.13)

56

4 Quasi-Modes and Spectral Instability in One Dimension

Proof of (4.1.11) It suffices to treat the case when P = a(x)(hDx )k , Q = b(x)(hDx ) . Leibnitz’ formula gives P ◦ Qu(x) = a(x)(hDx )k (b(x)(hDx ) u(x)) =

k 

a(x)

j =0

k! ((hDx )j b(x))(hDx )k−j + u(x). j !(k − j )!

Thus the symbol of P ◦ Q is k  hj j =0

j!

 hj j k! j j ξ k−j Dx b(x)ξ  = ∂ (a(x)ξ k )Dx (b(x)ξ ) (k − j )! j! ξ k

a(x)

j =0

=

∞  hj j =0

j!

j

j

(∂ξ P )(x, ξ )Dx Q(x, ξ ).  

Here is a more sophisticated, but perhaps more intuitive proof of (4.1.11): We observe that e−ixξ/ h ◦ P (x, hDx ; h) ◦ eixξ/ h = P (x, ξ + hDx ; h) =

∞  hj j =0

j!

j

∂ξ P (x, ξ ; h)(hDx )j ,

where the sum is finite. In particular, P (x, ξ ; h) = P (x, ξ + hDx ; h)(1). Moreover, R(x, ξ + hDx ) = e−ixξ/ h ◦ (P (x, hD)Q(x, hD)) ◦ eixξ/ h = P (x, ξ + hDx ; h)Q(x, ξ + hDx ; h), and in particular, R(x, ξ ; h) = P (x, ξ + hDx )Q(x, ξ + hDx ; h)(1) = P (x, ξ + hDx ; h)(Q(x, ξ ; h)) =

∞  1 j j (∂ P )(x, ξ ; h)hj Dx Q(x, ξ ; h), j! ξ j =0

which proves (4.1.11). If φ ∈ C ∞ ( ]α, β[ ), we have e−iφ(x)/ h ◦ hDx ◦ eiφ(x)/ h = hDx + φ  (x),

4.1 Asymptotic WKB Solutions

57

which is a semi-classical differential operator of order 1 with full symbol ξ + φ  (x). From the composition result above we see that, more generally, e−iφ(x)/ h ◦ (hDx )k ◦ eiφ(x)/ h = (hDx + φ  (x))k ,

k ∈ N,

is a semi-classical differential operator of order k. We can apply this to the conjugated operator P φ := e−iφ(x)/ hP (x, hDx ; h)eiφ(x)/ h =

m 

bk (x; h)(hDx + φ  (x))k

(4.1.14)

k=0

to see that P φ is also a semi-classical differential operator of order m, with symbol P φ (x, ξ ) ∼

∞ 

hν pνφ (x, ξ ),

(4.1.15)

ν=0

where pν (x, ξ ) is a polynomial of order m in ξ with smooth coefficients. Moreover, pφ (x, ξ ) := p0 (x, ξ ) = p(x, φ  (x) + ξ ). φ

(4.1.16)

Now assume that we have found a function φ as above such that p(x, φ  (x)) = 0, x ∈ ]α, β[

(4.1.17)

pξ (x, φ  (x)) = 0, on ]α, β[ ,

(4.1.18)

and, in addition,

where we use the notation pξ = ∂ξ p. Then for the conjugated operator above, we get pφ (x, 0) = 0, We now look for a(x; h) ∼ that

∞

∂ξ pφ (x, 0) = 0.

ν ν=0 aν (x)h

(4.1.19)

with a0 (x) non-vanishing on ]α, β[ , such

P φ (x, hDx ; h)(a(x; h)) ∼ 0,

(4.1.20)

in the sense of asymptotic sums and for that we write P φ as an asymptotic sum of hindependent differential operators of order ≤ m, P φ (x, hDx ; h) ∼ Q0 (x, Dx ) + hQ1 (x, Dx ) + h2 Q2 (x, Dx ) + · · ·

(4.1.21)

58

4 Quasi-Modes and Spectral Instability in One Dimension

Here Q0 = pφ (x, 0) = p(x, φ  (x)) = 0 and Q1 = ∂ξ pφ (x, 0)Dx + p1 (x, 0) = pξ (x, φ  (x))Dx + p1 (x, 0), φ

φ

(4.1.22)

 ν where we used (4.1.16). Applying (4.1.21) to a(x; h) ∼ ∞ ν=0 aν (x)h and regrouping the terms in powers of h, we get the sequence of transport equations Q1 a0 = 0, Q1 a1 + Q2 a0 = 0, Q1 a2 + Q2 a1 + Q3 a0 = 0.

(4.1.23)

... Each equation is of first order and can be solved explicitly. From the first one we get 



a0 (x) = C0 exp −i

x x0

 φ p1 (t, 0) dt , ∂ξ pφ (t, 0)

(4.1.24)

where C0 ∈ C is an arbitrary integration constant and x0 an arbitrary point in ]α, β[ . We fix C0 = 0. Remark 4.1.2 Let a(x; h) ∼ a0 (x) + ha1 (x) + · · · where a0 , a1 , . . . are obtained by solving the transport equations (4.1.23) and where a0 is non-vanishing. Then the general solution is of the form  a (x; h) ∼ c(h)a(x; h), where c(h) ∼ c0 + hc1 + · · · for any complex numbers c0 , c1 , . . . Example 4.1.3 Let  P = −h2

d dx

2 + V (x) − E,

(4.1.25)

P (x, ξ ) = p(x, ξ ) = ξ + V (x) − E, 2

where V ∈ C ∞ (]α, β[; R) and E ∈ R. (a) Assume that V (x) < E on ]α, β[ . We are in the classically allowed region and √ the equation p(x, ξ ) = 0 has two real solutions ξ = ξ± = ± E − V (x), and have two solutions ±φ(x) of the eikonal equation (4.1.17), given by φ(x) = we x √ x0 E − V (y)dy. The WKB constructions above can be applied and we find two oscillatory solutions u± (x; h) = a ± (x; h)e±iφ(x)/ h,

(4.1.26)

4.1 Asymptotic WKB Solutions

59

with u− = u+ , that satisfy P u± = O(hN )

(4.1.27)

locally uniformly on the interval for every N ∈ N together with all their derivatives. (We will express this more briefly by saying that P u± = O(h∞ ) locally uniformly together with all its derivatives.) In this case, we have P φ = (hDx +φ  )2 +V −E = h(2φ  (x)Dx + 1i φ  (x))+(hDx )2

and we get a0 = C(φ  )− 2 . (b) Let V (x) > E on ]α, β[ . We are in the classically forbidden region (in the sense that there are no points in the real phase space where p vanishes). Now the equation √ p(x, ξ ) = 0 has the two non-real solutions ξ = ±i V (x) − E. Let ψ(x) = x √ x0 V (y) − Edy. The WKB method produces two functions, 1

v± (x; h) = b± (x; h)e±ψ(x)/ h, such that P (x, hDx ; h)(b± (x; h)eψ(x)/ h) = r ± (x; h)e±ψ(x)/ h, where r ± (x; h) = O(h∞ ) locally uniformly with all their derivatives. 1 Again, b0 = C(ψ  )− 2 . Notice that if for some (x0 , ξ0 ) ∈ ]α, β[ ×C p(x0 , ξ0 ) = 0, pξ (x0 , ξ0 ) = 0,

(4.1.28)

then by the implicit function theorem there is a small open interval I ⊂ ]α, β[ containing x0 such that the equation p(x, ξ ) = 0 has a solution ξ = ξ(x) ∈ C depending smoothly on x ∈ I with ξ(x0 ) = ξ0 , pξ (x, ξ(x)) = 0. If we let φ ∈ C ∞ (I ) be a primitive of ξ(x), so that φ  (x) = ξ(x), then we get p(x, φ  (x)) = 0,

pξ (x, φ  (x)) = 0.

Summarizing the discussion, we have Theorem 4.1.4 Consider the semi-classical differential operator P = P (x, hD; h) =

m  k=0

bk (x; h)(hD)k ,

x ∈ ]α, β[ ,

(4.1.29)

60

4 Quasi-Modes and Spectral Instability in One Dimension

where bk (x; h) ∼ symbol

∞

ν ν=0 bk,ν (x)h ,

bk,ν ∈ C ∞ ( ]α, β[ ) and its semi-classical principal

p(x, ξ ) =

m 

bk,0 (x)ξ k .

(4.1.30)

k=0

Let (x0 , ξ0 ) ∈ ]α, β[ ×C be a point where (4.1.28) holds. Then there exist an open interval I with x0 ∈ I ⊂ ]α, β[ and a function φ ∈ C ∞ (I ) such that p(x, φ  (x)) = 0,

pξ (x, φ  (x)) = 0,

φ  (x0 ) = ξ0 .

This function is uniquely determined up to a constant.  ν For any such (I, φ) there exists a(x; h) ∼ ∞ 0 aν (x)h with a0 = 0 on I such that P (x, hD; h)(eiφ(x)/ h a(x; h)) = r(x; h)eiφ(x)/ h,

r = O(h∞ ),

locally uniformly on I with all its derivatives.

4.2

Quasimodes in One Dimension

Davies [33] observed that for the one-dimensional Schrödinger operator we can construct quasimodes for values of the spectral parameter that may be quite far from the spectrum of the operator. Zworski [161] observed that this result can be viewed as a special case of more general and older results of Hörmander [78, 79] and generalized the result of Davies to more general operators in the semi-classical limit in arbitrary dimension, by adaptation of Hörmander’s results via a known reduction. In [40] a direct proof was given. (See Chap. 9.) Here we explain the result in the simpler one-dimensional case. Consider an operator as in (4.1.7). The formal complex adjoint is given by P (x, hD; h)∗ =

m  0

(hDx )k bk (x; h) =

m 

ck (x; h)(hDx )k ,

(4.2.1)

0

where ck (x; h) ∼ ck,0 (x) + hck,1 (x) + · · · in C ∞ ( ]α, β[ ) and ck,0 (x) = bk,0 (x). The (semi-classical) principal symbol of P ∗ is equal to p(x, ξ ) (when ξ is real), where p(x, ξ ) is the one of P . Motivated by the notion of normal operators, we are interested in the commutator [P , P ∗ ].

4.2 Quasimodes in One Dimension

61

Proposition 4.2.1 Let Q(x, hD; h) =

m  

ck (x; h)(hDx )k , (4.2.2)

0

ck (x; h) ∼ ck,0 (x) + hck,1 (x) + · · · in C ∞ ( ]α, β[ ), be a second semi-classical differential operator of the same type as P . Then [P , Q] = h

m+ m−1 

dk (x; h)(hDx )k =: hR(x, hDx ; h),

(4.2.3)

0

where R is a semi-classical differential operator of order m + m  − 1 of the same type and whose principal symbol is given by r(x, ξ ) =

1 1 {p, q} = (pξ qx − px qξ ). i i

(4.2.4)

This result follows easily from the composition formula (4.1.11) and especially from (4.1.12), (4.1.13). Notice that the Poisson bracket {a, b} := aξ (x, ξ )bx (x, ξ ) − ax (x, ξ )bξ (x, ξ ) of two smooth functions a and b is anti-symmetric {a, b} = −{b, a},

(4.2.5)

{a, a} = 0.

(4.2.6)

and in particular

From the proposition and the fact that the principal symbol of P ∗ is equal to p(x, ξ ), we get [P , P ∗ ] = hR,

r=

1 {p, p}, i

(4.2.7)

where r denotes the principal symbol of R. Notice that r is real-valued (reflecting the fact that [P , P ∗ ] is formally self-adjoint). We also have the formula 1 {p, p} = −2{p, p}. i

(4.2.8)

62

4 Quasi-Modes and Spectral Instability in One Dimension

Example 4.2.2 Let  P = −h2

d dx

2

+ V (x), V ∈ C ∞ ( ]α, β[ ).

(4.2.9)

Then p(x, ξ ) = ξ 2 + V (x) and an easy computation gives 1 {p, p} = −4ξ V  (x). i

(4.2.10)

Back to the general case, let (x0 , ξ0 ) ∈ ]α, β[ ×R = T ∗ ]α, β[ be a point where 1 {p, p} > 0. i

p(x0 , ξ0 ) = 0,

(4.2.11)

In particular, pξ (x0 , ξ0 ) = 0,

(4.2.12)

so we are in the situation of Theorem 4.1.4. Let ξ(x) ∈ C ∞ (neigh(x0 , ]α, β[ )) (using the notation “neigh(x, A)” for “some neighborhood of x in A”) be the solution of p(x, ξ(x)) = 0 with ξ(x0 ) = ξ0 and pξ (x, ξ(x)) = 0. Let  φ(x) =

x

ξ(y)dy

(4.2.13)

x0

be the corresponding solution of p(x, φ  (x)) = 0,

φ(x0 ) = 0, φ  (x0 ) = ξ0 .

(4.2.14)

Proposition 4.2.3 In the situation above, φ  (x0 ) > 0. Proof We differentiate (4.2.14) and get for x = x0 : px (x0 , ξ0 ) + pξ (x0 , ξ0 )φ  (x0 ) = 0, φ  (x0 ) = −

px pξ px (x0 , ξ0 ) = − . pξ (x0 , ξ0 ) |pξ |2

(4.2.15)

4.2 Quasimodes in One Dimension

63

Hence, 1 (−px p ξ + pξ px ) 2i 1 = |pξ |−2 {p, p}(x0 , ξ0 ) > 0. 2i

φ  (x0 ) = |pξ |−2

  Let 1

f (x; h) = h− 4 a(x; h)eiφ(x)/ h,

x ∈ neigh (x0 , ]α, β[) =: J

(4.2.16)

be an asymptotic solution of P (x, hDx ; h)f (x; h) = O(h∞ )eiφ(x)/ h

(4.2.17)

with a ∼ a0 (x) + ha1(x) + · · · in C ∞ (J ) and a0 non-vanishing. It follows from (4.2.14)– (4.2.16) that after an arbitrarily small shrinking of J , there exists a constant C > 0 such that 1 ≤ f (x; h)2L2 (J ) ≤ C, C

(4.2.18)

and for every δ > 0, 

1

|x−x0 |≥δ

δ2

|f (x; h)|2dx = O(h− 2 e− Ch ),

0 0, we have 0 ∈ σhN (P ) when h is sufficiently small depending on N.

64

4 Quasi-Modes and Spectral Instability in One Dimension

We can have other values than 0 for the spectral parameter. It suffices to find ρ(z) = (x(z), ξ(z)) ∈ T ∗ I such that p(x(z), ξ(z)) = z,

1 {p, p}(x(z), ξ(z)) > 0. i

(4.2.21)

Here we observe that if we identify p with the smooth map F : T ∗ I  (x, ξ ) → (p(x, ξ ), p(x, ξ )) ∈ R2 , then dF (x, ξ ) is bijective precisely at the points where i −1 {p, p}(x, ξ ) = 0. By the implicit function theorem, the points for which we can solve (4.2.21) form an open set + ⊂ C and locally we can choose the solution (x(z), ξ(z)) to depend smoothly on z ∈ + . The preceding theorem can be generalized as follows: Theorem 4.2.5 For every N > 0 and every compact set K ⊂ + , there exists h = h(N, K) > 0 such that K ⊂ σhN (P ) for 0 < h ≤ h(N, K). Exercise Show that if z = p(x, ξ ), then there exists u ∈ C0∞ (R) which is normalized in L2 , such that (P − z)u = O(h1/2 ). Identifying p again with the map T ∗ I  (x, ξ ) → (p, p) ∈ R2 , we see by Sard’s theorem that the image N under p of {ρ ∈ T ∗ I ; i −1 {p, p} = 0} is of (Lebesgue) measure 0. We get Proposition 4.2.6 Let ± = {p(ρ); ρ ∈ T ∗ I, ±i −1 {p, p}(ρ) > 0}. Then R(p) = + ∪ − ∪ N ,

(4.2.22)

where N is of measure 0. If p(x, ξ ) is an even function of ξ , then i −1 {p, p} is odd and we have + = − , so (4.2.22) becomes R(p) = + ∪ N . In particular, this is the case for the semi-classical Schrödinger operator in Example 4.2.3. Noticing that var argγ (p − z) = ±2π if γ = ∂D(ρ, ) is the positively oriented boundary of a small disc centered at a point ρ ∈ p−1 (z) where ±i −1 {p, p} is > 0, it is possible to show that + = − under more general assumptions. Now return to the non-self-adjoint harmonic oscillator P = Pc in (2.5.1) with c ∈ C\ ] − ∞, 0]. We have p(x, ξ ) = ξ 2 + cx 2 and := R(p) = {t + sc; t, s ≥ 0} and it is ◦

easy to see that + = − = is the interior of . Assume that c > 0 in order to fix the ideas. Davies [34] showed that if 0 < θ < arg (c), then in the case h = 1, we have (Ph=1 − E)−1  → +∞, E = eiθ F,

F → +∞.

4.2 Quasimodes in One Dimension

65

It is now easy to see that we have the following improvement when E tends to infinity along the half-ray eiθ [0, +∞[ : For every N ∈ N, there exists c = c(N, θ ) > 0, such that (Ph=1 − E)−1  ≥ c|E|N , E = eiθ F, F ≥ 1.

(4.2.23)

Here, we adopt the convention that (Ph=1 − E)−1  = +∞ when E ∈ σ (Ph=1 ). In fact, the equation (Ph=1 − E)u = v reads (Dx2 + cx 2 − E)u = v, and the change of variable x = |E|1/2 x (which changes the L2 norms of u and v by the same factor) gives 2 x 2 − E)u = v, (|E|−1 D x + c|E|

which we can write 2 ((hD x 2 − eiθ )u = hv, x ) + c

h = 1/|E|.

Since eiθ ∈ + , we can find u ∈ B 2 with u = 1 such that v ≤ CN hN , and (4.2.23) follows. Notes Section 4.1 is largely classical. As already mentioned, Davies [33] constructed quasimodes for the non-self-adjoint one-dimensional Schrödinger operator and Zworski [161] observed that this result can be viewed as a special case of more general and older results of Hörmander [78, 79] and deduced from that a generalization to more general semi-classical operators. A direct proof was given in [40]. (See Chap. 9.) In the analytic framework we obtain exponentially accurate quasimodes (cf. [40, 132]), implying that we can replace c|E|N by c exp |E| in (4.2.23). In the one-dimensional case more precise results are possible, using the complex WKB-method. Cf. [39]. See also [63,64] for related results.

5

Spectral Asymptotics for More General Operators in One Dimension

In this chapter, we generalize the results of Chap. 3. The results and the main ideas are close, but not identical, to the ones of Hager [55]. We will use some h-pseudodifferential machinery, see for instance [41, Ch. 7].

5.1

Preliminaries for the Unperturbed Operator

We will work in L2 (R). Only minor modifications are required if we wish to replace R by the compact manifold T = S 1 = R/(2πZ). Actually, the discussion in this section extends to the case of Rn and general smooth compact manifolds, respectively. Let m ∈ C ∞ (R2x,ξ ; ]0, +∞[ ) be an order function in the sense that for some fixed C0 ≥ 1, N0 ≥ 0, m(X) ≤ C0 X − Y N0 m(Y ), X, Y ∈ R2 .

(5.1.1)

Here, for X = (x, ξ ), we write X = (1 + X2 )1/2 , X2 = x 2 + ξ 2 . Basic examples of such functions are • m(X) = ξ m0 , where ξ  = (1 + ξ 2 )1/2 , • m(X) = Xm0 , where m0 ∈ R.

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_5

67

68

5 Spectral Asymptotics for More General Operators in One Dimension

We say that P ∈ C ∞ (R2 ) belongs to the symbol space S(m), where m is an order function, if for all α, β ∈ N, there exists Cα,β = Cα,β (P ) such that β

|∂xα ∂ξ P (x, ξ )| ≤ Cα,β m(x, ξ ), ∀(x, ξ ) ∈ R2 .

(5.1.2)

When P depends on additional parameters (like the semi-classical parameter h ∈ ]0, h0 ], h0 > 0), we say that P ∈ S(m) if (5.1.2) holds uniformly. S(m) is a Fréchet space with the smallest possible constants Cα,β (P ) as seminorms. We say that P ∈ hN S(m) = S(hN m), when P depends on h and (5.1.2) holds uniformly with m there replaced by hN m. If P ∈ S(m), we define a corresponding h-pseudodifferential operator by using the Weyl quantization:  i x +y 1 , θ )u(y)dydθ P (x, hD)u(x) = e h (x−y)·θ P ( 2πh 2  1 x+y , hθ )u(y)dydθ. = ei(x−y)·θ P ( 2π 2

(5.1.3)

As explained for instance in [41, Lemma 7.8], this gives a continuous operator P : S(R) → S(R), which extends to S  (R) → S  (R), and we use the same letter to denote the extension. The semi-classical version of the Calderón-Zygmund theorem tells us that when m = 1, then P : L2 (R) → L2 (R) is bounded and the norm is bounded by a constant times a finite sum of the Cα,β (P ). In the h-dependent case, we say that P = Ph ∈ S(m) is a classical symbol, and write P ∈ Scl (m), if there exist p0 , p1 , . . . ∈ S(m) independent of h, such that P ∼

∞ 

hk pk

in S(m),

(5.1.4)

0

in the sense that for every N ∈ N, P−

N 

hk pk ∈ hN+1 S(m).

(5.1.5)

0

The leading term p := p0 is called the semi-classical principal symbol. We assume that m≥1

(5.1.6)

∃z0 ∈ C such that p − z0 is elliptic

(5.1.7)

and

5.1 Preliminaries for the Unperturbed Operator

69

in the sense that (p(X) − z0 )−1 = O(1/m(X)) uniformly for X ∈ R2 . Then it is standard to check that (p − z0 )−1 ∈ S(1/m) (noticing that 1/m is an order function). We also know (cf. [41, Ch. 8]) that when h > 0 is small enough, P − z0 is bijective S → S, S  → S  and the inverse Q = (P − z0 )−1 is a classical h-pseudodifferential operator with symbol Q ∼ q0 + hq1 + · · · in S(1/m), q := q0 =

1 . p−z

For h small enough, we can define the semi-classical Hilbert space H (m) = (P − z0 )−1 L2 (R), which does not depend on the choice of P − z0 as above: After regularization and without changing the order of magnitude of m, we can assume that m ∈ S(m), then define H (m) = m−1 L2 (R), where m = m(x, hD) and m−1 = (m(x, hD))−1 . We also have Proposition 5.1.1 For h small enough, the closure of P as an unbounded operator in L2 with domain S is given by P : H (m) → H (1) = L2 with domain D(P ) = H (m). (We use the same letter to denote the closure.) Again, this follows from standard elliptic theory of h-pseudodifferential operators. Having now a closed operator in L2 , we can consider its spectrum σ (P ). Let

= (p) = p(R2 ).

(5.1.8)

Proposition 5.1.2 • p − z is elliptic for every z ∈ C \ (p), • If K ⊂ C is compact, h-independent and disjoint from (p), then for h small enough, P − z : H (m) → L2 is bijective for every z ∈ K. Here, the first property is elementary and quite easy to verify. The second property is a standard elliptic result (as in Proposition 5.1.1). Define

∞ (p) = the set of accumulation points of p(ρ), ρ → ∞,

(5.1.9)

so that

∞ (p) ⊂ (p).

(5.1.10)

70

5 Spectral Asymptotics for More General Operators in One Dimension

Proposition 5.1.3 Let W  C be independent of h with smooth boundary and assume that • W is open and simply connected, • W ⊂

, • W ∩ ∞ = ∅. Then, for h small enough, the spectrum of P in W is discrete, i.e., it consists of a discrete set (in W ) of eigenvalues of finite algebraic multiplicity. Proof Let z0 ∈ W \ and let κ : C \ {z0 } → C \ {z0 } be a diffeomorphism onto its image such that • κ(W \ {z0 }) ∩ W = ∅,  , where W  is a neighborhood of W that we can choose arbitrarily • κ(z) = z, for z ∈ C\W small.  with its closure disjoint from ∞ . Then p  := p ◦ κ is equal to p outside a We choose W  compact set and p  − z is elliptic for z ∈ W . Let (x, hD) = P (x, hD) + ( P p − p)(x, hD). The symbol p  − p has compact support and we know (cf. e.g., [41, Ch. 9]) that the corresponding operator is compact and even of trace class. For h > 0 small enough, P − z : H (m) → H (1) is bijective for every z ∈ W . Write on the operator level P − z = (1 + (p − p )(P − z)−1 )(P − z), z ∈ W. Here W  z → 1 + (p − p )(P − z)−1 is a holomorphic family of Fredholm operators, invertible for at least one value of z (in W \ ). By analytic Fredholm theory, reviewed in Chap. 8, we know that (for each h small enough) 1 + (p − p )(P − z)−1 is bijective for z ∈ W away from a discrete set (h), and since the spectrum of P in W has to be contained in (h), it is a discrete set. Moreover, the singularities of 1 + (p − p )(P − z)−1 at (h) are poles of finite order with finite-rank Laurent coefficients, so we have the same property for (P − z)−1 = (P − z)−1 (1 + (p − p )(P − z)−1 )−1 , and in particular the spectral projections (z) =

1 2πi



(z − P )−1 dz, 0 <

∂D(z, )

1,

5.2 The Result

71

have finite rank (equal to the algebraic multiplicity of z as an eigenvalue of P ) for every z ∈ (h).  

5.2

The Result

We treat the case of operators on R and indicate later the modifications to be made in the case of T. Let W be as in Proposition 5.1.3. Then p−1 (W ) is compact. Let  ⊂ W be open, h-independent, simply connected with smooth boundary. We assume 1  ⊂ , and for every ρ ∈ p−1 (), we have {p, p}(ρ) = 0. i

(5.2.1)

This means that dp, and dp are linearly independent at every point of p−1 (), and by the implicit function theorem, we conclude that p−1 (z) is a finite set for every z ∈ , whose cardinality is independent of z. Since  is simply connected, we can write p−1 (z) = {ρ1 (z), . . . , ρM (z)}, where ρj (z) are mutually distinct and depend smoothly on z ∈ . Proposition 5.2.1 The cardinality of p−1 (z) is even = 2N for every z ∈ , and we can write + − (z), ρ1− (z), . . . , ρN (z)}, p−1 (z) = {ρ1+ (z), . . . , ρN

where ρj± (z) depend smoothly on z and ±i −1 {p, p}(ρj± (z)) > 0. Proof We only have to prove that the number points in p−1 (z) with i −1 {p, p}(ρ) > 0 is equal to the number of points with i −1 {p, p}(ρ) < 0. For that, we notice that if ρ ∈ p−1 (z) and γ is a simple closed positively oriented loop around ρ in R2 \ {ρ}, confined to a small neighborhood of ρ, then   1 1 var argγ (p − z) = sign {p, p}(ρ) . 2π i

(5.2.2)

Here, to define positive orientation of a closed contour, we take for granted that notion in the case of contours in C and make the corresponding convention that (2i)−1 dz ∧ dz = dy ∧ dx > 0. As for R2x,ξ , the corresponding positive form is the symplectic one, σ = dξ ∧ dx. (Or more down-to-earth, we identify R2x,ξ  R2z,z  C.) Then (5.2.2) follows by direct calculation, or by the identity, 1 1 dp ∧ dp = {p, p}dξ ∧ dx, 2i 2i cf. Proposition 3.3.3.

(5.2.3)

72

5 Spectral Asymptotics for More General Operators in One Dimension

Let γ be the positively oriented boundary of ∂BR2 (0, R), for R ( 1. Then,  1 1 var argγ (p − z) = var argγj (p − z), 2π 2π M

(5.2.4)

1

where γj are the oriented boundaries of BR2 (ρj (z), ), 0 <

p=p  along γ when R is large enough, so

1. On the other hand,

1 1 var argγ (p − z) = var argγ ( p − z) = 0. 2π 2π

(5.2.5)

. The result follows Indeed, BR2 (0, R) is contractible and z is not in its image under p from (5.2.2), (5.2.4), (5.2.5).   From the proposition and its proof, in particular (5.2.3), we get p∗ (|σ |) =

N  1



 2i 2i L(dz), − {p, p}(ρj+ (z)) {p, p}(ρj− (z))

(5.2.6)

where L(dz) = |dz∧dz| is the Lebesgue measure and |σ | = |dξ ∧dx| is the symplectic volume element.  Let L > 0 be large enough, so that πx p−1 () ⊂ ] − πL, πL[, where πx : R2x,ξ → Rx is the natural projection. Let χ ∈ C0∞ (R; [0, 1]) be equal to 1 on [−πL, πL]. Our perturbed operator is given by Pδ = P (x, hD; h) + δQω , Qω u(x) =



α,k (ω)(χu|ek )(χe )(x),

(5.2.7) (5.2.8)

C |k|, ||≤ h1

where C1 > 0 is sufficiently large, ek (x) = (2πL)−1/2 eikx/L , k ∈ Z, and αj,k ∼ NC (0, 1) are independent complex Gaussian random variables. Notice that the functions ek form an orthonormal family in L2 ([−πL, πL[ ). Alternatively, we could have taken Qω u =



α,k (ω)(u| ek ) e ,

C k,≤ h1

√ where  ej (x; h) = h−1/4 ej (x/ h) and ej is the sequence of L2 -normalized Hermite functions (reviewed in Sect. 2.5), so that  ej is an orthonormal family of eigenfunctions of the semi-classical harmonic oscillator.

5.2 The Result

73

Theorem 5.2.2 Let    have smooth boundary, and left e− 0 / h ≤ δ h7/2 where

0 > 0 is a sufficiently small constant. For 2Nh ln(1/δ) ≤ ≤ 1, we have with probability ≥ 1 − O( −1/2 )e− /2h that   √ −1   #(σ (Pδ ) ∩ ) − vol p ()  ≤ O(1) .  2πh  h

(5.2.9)

If instead we let  vary in a set of subsets that satisfy the assumptions uniformly, then with probability ≥ 1 − O( −1 )e− /2h we have (5.2.9) uniformly for all  in that subset. The remainder of the chapter is devoted to the proof of this result that will follow the general strategy in Chap. 3, with an improvement of the probabilistic discussion. Example 5.2.3 Let Pc be the semi-classical variant of the non-self-adjoint harmonic oscillator in (2.5.1), Pc = (hD)2 + cx 2 , with c > 0. This operator fulfills the general assumptions with order function m(x, ξ ) = 1 + x 2 + ξ 2 and

∞ = ∅, = [0, +∞[+[0, +∞[×c. Let  be open, bounded, simply connected, h-independent with smooth boundary, such that ◦

 ⊂ . Then we can find W as in Proposition 5.1.3, containing . If z ∈ , then p−1 (z) consists of the four points (±x, ±ξ ), where x = x(z) > 0, and ξ = ξ(z) > 0 are determined by (c)x 2 = z, ξ 2 + (c)x 2 = z. The Poisson bracket {p, p}/(2i) is positive at two of these points, namely at (x, −ξ ) and (−x, ξ ), and negative at the other two. Theorem 5.2.2 shows that after adding a small random perturbation, the eigenvalues will obey a Weyl law in , while the eigenvalues of the unperturbed operator are confined to the bisector of , as we saw in Proposition 2.5.1.

74

5 Spectral Asymptotics for More General Operators in One Dimension

5.3

Preparations for the Unperturbed Operator

To each point ρj+ (z) = (xj+ (z), ξj+ (z)), z ∈ , we associate a quasimode ej (x, z; h) = i

a j (x, z; h)e h φ

j (x,z)

as in Theorem 4.1.4, Proposition 4.2.3, so that (P − z)(ej ) = O(h∞ ) in L2 (neigh (xj+ )).

(5.3.1)

Here φ j satisfies the eikonal equation p(x, ∂x φ j (x)) − z = 0, x ∈ neigh (xj+ ),

(5.3.2)

and φ j (xj+ (z), z) = 0, ∂x φ j (xj+ (z), z) = ξj+ (z), φ j (x, z) " (x − xj+ (z))2 .

(5.3.3)

Actually, this follows from the quoted results only when P is a semi-classical differential operator and we need to appeal to some more microlocal analysis in the pseudodifferential case. Equation (5.3.2) makes sense when p is analytic in ξ , since ∂x φ j is complex-valued, and for a more general pseudodifferential operator considered here, we have to weaken that equation to   p(x, ∂x φ j (x)) − z = O (x − xj+ (z))∞ ,

(5.3.4)

in the sense of formal Taylor expansions. We refer to [41, 102] for more details. To get a globally well-defined quasimode we let χ be a standard smooth cutoff, equal to one near 0, and notice that by the last property in (5.3.3), the above properties remain unchanged if we insert the cutoff χ(x − xj+ (z)) and obtain ej (x, z; h) = χ(x − xj+ (z))h−1/4 a j (x, z; h)e h φ i

j (x,z)

.

(5.3.5)

We can then arrange so that a j is a classical symbol of order 0, with ej L2 (R) = 1,

(5.3.6)

(P − z)(ej ) = O(h∞ ) in L2 (R).

(5.3.7)

and so that (5.3.1) holds globally:

In the construction, we can arrange so that ∂zk ∂z ej = O(h−k− ) in L2 .

(5.3.8)

5.3 Preparations for the Unperturbed Operator

75

Naturally, we have the analogous constructions of quasimodes for (P − z)∗ , where we replace the ρj+ by the ρj− . We have quasimodes of the form f j (x, z; h) = χ(x − xj− )h− 4 bj (x, z; h)e h ψ 1

i

j (x,z)

,

(5.3.9)

where bj is a classical symbol of order 0, p∗ (x, ∂x ψ j ) − z = 0 near xj− ,

(5.3.10)

ψ j (xj− (z), z) = 0, ∂x ψ j (xj− (z), z) = ξj− (z), ψ j (xj− (z), z) " (x − xj− (z))2 , (5.3.11) (5.3.12) fj L2 = 1, ∂zk ∂z f j = O(h−k− ) in L2 , (P − z)∗ (f j ) = O(h∞ ) in L2 .

(5.3.13)

Here we write p∗ (x, ξ ) = p(x, ξ ), to be understood in the sense of Taylor series, when p is not analytic. Also when p is not analytic in ξ , (5.3.10) should be weakened to p∗ (x, ∂x ψ j (x, z), z) − z = O((x − xj− )∞ ),

(5.3.14)

in the sense of Taylor expansions at xj− (z). By construction, (ej |ek ), (fj |fk ), (eν |fμ ) = O(h∞ ), j = k,

(5.3.15)

N so approximately we can say that {ej }N 1 , {fj }1 are orthonormal families, which are mutually orthogonal. Define R+ : L2 → CN , R− : CN → L2 , by

R+ u(j ) = (u|ej ), u ∈ L2 , R− u− =

N 

u− (j )fj , u− ∈ CN .

(5.3.16)

(5.3.17)

1 ∗ − 1, R ∗ R − 1 are O(h∞ ) in L(CN , CN ) and that, in particular, Note that R+ R+ − − R+ L(L2 ,CN ) = 1 + O(h∞ ), R− L(CN ,L2 ) = 1 + O(h∞ ). Define   P − z R− P(z) = (5.3.18) : H (m) × CN → L2 × CN , R+ 0

76

5 Spectral Asymptotics for More General Operators in One Dimension

and notice that P(z) is uniformly bounded in L(H (m) × CN , L2 × CN ) when h is small and z varies in any fixed compact subset of . Proposition 5.3.1 P(z) is bijective for z ∈ , with an inverse 

 E(z) E+ (z) E(z) = , E− (z) E−+ (z)

(5.3.19)

which satisfies, E(z) = O(h− 2 ) : L2 → H (m), 1

∗ (z) = O(h∞ ) : CN → H (m), E+ (z) − R+ ∗ E− (z) − R− (z) = O(h∞ ) : L2 → CN ,

(5.3.20)

E−+ (z) = O(h∞ ) : CN → CN . The proof involves some microlocal analysis that we shall not develop in detail here. Using the h-pseudodifferential calculus [41] we first see that Q+ := (P − z)∗ (P − z), Q− := (P − z)(P − z)∗ are essentially self-adjoint with domain H (m2 ). Here we make use of the ellipticity assumption (5.1.7). (Here and in what follows, it is tacitly assumed that h is small enough.) Moreover, because of the ellipticity assumption and the fact that m ≥ 1, we know that these operators have purely discrete spectrum in some h-independent neighborhood of 0. We also notice that Q± ≥ 0, so the spectra are contained in [0, +∞[ . For a given fixed constant C > 0, the eigenvalues in [0, Ch[ have complete asymptotic expansions in integer powers of h and can be described in the following way: The operators Q± have the same principal symbol q(x, ξ, z) = |p(x, ξ ) − z|2 , which vanishes precisely at the points ρj± , j = 1, 2, . . . , N, that we will also label ρν , 1 ≤ ν ≤ 2N whenever convenient. These points are also nondegenerate minima for q(·, z). Therefore, as for semi-classical Schrödinger operators [59, 129] and in later works for more general pseudodifferential operators [65, 67], each such point generates a sequence of eigenvalues of Q± , 1,± 2 2,± λ± ν,k (z; h) ∼ hλν,k (z) + h λν,k (z) + · · · , k = 0, 1, 2, . . . 1,± where λν,k (z) are the eigenvalues (arranged in increasing arithmetic progression) of a harmonic oscillator, appearing as a quadratic approximation of Q± at ρν . These quadratic approximations are Pν∗ Pν and Pν Pν∗ for Q+ and Q− , respectively, where

5.3 Preparations for the Unperturbed Operator

77

Pν = ∂x p(ρν )x + ∂ξ p(ρν )Dx . When ρν = ρj+ , then Pj is of annihilation type (surjective H ((x, ξ )) → L2 with a one-dimensional kernel, generated by a Gaussian function), 1,+ 1,+ = 0, λν,k > 0 for while Pj∗ is of creation type (injective), and we then know that λν,0 1,− k ≥ 1, while λν,k > 0 for k ≥ 0. When ρν = ρj− , then Pj∗ , Pj are annihilation and

1,+ 1,− 1,− creation operators, respectively, and we get λν,k > 0 for k ≥ 0, λν,0 = 0, λν,k > 0 for k ≥ 1. Thus, if C > 0 is large enough, each of Q+ , Q− has precisely N eigenvalues in [0, h/C] and these eigenvalues are O(h2 ). However, the existence of approximately N ∞ ∞ orthonormal systems {ej }N 1 , {fj }1 with Q+ ej = O(h ), Q+ fj = O(h ), shows that ∞ these eigenvalues are actually O(h ). Let S± be the N-dimensional subspace of L2 corresponding to the eigenvalues of Q± that are O(h∞ ). We observe that S± ⊂ H (mk ), ∀ k ∈ N, by the ellipticity of Q± near ∞. Let t12 ≤ t22 ≤ · · · ≤ tN2 , with 0 ≤ tj ≤ O(h∞ ) be the corresponding eigenvalues of Q+ and let 0 ≤ N0 ≤ N be the number of zero eigenvalues, so that (when N0 ≥ 1) 0 = t1 = t2 = · · · = tN0 < tN0 +1 ≤ · · · ≤ tN . N0 is then also equal to the dimension of the kernel of P − z. By Fredholm theory and ellipticity, we know that P − z : H (m) → H (1) is Fredholm of index 0 for z ∈  and this implies that dim Ker(P ∗ −z) = dim Ker(P −z). It follows that Q− has precisely N0 vanishing eigenvalues. We have the intertwining property

Q− (P − z) = (P − z)Q+ , and a similar one with (P ∗ − z). It follows that if Q+ j = tj2 j for some j ≥ N0 + 1, then Q− (P − z) j = tj2 (P − z) j , and hence (P − z) j is an eigenvector for Q− with the same eigenvalue tj2 . Since P − z is injective on the eigenspace corresponding to (Q+ , tj2 ), we see that the multiplicity of tj2 as an eigenvalue Q− is ≥ that of tj2 as an eigenvalue of Q+ . The sum of all the multiplicities is equal to N for each of Q+ and Q− , so the multiplicities have to agree. Pursuing this argument, we see that we can find orthonormal bases { j }N 1 in S+ and {γj }N in S such that − 1 (P − z) j = tj γj , (P − z)∗ γj = tj j . N Let { j }N 1 and {γj }1 be orthonormal bases in S+ and S− , respectively. For instance, we can take N { j }N 1 equal to the Gram orthonormalization of {S+ ej }1 , N {γj }N 1 equal to the Gram orthonormalization of {S− fj }1 ,

(5.3.21)

78

5 Spectral Asymptotics for More General Operators in One Dimension

where S± denotes the orthogonal projection onto S± . With that choice, we have

j − ej , γj − fj = O(h∞ ) in L2 .

(5.3.22)

+ : L2 → CN , R − : CN → L2 by With j , γj as in (5.3.21), (5.3.22) we define R + u(j ) = (u| j ), R − u− = R

N 

u− (j )γj ,

(5.3.23)

1

so that + L(L2 ,CN ) , R− − R − L(CN ,L2 ) = O(h∞ ). R+ − R

(5.3.24)

⊥ , L2 = S ⊕ S ⊥ , we see Using the spectral and orthogonal decompositions L2 = S+ ⊕ S+ − − that   − P − z R  = (5.3.25) : H (m) × CN → L2 × CN P(z) + 0 R

is bijective, with an “explicit” inverse 

  + (z) E(z) E  = E(z) , −+ (z) − (z) E E

(5.3.26)

where ⊥ ⊥   (P − z)−1 : S− → S+ , E(z) ∗ ± (z) = R ± E ,  −+ (z) = ((P − z) k |γj ) E

N j,k=1

 P − z : S+ → S− .

It follows that  = O(h− 2 ) : L2 → H (m), E 1

+ = O(1) : CN → H (m), E − = O(1) : H (1) → CN , E −+ = O(h∞ ). E Using this together with (5.3.24), we see that P(z) has an inverse E(z) such that E − E = O(h∞ ), and we get Proposition 5.3.1.

5.4 d-Bar Equation for E−+

79

d-Bar Equation for E−+

5.4

Here we generalize the discussion in Sect. 3.3. The proof of Proposition 3.3.1 leads to the d-bar equation, ∂z E−+ + E−+ (∂z R+ )E+ + E− (∂z R− )E−+ = 0.

(5.4.1)

We are interested  in the eigenvalues of P as the zeros of det E−+ . Using the identity  −1 ∂z ln det E−+ = tr E−+ ∂z E−+ , the cyclicity of the trace and (5.4.1), we get ∂z (ln det E−+ ) + f (z) = 0, where

(5.4.2)

f (z) = tr ((∂z R+ )E+ + E− ∂z R− ) , away from the zeros of det E−+ (z), which can also be written, ∂z (det E−+ (z)) + f (z) det E−+ = 0,

(5.4.3)

valid also near the zeros of det E−+ , either by continuity, or by direct computation without explicit use of logarithms. Again, eF det E−+ is holomorphic, if F solves ∂z F = f.

(5.4.4)

The multiplicity m(z0 ) of a zero z0 of E−+ is by definition equal to the multiplicity of z0 as a zero of eF det E−+ . We have (using an observation of Vogel [155]),  m(z0 ) = lim

→0 ∂D(z0 , )

since



∂D(z0 , ) ∂z F dz

∂z (eF det E−+ ) dz = lim

→0 eF det E−+ 2πi

 ∂D(z0 , )

∂z (det E−+ ) dz , det E−+ 2πi (5.4.5)

→ 0, → 0. On the other hand, in view of the standard identity

(z − P )−1 = −E(z) + E+ (z)E−+ (z)−1 E− (z), the algebraic multiplicity m (z0 ) of z0 as an eigenvalue of P is equal to 

 dz dz = tr lim (z − P ) E+ (z)E−+ (z)−1 E− (z) tr lim

→0 ∂D(z0 , )

→0 ∂D(z0 , ) 2πi 2πi      dz  dz −1 −1 = lim . = lim tr E+ E−+ E− tr E−+ E− E+

→0 ∂D(z0 , )

→0 ∂D(z0 , ) 2πi 2πi (5.4.6) −1

80

5 Spectral Asymptotics for More General Operators in One Dimension

Now by the same proof as for (5.4.1), E− E+ = ∂z (E−+ ) + E−+ (∂z R+ )E+ + E− (∂z R− )E−+ , and using this in the last integral together with the cyclicity of the trace in one of the resulting terms, we get  m (z0 ) = lim

→0

 dz  −1 = m(z0 ). tr E−+ ∂z E−+ 2πi ∂D(z0 , )

(5.4.7)

Here we also used the classical identity,   −1 ∂z E−+ . (det E−+ )−1 ∂z (det E−+ ) = tr E−+ The formula (3.3.4) and its derivation have straight forward generalizations, and we get Proposition 5.4.1 We have (5.4.2), so that eF a bit closer det E−+ is holomorphic when (5.4.4) holds. The zeros of det E−+ coincide with the eigenvalues of P in  and the multiplicities agree, cf. (5.4.5)–(5.4.7). Moreover,   N 1 2i 2i F (z) = − + O(1). h {p, p}(ρj+ (z)) {p, p}(ρj− (z)) j =1

5.5

(5.4.8)

Adding the Random Perturbation

We still have Proposition 3.4.2 and we make the assumption (3.4.5), 0≤δ

3

h2 ,

(5.5.1)

to be strengthened later on. Proposition 3.4.2 is applicable and we shall work under the assumption Qω H S ≤ C/ h. With R± as above, we introduce the perturbed Grushin matrix, 

Pδ − z R− Pδ (z) = R+ 0

 : H (m) × CN → L2 × CN ,

(5.5.2)

5.5 Adding the Random Perturbation

81

which is bijective with a bounded inverse (cf. (3.4.6)),  Eδ =

δ E δ E+ δ δ E− E−+

 ,

such that δ ) = O(h−1/2 ) in L(L2 , H (m)), h2 δ δ E+ = E+ + O( 3/2 ) = O(1) in L(CN , H (m)), h δ δ E− = E− + O( 3/2 ) = O(1) in L(L2 , CN ), h E δ = E + O(

δ E−+ = E−+ − δE− QE+ + O(

(5.5.3)

δ2 ) in L(CN , CN ). h5/2

δ and the multiplicities The eigenvalues of Pδ in  coincide with the zeros there of det E−+ agree. We have the d-bar equation,

 δ δ (z) + f δ det E−+ (z) = 0, ∂z det E−+  δ δ + E− ∂z R− , f δ (z) = tr (∂z R+ )E+

(5.5.4)

and from (5.5.3) and the boundedness of E± , we have  f (z) = f (z) + O δ

δ



5

.

(5.5.5)

h2

δ

δ holomorphic) with (cf. (3.4.7)) We can solve ∂z F δ = f δ (making eF det E−+

Fδ = F +

  δ 1 O . 3 h h2

(5.5.6)

δ in (5.5.3). From (5.3.20), We next look at the second term in the expression for E−+ (5.3.16), (5.3.17), we get for the matrix E− QE+ ,

E− QE+ (j, k) = (Qek |fj ) + O(h∞ ).

(5.5.7)

82

5 Spectral Asymptotics for More General Operators in One Dimension

Here, by (5.2.8), 

(Qek |fj ) =

αν,μ (ω)(χek |eμ )(eν |χfj )

C |ν|, |μ|≤ h1



=

|ν|, |μ|≤

&j (ν), αν,μ (ω)& χ ek (μ)χf C1 h

where the last equality is an implicit definition of the Fourier coefficients of χek and χfj with respect to the orthonormal family {eν }, where we can assume that ej and fj are supported in ] − πL, πL[ . We have seen that χej , . . . , χeN , χf1 , . . . , χfN is an orthonormal system up to O(h∞ ) and if C1 is large enough, we get the same properties &1 , . . . , χf &N in C1+2[C1 / h] . Then χ &j is an orthonormal &N and χf & ek ⊗ χf for χ& ej , . . . , χe M ∞ 2 family in C up to O(h ), where M := (1 + 2[C1 / h]) . Let E1 , . . . , EN 2 be the Gram orthonormalization of this family, and complete it to an orthonormal basis E1 , . . . , EM in CM . Write, 2

α=

N  1

αj Ej +

M 

αj Ej ,

α  = (α1 , . . . , αN 2 ), α  = (αN 2 +1 , . . . , αM ).

N 2 +1

By (5.5.7) and the following discussion, we have with Q = Q(α) in (5.2.8), E− QE+ = α  + O(h∞ ) in CN . 2

(5.5.8)

Our change of coordinates depends on z, but the estimates are uniform. For notational reasons, we change the sign of Q, so we get rid of the minus sign in the last equation in (5.5.3), which we then write as δ = E−+ + δA, E−+     δ δ  ∞ A = E− QE+ + O = α + O h + 5/2 , h5/2 h

(5.5.9)

for the new α-coordinates, obtained from the old ones by a z-dependent unitary transformation. Applying the Cauchy inequality to complex lines in CM , we get     δ 2 dα A = dα  + O h∞ + 3/2 in L CM ; CN . h

(5.5.10)

By the implicit function theorem, the map BCM (0, C/ h)  α → (A, α  ) ∈ CM

(5.5.11)

5.5 Adding the Random Perturbation

83

 which is sandwiched between is a holomorphic diffeomorphism onto its image B, BCM

      C C δ δ ∞ ∞ 0, − O h + 5 and BCM 0, + O h + 5 . h h h2 h2

It also follows from (5.5.10) that    δ L(dα) = L(dα  )L(dα  ) = 1 + O h∞ + 3 L(dA)L(dα ), 2 h

(5.5.12)

where L(dα), L(dα  ), and L(dα  ) denote the Lebesgue measure on CM , CN and CM−N , respectively. By (5.5.9), we have 2

  δ |α|2 = |A|2 + |α  |2 + O h∞ + 7 . h2

2

(5.5.13)

Now, strengthen the assumption (5.5.1) to δ

7

(5.5.14)

h2 .

Then from (5.5.12), (5.5.13), we get π

−M −|α|2

e

   δ 2  2 ∞ L(dα) = 1 + O h + 7 π −M e−|A| −|α | L(dA)L(dα ). h2 (5.5.15)

The probability distribution μ of A is the direct image of π −M e−|α| L(dα) under the map 2

2

BCM (0, C/ h)  α → A(α) ∈ CN .  its projection  denote the image of B(0, 1/ h) under α → (A(α), α  ) and πA (B) Letting B 2 N in CA , we conclude that if f (A) is a continuous function, then 

f (A)μ(dA) = π −M



f (A(α))e−|α| L(dα) 2

|α|≤C/ h

   δ 2  2 e−|A| −|α | L(dA)L(dα ) f (A) 1 + O h∞ + 7/2 h  (A,α  )∈B     δ 2 2 L(dA), f (A)e−|A| 1 + O h∞ + 7/2 ≤ π −N h  πA (B)

= π −M



84

5 Spectral Asymptotics for More General Operators in One Dimension

and we conclude that    δ −N 2 −|A|2 1πA (B) e L(dA). μ ≤ 1 + O h∞ + 7/2  (A)π h

(5.5.16)

Combining this with (5.A.21) below, we see that P (ln |det(D + A)|) ≤ a) ≤ O(1)ea/2 + e−1/(Ch ) , for a ≤ 0, 2

(5.5.17)

2

uniformly with respect to D ∈ CN . Equivalently, ln |det(D + A)| ≥ a, with probability ≥ 1 − O(1)ea/2 − e−1/(Ch ) . 2

δ = δ(A + δ −1 E Applying this to E−+ −+ ), we see that for every z ∈ , δ (z)| − N ln δ ≥ a with probability ≥ 1 − O(1)ea/2 − e−1/(Ch ) , ln |det E−+ 2

i.e., δ ln |det E−+ (z)| ≥ a − N ln

1 2 with probability ≥ 1 − O(1)ea/2 − e−1/(Ch ) , δ

(5.5.18)

uniformly for z ∈ , a ≤ 0. On the other hand, by the last equation in (5.5.3), we know that with probability ≥ 2 1 − e−1/(Ch ) , we have δ ln |det E−+ (z)| ≤ 0, for all z ∈ .

(5.5.19)

Now, consider the holomorphic function uδ (z; h) = eF

δ (z)

δ det E−+ (z),

(5.5.20)

and recall (5.5.6). Then with probability ≥ 1 − e−1/(Ch ) , 2

ln |uδ (z)| ≤ F (z) +

O(δ) , z ∈ . h5/2

Moreover, for every z ∈ , b ≥ 0 we have   1 O(δ) ln |u | ≥ F (z) − b + N ln + 5/2 , δ h δ

(5.5.21)

5.A Appendix: Estimates on Determinants of Gaussian Random Matrices

85

with probability ≥ 1 − O(1)e−b/2 − e−1/(Ch ) . By the assumption (5.5.14), we have N ln(1/δ) ( δ/ h5/2 , and restricting b to the interval 2N ln(1/δ) ≤ b ≤ 1/ h, we get for every z ∈ : 2

ln |uδ | ≥ F (z) − 2b, with probability ≥ 1 − O(1)e−b/2 − e−1/(Ch ) . 2

(5.5.22)

Let    be independent of h and have smooth boundary. In view of (5.5.21), (5.5.22), we can apply Proposition 3.4.6 to u = uδ , with φ(z)/ h = F (z) + b, = hb and with " −1/2 boundary points, where √ the lower bound 2(5.5.22) is required. We conclude that with probability ≥ 1 − O(1/ hb)e−b/2 − e−1/(Ch ) we have √        # (uδ )−1 (0) ∩  − 1  ≤ O(1) √b . F (z)L(dz)   2π  h

(5.5.23)

By (5.2.6) and (5.4.8), F (z)L(dz) =

1 p∗ (|σ |) + O(1), h

so (5.5.23) gives, √       # (uδ )−1 (0) ∩  − 1 volT ∗ R p−1 () ≤ O(1) √b ,   2πh h

(5.5.24)

√ √ where we also used that b/ h ( 1, since 2N ln(1/δ) ≤ b ≤ 1/ h. Now, (uδ )−1 (0)∩ = σ (Pδ ) ∩  and we get Theorem 5.2.2 with = bh. 

5.A

Appendix: Estimates on Determinants of Gaussian Random Matrices

In this appendix, we follow Section 7 in [56]. Consider first a random vector u(ω)t = (α1 (ω), . . . , αN (ω)) ∈ CN ,

(5.A.1)

where α1 , . . . , αN are independent complex Gaussian random variables with a NC (0, 1) law: (αj )∗ (P ) =

1 −|z|2 e L(dz) =: f (z)L(dz). π

(5.A.2)

86

5 Spectral Asymptotics for More General Operators in One Dimension

The distribution of u is u∗ (P ) =

1 −|u|2 e LCN (du). πN

(5.A.3)

If U : CN → CN is unitary, then U u has the same distribution as u. We next compute the distribution of |u(ω)|2 . The distribution of |αj (ω)|2 is μ(r)dr, where μ(r) = −H (r)

d −r e = e−r H (r), dr

and H (r) = 1[0,∞[ (r). The Fourier transform of μ is given by   μ(ρ) =

∞

e−r e−irρ dr =

0

1 . 1 + iρ

 2 2 We have |u(ω)|2 = N 1 |αj (ω)| , and since |αj (ω)| are independent and identically 2 distributed, the distribution of |u(ω)| is μ ∗ · · · ∗ μ dr = μ∗N dr, where ∗ indicates convolution. For r > 0, we get  1 1 μ∗N (r) = dρ eirρ 2π (1 + iρ)N      1 1 d N−1 1 = eirρ − dρ (N − 1)!2π γ i dρ 1 + iρ      1 d N−1 irρ 1 1 dρ (e ) = (N − 1)!2π γ i dρ 1 + iρ  r N−1 1 dρ = eirρ (N − 1)!2π γ 1 + iρ =

r N−1 e−r , (N − 1)!

where γ is a small simple positively oriented loop around the pole ρ = i. Hence μ∗N dr =

r N−1 e−r H (r)dr. (N − 1)!

Recall here that 

∞ 0

so μ∗N is indeed normalized.

r N−1 e−r dr = (N) = (N − 1)!,

(5.A.4)

5.A Appendix: Estimates on Determinants of Gaussian Random Matrices

87

The expectation value E(|αj |2 ) = |αj |2  of each |αj (ω)|2 is equal to 1, so |u(ω)|2  = N.

(5.A.5)

We next estimate the probability that |u(ω)|2 is very large. It will be convenient to pass to the variable ln(|u(ω)|2 ) which has the distribution obtained from (5.A.4) by replacing r by t = ln r, so that r = et , dr/r = dt. Thus ln(|u(ω)|2 ) has the distribution dr eNt −e r N e−r H (r) = dt =: νN (t)dt. (N − 1)! r (N − 1)! t

(5.A.6)

Now consider a random matrix 

 u1 u2 . . . uN ,

(5.A.7)

where uk (ω) are random vectors in CN (here viewed as column vectors) of the form uk (ω)t = (α1,k (ω), . . . , αN,k (ω)), and all the αj,k are independent with the same law (5.A.2). Then det(u1 u2 . . . uN ) = det(u1  u2 . . . uN ),

(5.A.8)

where  uj are obtained in the following way (assuming the uj to be linearly independent, as they are almost surely):  u2 is the orthogonal projection of u2 in the orthogonal complement u3 is the orthogonal projection of u3 in (u1 , u2 )⊥ = (u1 , u2 )⊥ , etc. (u1 )⊥ ,  u2 can be viewed as a random vector in CN−1 of the If u1 is fixed, then  type (5.A.1), (5.A.2), and with u1 , u2 fixed, we can view  u3 as a random vector of N−2 , etc. On the other hand the same type in C | det(u1 u2 . . . uN )|2 = |u1 |2 | u2 |2 · · · | uN | 2 .

(5.A.9)

The squared lengths |u1 |2 , | u2 |2 , . . . , | uN |2 are independent random variables with distri∗N ∗(N−1) butions μ dr, μ dr, . . . , μdr. This reduction plays an important role in [47]. Taking the logarithm of (5.A.9), we get in the right-hand side a sum of independent random variables with distributions νN dt, . . . , ν1 dt, so the distribution of the random variable ln |det(u1 u2 · · · uN )|2 is equal to (ν1 ∗ ν2 ∗ · · · ∗ νN )dt, with νj defined in (5.A.6).

(5.A.10)

88

5 Spectral Asymptotics for More General Operators in One Dimension

We have νN (t) ≤  νN (t) :=

1 eNt . (N − 1)!

Choose x(N) ∈ R such that 

x(N) −∞

 νN (t)dt = 1.

(5.A.11)

More explicitly, we have x(N) ≥ 0 and 1 Nx(N) e = 1, N!

x(N) =

1 1 ln(N!) = ln (N + 1). N N

(5.A.12)

In [56] we used Stirling’s formula, to get x(N) = ln N +

C0 1 1 ln N − 1 + + O( 2 ), 2N N N

(5.A.13)

where C0 = (ln 2π)/2 > 0. Here we shall not need the large-N limit. With this choice of x(N), we put νN (t), ρN (t) = 1]−∞,x(N)] (t) so that ρN (t)dt is a probability measure “obtained from νN (t)dt by transferring mass to the left”, in the sense that 

 f νN dt ≤

fρN dt,

(5.A.14)

whenever f is a bounded decreasing function. Equivalently, g ∗ νN ≤ g ∗ ρN , when g is a bounded increasing function. Now, for such a g, both g ∗ νN and g ∗ ρN are bounded increasing functions, so by iteration, g ∗ ν1 ∗ · · · ∗ νN ≤ g ∗ ρ1 ∗ · · · ∗ ρN . In particular, taking g = H we get 

t −∞

 ν1 ∗ · · · ∗ νN (s)ds ≤

t −∞

ρ1 ∗ · · · ∗ ρN (s)ds, t ∈ R.

(5.A.15)

5.A Appendix: Estimates on Determinants of Gaussian Random Matrices

89

We have by (5.A.12)  ρ N (τ ) =

x(N)

−∞

1 1 et (N−iτ ) dt = eNx(N)−ix(N)τ (N − 1)! (N − 1)!(N − iτ )

e−ix(N)τ = . 1 − i Nτ

(5.A.16)

This function has a pole at τ = −iN. Similarly, 1 ]−∞,a] (τ ) =

i e−iaτ . τ + i0

(5.A.17)

By Parseval’s formula, we get 

a

1 ρ1 ∗ · · · ∗ ρN dt = 2π −∞



∞ −∞

F (ρ1 ∗ · · · ∗ ρN )(τ )F 1]−∞,a] (τ )dτ 1 = 2π



+∞ −∞

e

−iτ (

N 1

−i  1 dτ. τ − i0 1 − iτj N

x(j )−a)

1

We deform the contour to τ = −1/2 (half-way between R and the first pole in the lower  half-plane). It follows that for a ≤ N 1 x(j ) : 

a −∞

ρ1 ∗ · · · ∗ ρN dt ≤ C(N)ea/2 .

(5.A.18)

In view of (5.A.15), the right-hand side is an upper bound for the probability that ln | det(u1 · · · uN )|2 ≤ a. Hence, for a ≤ 0, P(ln | det(u1 . . . uN )|2 ≤ a) ≤ C(N)ea/2 .

(5.A.19)

We shall next extend our probabilistic bounds to determinants of the form det(D + Q) where Q = (u1 . . . uN ) is as before, and D = (d1 . . . dN ) is a fixed complex N ×N matrix. As before, we can write u2 |2 · · · |dN +  uN | 2 , | det((d1 + u1 ) . . . (dN + uN ))|2 = |d1 + u1 |2 |d2 + 

90

5 Spectral Asymptotics for More General Operators in One Dimension

2 =  where d d2 (u1 ),  u2 =  u2 (u1 , u2 ) are the orthogonal projections of d2 , u2 on ⊥   u3 =  u3 (u1 , u2 , u3 ) are the orthogonal projections of d3 , (d1 + u1 ) , d3 = d3 (u1 , u2 ),  u3 on (d1 + u1 , d2 + u2 )⊥ , and so on. (N) Let νd (t)dt be the probability distribution of ln |d + u|2 , when d ∈ CN is fixed and (N) u ∈ CN is random as in (5.A.1), (5.A.2). Notice that ν0 (t) = ν (N) (t) is the density we have already studied. Lemma 5.A.1 For every a ∈ R, we have 

a



(N)

−∞

νd (t)dt ≤

a

−∞

ν (N) (t)dt.

Proof Equivalently, we have to show that P(|d + u|2 ≤  a ) ≤ P(|u|2 ≤  a ) for every  a > 0. For this, we may assume that d = (c, 0, . . . , 0), c > 0. We then only have to prove that P(|c + u1 |2 ≤ b2) ≤ P(|u1 |2 ≤ b 2 ), b > 0, and here we may replace P by the corresponding probability density 1 2 μ(t)dt = √ e−t dt π for μ1 . Thus, we have to show that 1 √ π

 |c+t |≤b

e

−t 2

1 dt ≤ √ π



e−t dt. 2

|t |≤b

(5.A.20)

Fix b and rewrite the left-hand side as 1 I (c) = √ π



b−c −b−c

e−t dt. 2

The derivative satisfies 1 2 2 I  (c) = √ (e−(b+c) − e−(b−c) ) ≤ 0. π hence c → I (c) is decreasing and (5.A.20) follows, since it holds when c = 0.

 

5.A Appendix: Estimates on Determinants of Gaussian Random Matrices

91

Now consider the probability that ln|det(D + Q)|2 ≤ a. If χa (t) = H (a − t), this probability becomes 

 ···

 P(du1 ) · · · P(duN )χa ln |d1 + u1 |2 + ln |d2 (u1 ) +  u2 (u1 , u2 )|2 + · · ·

uN (u1 , · · · , uN )|2 . + ln |dN (u1 , · · · , uN−1 ) +  Here we first carry out the integration with respect to uN , noticing that with the other u1 , . . . , uN−1 fixed, we may consider dN (u1 , . . . , uN−1 ) as a fixed vector in C  (d1 + u1 , . . . , dN−1 + uN−1 )⊥ and  uN as a random vector in C. Using also the lemma, we get P(ln|det(D + Q)|2 ≤ a)   (1) (tN )dtN P(duN−1 ) · · · P(du1 )× = · · · ν d 

N

χa ln |d1 + u1 |2 + · · · + ln |dN−1 (u1 , · · · , uN−2 ) +  uN−1 (u1 , · · · , uN−1 )|2 + tN   ≤ · · · ν (1) (tN )dtN P(duN−1 ) · · · P(du1 )×



  χa ln |d1 + u1 |2 + · · · + ln |dN−1 (u1 , · · · , uN−2 ) +  uN−1 (u1 , · · · , uN−1 )|2 + tN . We next estimate the uN−1 - integral in the same way, and so on. Eventually, we get Proposition 5.A.2 Under the assumptions above, P(ln|det(D + Q)|2 ≤ a)   ≤ · · · χa (t1 + · · · + tN )ν (1) (tN )ν (2) (tN−1 ) · · · ν (N) (t1 ) = P(ln |det Q|2 ≤ a). In particular, the estimate (5.A.19) extends to random perturbations of constant matrices: P(ln |det(D + Q)|2 ≤ a) ≤ C(N)ea/2 , for a ≤ 0.

(5.A.21)

Notes In this chapter we have generalized the results of Chap. 3 and tried to explain the role of the points ρ in the cotangent space where i −1 {p, p}(ρ) is positive or negative. In higher dimension this simple reduction via a Grushin seems to break down. The results are close to the ones in [55], but we have not included the case of multiplicative random perturbations since we will deal with that case in later chapters, directly in higher dimensions. It was convenient to use some elements from later works, like singular values. In particular, in Sect. 5.A, we have followed [56].

6

Resolvent Estimates Near the Boundary of the Range of the Symbol

6.1

Introduction and Statement of the Main Result

The purpose of this chapter is to give quite explicit bounds on the resolvent near the boundary of (p) (or more generally, near certain “generic boundary-like” points). The result is due (up to a small generalization) to Montrieux [18] and improves earlier results by Martinet [98] about upper and lower bounds for the norm of the resolvent of the complex Airy operator, which has empty spectrum [7]. There are more results about upper bounds, and some of them will be recalled in Chap. 10 when dealing with such bounds in arbitrary dimension. To fix the ideas, we will consider operators on R and indicate later the minor modifications needed for operators on S 1 . Let P ∈ S(m) be as in Chap. 5 and assume Eqs. (5.1.1), (5.1.2), (5.1.4)–(5.1.7). Depending on the context, P may also denote the corresponding pseudodifferential operator in (5.1.3). Define = (p), ∞ = ∞ (p) as in (5.1.8), (5.1.9) with p = p0 in (5.1.4), and recall (5.1.10). Let z0 ∈ (p) \ ∞ (p) and assume that for every ρ ∈ p−1 (z0 ), we have

1 {p, p}(ρ) = 0, {p, {p, p}}(ρ) = 0. 2i

(6.1.1)

This is in some sense the generic situation for z0 ∈ ∂ (p) \ ∞ (p). Example 6.1.1 a) Let P = hD +g(x) be Hager’s operator, g ∈ C ∞ (S 1 ), and assume that g has a unique minimum xmin ∈ S 1 and that this minimum is nondegenerate. Notice that the extension of the definition of and ∞ to the case of semi-classical differential operators on compact manifolds is straightforward. In our case p(x, ξ ) = ξ + g(x), either as a © Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_6

93

94

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

symbol on T ∗ R, or as one on T ∗ S 1 . In both cases, = R + i[min g, max g], and we can take m = ξ . We have ∞ = in the case of R and ∞ = ∅ in the S 1 -case. Let z0 ∈ ∂ belong to the lower part R + i min g of ∂ . When the underlying manifold is S 1 , then p−1 (z0 ) = {ρ0 }, where ρ0 = (xmin, ξ0 ), g(xmin ) = min g, and ξ0 = z0 − g(xmin ). We have 1 {p, p}(ρ0 ) = −g  (xmin ) = 0, {p, {p, p}}(ρ0 ) = −2ig  (xmin ) = 0. 2i Our main result (Theorem 6.1.4) applies in this case. When P acts on L2 (R), then xmin is unique up to a multiple of 2π, and using the periodicity, we can check that the conclusion of the main result still holds. b) The non-self-adjoint Airy operator, P = (hD)2 + ix, with symbol p = ξ 2 + ix. We have = [0, +∞[ +iR, ∞ = ∅, and we can take m = ξ 2 + x. Let z0 = iy0 ∈ ∂ , so that p−1 (z0 ) = {ρ0 }, where ρ0 = (y0 , 0). We have (2i)−1 {p, p}(x, ξ ) = −2ξ which vanishes at ρ0 . Further, {p, (2i)−1 {p, p}} = 2i = 0. c) The non-self-adjoint harmonic oscillator P = 12 ((hD)2 + ix 2 ). We have = [0, +∞[ +i[0, +∞[ , ∞ = ∅ and we can take m = 1 + x 2 + ξ 2 . The boundary of is the union of [0, +∞[ and i[0, +∞[ and by Fourier transform, the study near one of these half-rays is equivalent to that near the other. Let 0 = z0 ∈ ∂ and take z0 > 0 to fix the ideas. Then p−1 (z0 ) = {ρ+ , ρ− }, where ρ± = (0, ±ξ0 ) and ξ0 > 0 is given by ξ02 /2 = z0 . We have (2i)−1 {p, p}(x, ξ ) = −xξ which vanishes at ρ± . Further, {p, {p, p}} = −2i(ξ 2 − ix 2 ) is = 0 at ρ± . As we shall see below, the assumptions imply that p−1 (z0 ) is a finite set. Let ρ0 ∈ p−1 (z0 ). Then γ := {ρ ∈ neigh (ρ0 );

1 {p, p}(ρ) = 0} 2i

(6.1.2)

is a smooth curve, since 2i1 {p, p} is real-valued with a non-vanishing differential near ρ0 . Along γ the vectors Hp are Hp are collinear and never both equal to zero. Hence there is a smooth function γ  ρ → θ (ρ) ∈ R such that e−iθ(ρ) Hp is real and non-zero; θ (ρ) is unique up to a multiple of π. We can parametrize γ by   γ (t) = exp tH 1 {p,p} (ρ0 ). 2i

Then, d 1 p(γ (t)) = H 1 {p,p} (p)(γ (t)) = −{p, {p, p}}(γ (t)) = 0, 2i dt 2i

(6.1.3)

6.1 Introduction and Statement of the Main Result

95

so δ := p ◦ γ

(6.1.4)

is a smooth curve in neigh (z0 , C) and e

−iθ(γ (t )) ˙

δ(t) = −e

−iθ(γ (t ))

 Hp

 1 {p, p} ∈ R \ {0}. 2i

(6.1.5)

In order to simplify the local geometric discussion, it will be helpful to replace p by f ◦ p, where f : neigh (z0 , C) → neigh (0, C)

(6.1.6)

is almost holomorphic (cf. Appendix 6.A) along the curve δ: ∂z f = O(dist (z, δ)∞ ).

(6.1.7)

When p is real-analytic, δ is a real-analytic curve and we will be able to choose f holomorphic. Notice that Hf ◦p (ρ) = (∂z f )(p(ρ))Hp (ρ) + O(dist (ρ, γ )∞ ), 1 1 {f ◦ p, f ◦ p}(ρ) = |∂z f |2 {p, p}(ρ) + O(dist (ρ, γ )∞ ), 2i 2i so

1 2i {f

(6.1.8) (6.1.9)

◦ p, f ◦ p} vanishes on γ and

{f ◦ p,

1 1 {f ◦ p, f ◦ p}} = ∂z f |∂z f |2 {p, {p, p}} = 0 on γ . 2i 2i

(6.1.10)

Thus, p  = f ◦ p satisfies the same general assumptions as p, with z0 replaced by 0 = f (z0 ). Now, choose f mapping (the image of) δ to a real interval. Then p  is real-valued on γ , and since d p  is a complex multiple of a real differential when ρ ∈ γ , we conclude that d p  p = 0 on γ and is real, and hence that Hp is real on γ . By the choice of f , we know that  it follows that  p = O(dist (ρ, γ )2 ).

(6.1.11)

The natural parametrization in (6.1.3) induces an orientation of γ such that the region where (2i)−1 {p, p} > 0 is to the left when walking along γ in the positive direction. Replacing p by p  will change the parametrization in (6.1.3), but not the orientation of γ .

96

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

We now choose smooth local symplectic coordinates (x, ξ ) (i.e., for which the symplectic form is dξ ∧ dx) centered at ρ0 such that γ becomes the ξ -axis {(0, ξ )} with it natural orientation, namely that of increasing values of ξ . Then, since  p = O(x 2 ), we can write p (x, ξ ) = d(x, ξ ) + ir(x, ξ )

(6.1.12)

where d, r are smooth real-valued functions such that ∂ξ d(0, ξ ) = 0, r = O(x 2 ). We have 1 { p, p } = −{d, r} = −dξ rx + dx rξ = −dξ rx + O(x 2 ), 2i and at x = 0 0 = { p,

1  { p, p }} = −(dξ )2 rxx . 2i

(6.1.13)

Having chosen the symplectic coordinates so that the orientation of the ξ -axis is the same p, p} has the same as the one of γ (when identifying the two sets), we know that (2i)−1 { sign as −x. In order to fix the ideas, we can impose an additional condition on the map f , namely that the orientation of f ◦ δ = f ◦ p ◦ γ on R is the one of increasing real values. This means that p (0, ξ ) should be an increasing function of ξ , or in other terms, −1 p, p }, p }, i.e. that the quantity in (6.1.13) is < 0. Hence that 0 < {(2i) {  (0, ξ ) > 0. ∂ξ d(0, ξ ) > 0, rxx

(6.1.14)

We can refine the choice of symplectic coordinates above so that d(x, ξ ) = ξ . Then with a new r as in (6.1.14), we get p (x, ξ ) = ξ + ir(x, ξ ), 0 ≤ r(x, ξ ) " x 2

(6.1.15)

1 { p, p } = −∂x r, 2i

(6.1.16)

and

which is positive for x < 0 (i.e., to the left of γ ) and negative to the right. The map p  is orientation preserving to the left and orientation reversing to the right, when we adopt the

6.1 Introduction and Statement of the Main Result

97

standard orientations on R2x,ξ and on Cw ≡ R2w,w . Indeed, d p ∧ d p = −{ p,  p }dx ∧ dξ =

1 { p, p }dx ∧ dξ. 2i

We can say that p  has a simple fold along γ : The equation p (ρ) = w, for ρ ∈ 2 neigh ((0, 0), R ), w ∈ neigh (0, C) • has two simple solutions when w > 0, • has one degenerate solution when w = 0, • has no solution when w < 0. (ρ) = w when w = f (z) The equation p(ρ) = z, z ∈ neigh (z0 , C), is equivalent to p and locally, the inverse image under f of the upper half-plane is the region situated to the left of the oriented curve δ. p has a simple fold along γ and is orientation preserving to the left and orientation reversing to the right of γ as 2i1 {p, p} is positive to the left and negative to the right. The range of p is the region situated to the left of δ. Thus the equation p(ρ) = z • has two simple solutions when z ∈ δ is to the left of δ, • has one degenerate solution when z ∈ δ, • has no solution when z ∈ δ is to the right. For both equations we denote the solutions (when they exist) by ρ+ , ρ− with ρ+ and ρ− to the left and to the right of δ, respectively. (ρ) = w, so that ρ ± (f (z)) = ρ± (z). We get Let ρ ± be the corresponding solutions of p ρ ± (w) = (x± , ξ± ), where ξ± = w

(6.1.17)

r(x, w) = w, ∓x± ≤ 0.

(6.1.18)

√ x± = k(∓ w, w),

(6.1.19)

and x± are given by

We have

where k(s, t) is the smooth function given by √ r(k(s, t), t) = s

98

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

√ and r(x, ξ ) is the smooth branch of the square root of r which has the same sign as x. To leading order, k(s, t) =

s + O(s 2 ),  (0, t)/2)1/2 (rxx

so √  x± = ∓(rxx (0, w))−1/2 2w + O(w).

(6.1.20)

Naturally, (6.1.17), (6.1.19), (6.1.20) also describe ρ± . All we have to do, is to work in the same symplectic coordinates and represent z by the local coordinates (s, t) = (f, f ) = (w, w). Combining (6.1.10), where p  = f ◦ p, with (6.1.13), where now dξ = 1, we get  0 < rxx (0, ξ ) = −|∂z f |2 ∂z f {p,

1 {p, p}}(ρ), ∂z f = ∂z f (ρ), 2i

(6.1.21)

where ρ ∈ γ is determined by f (p(ρ)) = ξ . We may choose f so that |f  | = 1 on δ. Then (6.1.21) simplifies to     1  (0, ξ ) = (ξ ), (ξ ) := {p, {p, p}}(ρ) . rxx 2i

(6.1.22)

Combining (6.1.20), (6.1.22) with (6.1.15), we get for w ≥ 0, 1 { p, p }(ρ± ) = ±(2 (w))1/2(w)1/2 + O(w). 2i As in (6.1.21), we have 1 1 { p, p }(ρ) = |∂z f (p(ρ))|2 {p, p}(ρ) + O(dist (ρ, γ )∞ ), 2i 2i 1 {p, p}(ρ± ) = ±(2 (w))1/2(w)1/2 + O(w), w ≥ 0. 2i

(6.1.23)

Assume for the moment that p is analytic near ρ0 , so that f , p  are analytic also. If 2 w > 0, then {ρ ∈ neigh (ρ0 , C ); p(ρ) = z} is a complex curve (z) which intersects the real phase space at the two points ρ± (z) and we can therefore define the action  J (z) =

ξ dx, (z,ρ+ ,ρ)

(6.1.24)

6.1 Introduction and Statement of the Main Result

99

where (z, ρ+ , ρ− ) is a real curve in (z) which starts at ρ+ (z) and ends at ρ− (z), and which stays in a small neighborhood of ρ0 , so that all such curves are homotopic to one another. All curves in R2 are Lagrangian manifolds and this remains true when we pass to the complex-holomorphic category. Hence, ξ dx|(z) is closed (and even exact since we work locally) and the value of J (z) does not depend on the choice of (z, ρ+ , ρ− ). If (y, η) are some other real and analytic symplectic coordinates, then ξ dx − ηdy is closed and hence exact, so ξ dx − ηdy = dG, where G is real and analytic. Consequently, 

 ξ dx − (z,ρ+ ,ρ− )

(z,ρ+ ,ρ− )

ηdy = G(ρ− ) − G(ρ+ )

is real and we conclude that I (z) := J (z) is invariant under changes of real

(6.1.25)

and analytic symplectic coordinates. This means that we can work with p  in the form (6.1.15) and write I (z) = I(w), w = f (z), with the analogous definition of I(w). Parametrize z by w = f (z) = is + t, (s, t) ∈ neigh ((0, 0), R2 ),

(6.1.26)

and write I (z) = I (s, t). Working in the coordinates of (6.1.15), (z) is given by p (x, ξ ) = w, i.e., ξ + ir(x, ξ ) = w and  I (z) = 

x−

λ(x, w)dx,

(6.1.27)

x+

where λ is the solution of p (x, λ(x, w)) = w. By successive differentiations of this equation, we get λx = −

λxx = −

p x (x, λ) , p ξ (x, λ)

λw =

1 , p ξ (x, λ)

 (x, λ)  (x, λ) px (x, λ) p ξξ (x, λ) 2 pxξ x (x, λ)2 p p xx + − . p ξ (x, λ) p ξ (x, λ)2 p ξ (x, λ)3

(6.1.28)

100

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

In particular, λ(0, w) = w, λx (0, w) = 0,   pxx (0, w) = −irxx (0, w). λxx (0, w) = −

(6.1.29)

We get the leading Taylor expansion in x, i  λ(x, w) = w − rxx (0, w)x 2 + O(x 3 ), 2

(6.1.30)

 (0, w) = (w) for real w and we can extend this relation where we recall that rxx and (6.1.30) holomorphically in w. Thus, with w = t + is as in (6.1.26),

λ(x, w) = s −

(t) 2 x + O(s)x 2 + O(x 3 ). 2

(6.1.31)

By (6.1.20), ' x± = ∓

2s + O(s)

(t)

(6.1.32)

√ are smooth functions of ∓ s, for s ≥ 0 and we notice that (6.1.32) also follows √ from (6.1.31) and the fact that λ(x± , w) = 0. I (z) is a smooth function of s and  I (z) = O(s ) + 2

√ 2s/ (t )

  3

(t) 2 2 (2s) 2 x + O(s 2 ). s − dx = 1 √ 2 3 2 − 2s/ (t )

(t)

(6.1.33)

With z represented as in (6.1.26), we have √ I (z) = I (s, t) = ι( s, t),

(6.1.34)

where ι(σ, t) is smooth and = O(σ 3 ). We shall now drop the analyticity assumption and pause for a general discussion: Let p be a smooth function defined in a neighborhood of 0 ∈ R2 . We assume that p depends smoothly on a finite number of real parameters w ∈ neigh (0, Rk ) that we do not write out. Assume that dp = 0 and that (for w in a closed subset of the parameter space, containing 0), there are two real points x± ∈ R2 where p vanishes and that x± are Hölder continuous as functions of the parameter with x+ = x− = 0 for w = 0. Let p also denote an almost holomorphic extension of p to a complex neighborhood of 0 ∈ C2 . This means that ∂p(x) = O((x)∞ ), and we know (see e.g. [103] and Appendix 6.A) that such an extension is unique up to a term which is O((x)∞ ). After a permutation of the variables we may assume that ∂x2 p = 0. Then by the real implicit

6.1 Introduction and Statement of the Main Result

101

function theorem we have in a neighborhood of x = 0 in C2 p(x) = 0 ⇐⇒ x2 = λ(x1 ), where λ(x1 ) is a smooth function of x1 . Next, differentiating the equation p(x1 , λ(x1 )) = 0 we see that ∂x 1 λ = O((x1 , λ(x1 ))∞ ). Let ω(x)dx be a smooth 1-form and denote by the same symbol an almost holomorphic extension (as a (1,0) form). Consider  Jγ =

ω(x)dx,

(6.1.35)

γ

where γ is a smooth curve from x+ to x− inside the complex set p−1 (0), with the property that length (γ ) = O(|x+ − x− |). We have sup |γ | = O(|x+ − x− |), we can write γ : [0, 1]  t → (γ1 (t), λ(γ1 (t)))  to x−,1 with length (γ1 ) = O(|x+,1 − x−,1 |). In the given where γ1 is a curve from x+,1 coordinates, the simplest choice of γ would be to take γ = γ0 (t) = (γ0,1 (t), λ(γ0,1 (t)), where γ0,1(t) = (1 − t)x+,1 + tx−,1 . Using Stokes’ formula and the observation that d(ωdx|p−1 (0) ) = O(|x|∞ ), we get Proposition 6.1.2 In the general situation described above, and up to a term O(|x+ − x− |∞ ), J = Jγ does not depend on the choice of γ , nor on the choices of almost holomorphic extensions of p and ω. We have the same invariance under smooth changes of local coordinates in R2 . Now return to p in (6.1.1) and p  = f ◦ p in (6.1.15) and let “ωdx” above be the fundamental 1-form ξ dx. By taking almost holomorphic extensions we can still define the

102

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

almost complex curve1 (z) by p(ρ) = z as prior to (6.1.24), and when w ≥ 0 we define the action J (z) as in (6.1.24), choosing the curve from ρ+ to ρ− of length O(|ρ+ − ρ− |). Then J (z) is well defined up to O(|ρ+ − ρ− |∞ ), or equivalently up to O(|w|∞ ). The imaginary part I (z) is invariant under real canonical transformations (see e.g. [51, p. 100])  also denotes up to O(|w|∞ ) and we have (6.1.27), where λ solves (6.1.28), in which p an almost holomorphic extension. Proposition 6.1.3 Under the assumptions above, let w = f (z) be defined in (6.1.6) and after, so that |f  | = 1 on the curve δ = p ◦ γ . Recall the definition of (t)  (f ◦ p ◦ (t)) in (6.1.22). Then I (z) = J (w) is well defined (cf. (6.1.24), (6.1.27)) up to O(|w|∞ ) in √ the region w ≥ 0. Writing w = t + is, we have (6.1.34): I (s, t) = ι( s, t), where ι(σ, t) is a smooth function, well-defined mod (σ ∞ ), such that √ 4 2 σ3 ι(σ, t) = + O(σ 4 ) 3 (t) 12

(6.1.36)

We now start to formulate the main result of this chapter. Recall that z0 ∈ (p)\ ∞ (p) and that we work under the assumptions (5.1.1), (5.1.2), (5.1.4)–(5.1.7), and (6.1.1). It follows from the discussion above that the points of p−1 (z0 ) are isolated and hence that this set is finite, p−1 (z0 ) = {ρ01 , . . . , ρ0N }. j

(6.1.37) j

The arguments above then apply with ρ0 equal to ρ0 , 1 ≤ j ≤ N. Through ρ0 we have j a curve γj , defined as in (6.1.2), (6.1.1) with ρ0 replaced by ρ0 . Let δj = p ◦ γj be the image curves, all passing through z0 . We have smooth functions fj : neigh (z0 , C) → j = fj ◦ p is neigh (0, C), almost holomorphic at δj , such that |fj | = 1 on δj and p real-valued on γj . Put fj (z) = wj = tj + isj .

(6.1.38)

When sj ≥ 0 we have two real points ρj± ∈ neigh (ρ0 , T ∗ R) determined by p(ρj± ) = z, ±(2i)−1 {p, p}(ρj± ) ≥ 0, with strict inequality (and ρj+ = ρj− ) when sj > 0. As in (6.1.23), j

1√ 1 j {p, p}(ρ± ) = ±(2 j (tj )) 2 sj + O(sj ), sj ≥ 0, 2i

(6.1.39)

1 We say that the smooth submanifold  ⊂ Cn is an almost complex curve, if dim  = 2 and R dist (Tx , i(Tx )) = O(|x|∞ ) locally uniformly for x ∈ . See [103] for a more systematic discussion of almost holomorphic geometry.

6.1 Introduction and Statement of the Main Result

103

where j (t) = j (γj (t)) is defined as in (6.1.22),     1 

j (t) = {p, {p, p}}(γj (t)) . 2i

(6.1.40)

As in Proposition 6.1.3, we can define for sj ≥ 0, √ Ij (z) = ιj ( sj , tj ),

√ 4 2 σ3 ιj (σ, t) = + O(σ 4 ), 3 j (t)1/2

(6.1.41)

where ιj (σ, t) is smooth and real-valued. Theorem 6.1.4 We make the assumptions (5.1.1), (5.1.2), (5.1.4)–(5.1.7), (6.1.1), and j define the quantities ρj , ρ± , Ij , ιj as above. Let z ∈ neigh (z0 , C), so that we have the representation (6.1.38) for each j = 1, . . . , N. For C0 > 0 large enough, put ⎧ ⎪ (h2/3 + |sj |)−1 , when sj ≤ C0 h2/3 , ⎪ ⎪ ⎨  π 1/2 exp(Ij (z)/ h) Mj (z; h) = h 2/3 ⎪  1/4  1/4 , when sj ≥ C0 h . ⎪ ⎪ j j 1 1 ⎩ 2i {p, p}(ρ+ ) 2i {p, p}(ρ− ) (6.1.42) Assume that for some arbitrarily small and fixed δ0 > 0, −h

δ0

 2 1 3 ≤ sj ≤ O(1) h ln , j = 1, . . . , N. h

(6.1.43)

Put 2

N = {j ∈ {1, 2, . . . , N}; sj ≥ C0 h 3 }. Then for h > 0 small enough, we have uniformly with respect to z that P − z : H (m) → H (1) is bijective and (P − z)−1 H (1)→H (1)

⎧ ⎨= max

3/2 j ∈N (1 + O(h/sj ))Mj (z; h),

when N = ∅,

⎩≤ O(1) maxj Mj , in general.

(6.1.44) The lower bound in (6.1.43) has been introduced for convenience only and could probably be removed without too much extra work.

104

6.2

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

Preparations and Reductions

In this section, we discuss the situation locally. Let ρ0 ∈ T ∗ R = R2 , p ∈ C ∞ (neigh (ρ0 , R2 )) satisfy (6.1.1): 1 {p, p}(ρ0 ) = 0, 2i {p,

1 {p, p}}(ρ0 ) = 0. 2i

(6.2.1)

(6.2.2)

Let z0 = p(ρ0 ). We are interested in p − z (and in a corresponding pseudodifferential operator P (x, hD; h) − z) when z − z0 is small. Let f : neigh (z0 , C) → neigh (0, C) be the map for which p  = f ◦ p, where p  is as in (6.1.15). One could probably develop a functional calculus, allowing to work with “f (P )”, but we opt for a more direct approach,  by where we linearize f at z0 : f (z) = e−iθ(ρ0 ) (z − z0 ) + O((z − z0 )2 ) and, redefine p p − z = eiθ(ρ0 ) ( p (ρ) − ω), where p (ρ) = e−iθ(ρ0 ) (p(ρ) − z0 ),

ω = e−iθ(ρ0 ) (z − z0 ).

We are then interested in p (ρ) − ω when ω is small. p  satisfies the same assumptions, now with z0 replaced by 0. Remark 6.2.1 z0 = p(z0 ) corresponds to w = w0 = 0 and hence to ω = 0. By Taylor expansion we get ω = w + O(w2 ) for z ∈ neigh (z0 ). In the application later on, we will choose the point ρ0 ∈ γ and the corresponding point z0 ∈ δ as functions of the spectral parameter z. Thus, for a given z, we can choose the new base point ρ0 = ρ(z) ∈ γ and the new z0 = p(ρ(z)), so that w = w(z) has a vanishing real part. Then ω = iw + O((w)2 ). The image curve p  ◦ γ is tangent to R at 0 and oriented in the positive real direction at that point. It is of the form  δ : ω = k(ω), ω ∈ neigh (0), k  (0) = 0.

6.2 Preparations and Reductions

105

• The range under p  of neigh (ρ0 , R2 ) is equal to {ω ∈ neigh (0, C); ω ≥ k(ω)}. (It is situated to the left of p  ◦ γ .) • The map p  : neigh (ρ0 , R2 ) → neigh (0, C) has a simple fold along γ . It is a local diffeomorphism on each side of γ , orientation preserving to the left and orientation reversing to the right. Until further notice, we discuss p , but drop the tilde and simply write p. After composing p with an affine canonical transformation, we may assume that ρ0 = 0, dp(0) = dξ(0),

(6.2.3)

in addition to the fact that p(0) = 0. Then, p(x, ξ ) = ξ + ir(x, ξ ), r(x, ξ ) = O(x 2 + ξ 2 ).

(6.2.4)

Moreover, 1 {p, p} = −{p, p} = −∂x r + O(x 2 + ξ 2 ), 2i and (cf. (6.1.22)) ∂x2 r(0, 0) = (0) > 0.

(6.2.5)

Using Malgrange’s preparation theorem (see [55] for the use in the present context and for further references), we know that p(x, ξ ) − ω = a(x, ξ, ω)(ξ + g(x, ω)), (x, ξ, ω) ∈ neigh ((0, 0, 0), R2 × C),

(6.2.6)

where a, g are smooth functions (and g is basically equal to −λ, as discussed prior to (6.1.35) with n = 2, (x1 , x2 ) = (x, ξ )). Actually, for our purposes it suffices to have (6.2.6) at the level of formal Taylor series at (x, ξ, ω) = (0, 0, 0), and hence to settle for the more elementary fact that  p(x, ξ ) − ω = a(x, ξ, ω)(ξ + g(x, ω)) + O (x, ξ, ω)∞ .

(6.2.7)

Looking at the Taylor expansions for r, a, g up to second order, we first write r(x, ξ ) = r2,0 x 2 + r1,1 xξ + r0,2 ξ 2 + O((x, ξ )3 ),

(6.2.8)

106

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

and get after some calculations a(x, ξ, ω) = 1 + ir1,1 x + ir0,2 (ξ + ω) + O((x, ξ, ω)2 ),

(6.2.9)

g(x, ω) = −ω + ir2,0 x 2 + ir1,1 xω + ir0,2 ω2 + O((x, ω)3 ).

(6.2.10)

Here we know that r2,0 = (0)/2. Let P (x, hDx ; h) = hDx + ir(x, hDx ) + hQ(x, hDx ; h) be a classical h-pseudodifferential operator with symbol defined in a neighborhood of (0, 0). Using Malgrange’s preparation theorem repeatedly, we find classical symbols A(x, ξ, ω; h) ∼ a(x, ξ, ω) + ha1 (x, ξ, ω) + · · · , G(x, ω; h) ∼ g(x, ω) + hg1 (x, ω) + · · · ,

(6.2.11)

defined in a neighborhood of (x, ξ, ω) = (0, 0, 0), such that in the sense of formal composition of h-pseudodifferential operators, P (x, hDx ; h) − ω = A(x, hDx , ω; h)(hDx + G(x, ω; h)).

(6.2.12)

When using the softer version of Taylor series division, we have to add an error term to the right-hand side of the form S(x, hDx , ω; h), where S(x, ξ, ω; h) ∼ s0 (x, ξ, ω) + hs1 (x, ξ, ω) + . . . , sj (x, ξ, ω) = O((x, ξ, ω)∞ ), and this will still suffice for our purposes. We now concentrate on the region 2

|ω| ≤ ω ( h 3

(6.2.13)

B(x, hDx , ω; h) = hDx + G(x, ω; h).

(6.2.14)

and study the factor

We introduce the scaling ω = α ω, x =

√

α x,

(6.2.15)

6.2 Preparations and Reductions

107

where we let α " ω, so that  ω " 1, | ω| ≤  ω. Then, with  h = h/α 3/2

(6.2.16)

1, we get

1 √  B =  x,  B = α B, hD x , α ω ; h) =:  hD ω, α;  h). x + G( α x + G( α

(6.2.17)

From (6.2.11) we get  x,  G( ω , α;  h) ∼  g ( x,  ω , α) +

∞ 

α

√  x , α ω ), hk gk ( α

3k 2 −1

(6.2.18)

k=1

where  g ( x,  ω, α) :=

 j 1 √ g( α x , α ω ) ∼ − ω+ gj,k α 2 +k−1 xj  ωk , α j +k≥2

and we write the Taylor expansion 

g(x, ω) = −ω +

gj,k x j ωk .

j +k≥2

Recall here that, by (6.2.10), gj,k = irj,k for j + k = 2. Hence,  g ( x,  ω, α) ∼

∞ 



α2 g ( x,  ω),

(6.2.19)

 g1 ( x,  ω) = g3,0 x 3 + ir1,1 x ω,

(6.2.20)

0

where  g0 ( x,  ω) = − ω + ir2,0 x 2, and in general, for  ≥ 1,  g ( x,  ω) =

 j  2 +k=1+ 2

Recall that r2,0 = (0)/2.

gj,k  xj  ωk .

(6.2.21)

108

6.3

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

The Factor hDx + G(x, ω; h)

In this section we study hDx + G(x, ω; h) and its inverse. We mainly concentrate on the region (6.2.13), where we add the restriction that ω hδ , for δ > 0 arbitrarily small and fixed; this assumption will be strengthened later. From (6.2.10), (6.2.11) we see that Gxx " 1,

(6.3.1)

Gx = 2r2,0 x + (r1,1 )ω + O(|(x, ω)|2 + h),

(6.3.2)

for x in a small fixed neighborhood of 0. Hence G(·, ω) has a nondegenerate minimum at x = xmin (ω) = O(|ω| + h),

(6.3.3)

G(xmin, ω; h) = −ω + O(|ω|2 + h).

(6.3.4)

with

We can extend the definition of G to x ∈ R in such a way that (6.2.10), (6.2.11), (6.3.1), (6.3.2) become valid for x ∈ R and so that G = −ω+ir2,0 x 2 outside a small neighborhood of x = 0. We have G(x, ω) − G(xmin (ω), ω) " (x − xmin (ω))2 ,

(6.3.5)

∂x G(x, ω) " x − xmin(ω),

(6.3.6)

G(x, ω) = 0

(6.3.7)

and we see that the equation

has exactly two solutions, x = x± (ω), with 1

∓ (x± (ω) − xmin (ω)) " (ω) 2 .

(6.3.8)

Let ξ = ξ± (ω) = O(ω) be the solutions of ξ + G(x± (ω), ω) = 0, so that ξ± + G(x± (ω), ω) = 0.

(6.3.9)

6.3 The Factor hDx + G(x, ω; h)

109

Apart from the fact that ω 1, we are very much in the situation of Chap. 3. To remedy for the smallness of ω, we make the scalings (6.2.15)–(6.2.19) and work with  x,   = hD ω, α; h), B x + G( where we recall that, 1 1 √  x,  x , α ω ; h) =  g ( x,  ω) + α 2  G( ω, α;  h) = G( α hO(1). α

Here O(1) stands for a function which is bounded with all its  x -derivatives. (The asymptotic expansion (6.2.18) will be used in a region  x = O(1).) Then 2 ∂ x G " 1,

√   and G(·, ω ) has a nondegenerate minimum at  xmin( ω) = xmin (ω)/ α, so " ∂ ω). x − xmin ( xG Further, the equation  x,  G( ω) = 0,

(6.3.10)

√ has exactly two solutions  x = x± ( ω) = x± (ω)/ α, with 1

∓( x± ( ω) −  xmin ( ω)) " ( ω) 2 " 1.  x± ,  Let  ξ = ξ± ( ω, α) be the solution of  ξ + G( ω, α) = 0, so that  x± ,   ω) = 0. ξ± + G(

(6.3.11)

 h

(6.3.12)

Notice that  ξ± = ξ± /α = O(1). Under the assumption 2

1, i.e., α ( h 3 ,

 =   as in Sects. 3.1 and 3.2. Since we will work in the we can now study B hD x + G rescaled variables for a while, we will drop the tildes until further notice and simply recall their existence by addingafter each formula. As in Sect. 3.1, we have a function 1

i

ewkb(x) = h− 4 a(h)χ(x − x+ (ω, α))e h φ+ (x), 

(6.3.13)

110

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

such that χ ∈ C0∞ ( ] − C, x− (ω, α) −

C 1 2 [ , χ = 1 on ] − , x− (ω, α) − [ ,  C 2 C

(6.3.14)

where C ( 1 ,  a ∼ a0 (ω, α) + ha1 (ω, α) + · · · ,

a0 (ω, α) =

∂x2 φ+ (x+ ) π

 14

, 

ewkb L2 = 1,   φ+ (x) = φ+ (x, ω, α) = −

x

G(y, ω, α)dy, 

(6.3.15) (6.3.16) (6.3.17)

x+ (ω,α)

φ+ (x) " (x − x+ )2 , 

(6.3.18)

uniformly on any fixed compact subset of ] −∞, x− [ . By construction, (hDx + G)ewkb (x) =

1 i h  χ (x − x+ (ω, α))h− 4 a(h)e h φ+ (x),  i

(6.3.19)

which decays exponentially in L2 . Define the (ω, α) -dependent self-adjoint operators  = (hD + G)∗ (hD + G), † = (hD + G)(hD + G)∗ : L2 (R) → L2 (R)  with domain D() = D(† ) = {u ∈ L2 (R); x 2j (hD)k ∈ L2 (R), for j + k ≤ 2}.  These operators can be viewed as h -pseudodifferential operators with symbol in S(m),  2 m(x, ξ ) = ξ  + x2 . Again, σ () = σ († ) = {t02 , t12 , . . .},

0 ≤ tj # +∞ 

and Proposition 3.1.1 applies so that t02 is a simple eigenvalue which is exponentially small and t12 − t02 ≥ h/C . The second fact can be proved by modifying slightly the proof of Lemma 3.1.2, or else we can apply more general microlocal analysis as indicated after Proposition 5.3.1. Let 0  be the spectral projection corresponding to (, t02 )  and let e0 = 0 ewkb −1 0 ewkb be the normalized eigenstate with ( − t02 )e0 = 0 (unique up

6.3 The Factor hDx + G(x, ω; h)

111

to a factor eiθ , for some θ ∈ R). By (6.3.19), ewkb = r,

1 i h r = (hD + G)∗ χ  (x − x+ )h− 4 a(h)e h φ+ ,  i

(6.3.20)

and we see that for any given > 0, we can ensure that  1  r = O e− h (S0 − ) in L2 , 

(6.3.21)

by choosing the cutoff “wide enough”, i.e., with C in (6.3.14) large enough. Here S0 > 0 is the tunneling action between (x+ , ξ+ ) and (x− , ξ− ) , given by  S0 = −

x−

G(x, ω, α)dx ,

(6.3.22)

x+

or equivalently by  S0 = 

ξ dx, 

(6.3.23)

γ

where γ = {(x, ξ ); x+ ≤ x ≤ x− , ξ + G(x, ω, α) = 0} is oriented from (x+ , ξ+ ) to (x− , ξ− ) . For |z| = hδ , 0 < δ 1 , we write (z − )ewkb = zewkb − r,  leading to (z − )−1 ewkb = z−1 ewkb + z−1 (z − )−1 r.  Using that 0 =

1 2πi

 |z|=δh

(z − )−1 dz, 

we get 0 ewkb = ewkb +

1 2πi

 |z|=δh

z−1 (z − )−1 rdz = ewkb + O



Here we use (6.3.21) in the last step. It follows that 0 ewkb  = 1 + O(h−1 e− h (S0 − ) ),  1

1 − 1 (S0 − ) e h h

 in L2 . 

112

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

so 1

e0 = ewkb + O(h−1 e− h (S0 − ) ) in L2 . 

(6.3.24)

Similarly, we have a WKB state for (hD + G)∗ = hD + G given by 1

i

χ (x − x− (ω, α))e h φ− (x),  fwkb (x) = h− 4 b(h)  φ− (x) = φ− (x, ω, α) = −

x

(6.3.25)

G(y, ω, α)dy, 

(6.3.26)

x− (ω,α)

φ− (x) " (x − x− )2 , 

(6.3.27)

uniformly on any fixed compact subset of ]x+ , ∞[ ,  b ∼ b0 (ω, α) + hb1 (ω, α) + . . . , b0 (ω, α) =

∂x2 φ− (x− ) π

 14

, 

fwkb L2 = 1,  (hDx + G)fwkb (x) =

1 i h  χ  (x − x− (ω, α))h− 4 b(h)e h φ− (x),  i

(6.3.28) (6.3.29) (6.3.30)

which is exponentially decaying in L2 . The properties of χ (x − x− ) are analogous to those of χ(x − x+ ) . As above, we get a corresponding normalized eigenstate f0 of † : († − t02 )f0 = 0, 

(6.3.31)

f0 = fwkb + O(h−1 e− h (S0 − ) ) in L2 . 

(6.3.32)

with 1

We have (hD + G)e0 = α0 f0 , (hD + G)f0 = α 0 e0 , |α0 | = t0 . 

(6.3.33)

We now go beyond Chap. 3 and study the precise asymptotics of t0 . Use (6.3.33) to write α0 = ((hD + G)e0 |f0 ) = (χ+ (hD + G)e0 |f0 ) + (χ− (hD + G)e0 |f0 ), 

6.3 The Factor hDx + G(x, ω; h)

113

where χ± ∈ C ∞ (R) , supp χ+ ⊂ ] − ∞, x− [ , supp χ− ⊂ ]x+ , +∞[ , 1 = χ+ + χ− . We get α0 = (χ+ (hD + G)e0 |f0 ) + (χ− e0 |(hD + G)∗ f0 ) + ([χ− , hD]e0 |f0 )  = α0 ((χ+ f0 |f0 ) + (χ− e0 |e0 )) + ih(χ− e0 |f0 ).  Here we use (6.3.24), (6.3.32) and the exponential decay of fwkb and ewkb away from x− and x+ , respectively, to see that  1  (χ+ f0 |f0 ), (χ− e0 |e0 ) = O e− Ch .  Hence, 

 1  1 + O e− Ch α0 = ih(χ− e0 |f0 ). 

(6.3.34)

Using again (6.3.24), (6.3.32) and the fact that supp χ− is contained in a compact subset of ]x+ , x− [ , where ewkb and fwkb are exponentially decaying, we get 1

1

(χ− e0 |f0 ) = (χ− ewkb |fwkb ) + O(1)e− h (S0 + C ) .  Then (6.3.34) shows that α0 = ih(χ− ewkb |fwkb ) + O(1)e− h (S0 + C ) .  1

1

(6.3.35)

Recall that t0 ≥ 0 by construction and that we have inserted an x -independent factor of modulus one in front of f0 and fwkb . Combining (6.3.35), (6.3.25), (6.3.28), (6.3.13), (6.3.15), we get α0 = (1 + O(h))h

1 2



∂x2 φ+ (x+ ) π

 14 

∂x2 φ− (x− ) π

 14

 0 ei θ − h ,  θ ∈ R.  S

Here we also used that i

i

e h (φ+ (x)−φ− (x)) = e h

 x− x+

(−G(y))dy

where  θ ∈ R is independent of x and the fact that

t0 = |α0 | = (1 + O(h))h

1 2



∂x2 φ+ (x+ ) π

 14 



= e i θ − h S0 , 

 +∞ −∞

1

χ− (y)dy = 1 . It follows that

∂x2 φ− (x− ) π

 14

S0

e− h . 

(6.3.36)

114

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

From (6.3.17) and (6.3.26), we get ∂x2 φ+ (x+ ) = −∂x G(x+ ),  ∂x2 φ− (x− ) = −∂x G(x− ).  Introducing q = ξ + g(x, ω, α) , we get 1 {q, q}(ρ+ ) + O(h) " 1,  2i 1 ∂x2 φ− (x− ) = {q, q}(ρ− ) + O(h) " 1,  2i

∂x2 φ+ (x+ ) =

(6.3.37)

where ρ± = (x± , ξ± ) . Then (6.3.36) becomes  1  1  1 4 4 1 h 2 1 1 {q, q}(ρ+ ) {q, q}(ρ− ) e− h S0 .  t0 = (1 + O(h)) π 2i 2i

(6.3.38)

After replacing f0 , fwkb by eiθ f0 , eiθ fwkb for a suitable real θ , we may assume that α0 in (6.3.33) is equal to t0 . Proposition 3.2.1 carries over to the present situation, where we write out explicitly the various tildes. Proposition 6.3.1 Let m = ξ +x2 and define the semi-classical Sobolev space H h (m) + : H 1 (R) → C, R − : as in Sect. 5.1, with  h as the semi-classical parameter. Define R  h 2 C → L (R) by − u− = u− f0 . R

+ u = (u| e0 ), R Then   := P(z)

 R − hD + G + R 0

 2 : H h (m) × C → L × C

is bijective with the bounded inverse 

   = E E+ E(z) − E −+ E



  1 O( h− 2 ) O(1) : L2 × C → H = h (m) × C.  O(1) O(e−1/C h )

Here, − v = (v|f0 ), E −+ = − + v+ = v+ e0 , E t0 . E

6.3 The Factor hDx + G(x, ω; h)

115

We turn to the corresponding result for the unscaled operator hD +G in (6.2.12). Recall that  hDx + G = α( hD x + G),

 h = h/α 3/2 , x =

√ α x.

(6.3.39)

Thus the common lowest eigenvalue of  = (hD + G)∗ (hD + G) and † = (hD + t0 , and the corresponding normalized G)(hD + G)∗ is of the form t02 = α 2 t02 , where t0 = α eigenfunctions are f0 (x) = α − 4 f0 ( x ).

(6.3.40)

fwkb (x) = α − 4 fwkb ( x ).

(6.3.41)

1

e0 ( x ), e0 (x) = α − 4 

1

They are approximated by ewkb (x) = α − 4  ewkb ( x ), 1

1

The common gap t12 − t02 between the first two eigenvalues of , † is " α 2 h= Recalling the proof of Proposition 3.2.1 by spectral decomposition, we get

√

αh.

Proposition 6.3.2 With m as in the preceding proposition, we define the semi-classical Sobolev space H (m) as there, with h as the semi-classical parameter. Define R+ : H 1 (R) → C, R− : C → L2 (R) by R+ u = (u|e0 ),

R− u− = u− f0 .

Then  P(z) :=

hD + G R− R+ 0

 : Hh (m) × C → L2 × C

is bijective with the bounded inverse 

 E E+ E(z) = E− E−+   1 1 O(1) O(α − 4 h− 2 ) : L2 × C → Hh (m) × C. = α 3/2 1 1 − O(1) O(α 4 h 2 )e Ch Here, E+ v+ = v+ e0 , E− v = (v|f0 ), E−+ = −t0 .

116

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

Let us finally derive an asymptotic formula for t0 with the help of (6.3.38), where all the quantities carry invisible tildes. Now with the tildes written out, we recall that  x± are given by  x± ) = 0, G( This corresponds to x± =

√

 x+ <  x− .

α x± satisfying G(x± ) = 0. We get,

3  3  x−  S0 α 2 x− G(y) dy S0 α2   G( y )d y=− = , =− 1  h  h x+ α α2 h h x+

where,  S0 = −

x−

G(y)dy.

(6.3.42)

x+

 x x  + ( x) = −  Recalling that φ y )d y , we put φ+ (x) = − x+ G(y)dy. Then by the x+ G( change of variables above, we get  φ+ = α −3/2 φ+ (x), or equivalently,  φ+ φ+ . =  h h Similarly,  φ− φ− , =  h h where (cf. (6.3.26)), φ− (x) = −

x x−

G(y)dy. Clearly,

2 2 −2 2 ∂ ∂x φ± . x φ± = α∂x φ± = α 1

Let q = ξ + g,  q = ξ + g , so 1

{ q,  q }( ρ± ) = α − 2 {q, q}(ρ± ). We use this in (6.3.38) and get t0 = t0 = α 1  1     1  2 4 4 1 h h − 12 1 − 12 1 {q, {q, q}(ρ ) q}(ρ ) e − h S0 . α 1+O α α + − 3 3 2i 2i α2 πα 2

6.3 The Factor hDx + G(x, ω; h)

117

We already know that the precise choice of α " ω should not appear in the final result, and indeed the powers of α cancel out and we end up with 1  1    h  12  1 4 4 3 1 1 {q, q}(ρ+ ) {q, q}(ρ− ) e− h S0 . t0 = 1 + O h/(ω) 2 π 2i 2i 

Here ρ± and S0 are determined from the full symbol Q = ξ + G(x, ω; h) = q + O(h). 0 = ρ + O(h/√α) such that q(ρ 0 ) = 0, and moreover There are unique (real) points ρ± ± ± √ 0 0 ) + O(h/ α) = (1 + O(h/α)){q, q}(ρ± ). {q, q}(ρ± ) = {q, q}(ρ±

(6.3.43)

Similarly, we can compare S0 with  I =−

0 x−

0 x+

0 0 g(x, ω)dx, ρ± = (x± , ξ±0 ).

(6.3.44)

Since √ 0 x± − x± = O(h/ α),

0 |x± |"

√ α,

G − g = O(h),

we get 

 √ √ hα I − S0 = O(1) √ + h α = O(h α). α √ Here we also used that G = O(α) when x = O( α). Hence, √ e−S0 / h = (1 + O( α))e−I / h and here

√

(6.3.45)

α ≤ h/α 3/2 if we work with α in the range C0 h2/3 ≤ α ≤ h1/2 ,

(6.3.46)

which will be the case when in addition to (6.2.13), we have w  α O(1)(h ln(1/ h))2/3. Using (6.3.43) and (6.3.45) in the formula for t0 above, we get

≤

1  1   h  12  1   4 4 3 1 1 2 {q, q}(ρ+ ) {q, q}(ρ− ) e− h I , t0 = 1 + O h/(ω) π 2i 2i (6.3.47) where from now on ρ± denote the real zeros of q = ξ + g(x, ω).

118

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

As we have already seen, t0 " h 2 α 4 e−I / h 1

1

1

(6.3.48)

h2 ,

3

I " α2.

(6.3.49)

So far in this section, we have worked under the assumption (6.2.13) in conjunction with w ≤ O(1)(h ln(1/ h))2/3 . Let us consider two more regions: 1. Let |ω| ≤ O(h2/3 ).

(6.3.50)

We check that hDx + G : Hh (m) → L2 has the two-sided inverse i Ev(x) = h



x +∞

i

e h (φ(x)−φ(y))v(y)dy,

(6.3.51)

where φ is determined up to a constant by the eikonal equation φx + G(x, ω; h) = 0.

(6.3.52)

We have ⎧ ⎨= O(h2/3 ), | − φx | ⎩" h2/3 + x 2 ,

if |x| ≤ O(h1/3 ), if |x| ( h1/3 .

It follows that for y ≥ x, 1 1 (φ(x) − φ(y)) ≥ −O(1) + h Ch



y

2

(h 3 + t 2 )dt

x

  2 x3 1 y3 3 − = −O(1) + h (y − x) + Ch 3 3   2 1 ≥ −O(1) + (y − x) h 3 + x 2 + y 2 ,  Ch

 > 0. Thus in (6.3.51) for some C, C  i  2  h (φ(x)−φ(y)) − 1 (h 3 +|x|2 +|y|2 )(y−x) , e  ≤ O(1)e Ch

(6.3.53)

6.3 The Factor hDx + G(x, ω; h)

119

and applying Schur’s lemma to E we can conclude that (h2/3 + x 2 )E = O(1) : L2 → L2 .

(6.3.54)

Proof of (6.3.54) The distribution kernel of (h2/3 + x 2 )E is bounded in modulus by a constant times h2/3 + x 2 − 1 (h2/3 +x 2 +y 2 )(y−x) e Ch 1y≥x , h

K(x, y; h) =

and if K(h) denotes the corresponding integral operator, it suffices to show that the L(L2 , L2 )-norm K(h) is O(1), uniformly in h. We observe that x , h1/3 y ; h)h1/3 d y = K( x,  y ; 1)d y, K(h1/3 so K(h) = K(1). It then suffices to consider the case when h = 1. By Schur’s lemma, 1/2 1/2 K(1) ≤ mx my , where 



mx = supx

K(x, y; 1)dy, my = supy

K(x, y; 1)dx,

and it suffices to show that these quantities are bounded. Here  mx ≤ supx

+∞

1

(1 + x 2 )e− C (1+x

2 )(y−x)

dy ≤ O(1).

x

The estimate of my is slightly more difficult. Write,  my = supy

y

1

−∞

(1 + x 2 )e− C (1+x

2 +y 2 )(y−x)

dx ≤ I + II,

where  I = supy

y

(1 + y 2 )e− C (1+x 1

−∞

2 +y 2 )(y−x)

dx ≤ O(1),

as for mx , and  II = supy = supy

x

−∞  x −∞

|x 2 − y 2 |e− C (1+x 1

2 +y 2 )(y−x)

1

|x + y|(x − y)e− C (1+x

dx

2 +y 2 )(y−x)

dx.

120

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

Since |x + y| ≤ O(1)(1 + x 2 + y 2 ), the integrand in II is 1

≤ |x + y|(x − y)e− 2C (1+x

2 +y 2 )(y−x)

1

1

e− 2C (y−x) ≤ O(1)e− 2C (y−x),

and it follows that II = O(1). Hence my = O(1).

 

From (hD + G)E = 1, we see that hDE = O(1) : L2 → L2 . It follows that E = O(h−2/3 ) : L2 → Hh (m).

(6.3.55)

2. When ω −h2/3 , |ω| ≤ O(1)|ω|, we do essentially the same scaling as above, 1/2 x , α " −ω, to see that x=α  √  x , ω; h)), G(  x , ω; h) = 1 G( αx, ω; h), hD + G = α( hD x + G( α which is an elliptic  h-pseudodifferential operator. We conclude by elliptic theory that   + G : H (m) → L2 has a bounded inverse and, after returning to the xhD  x h coordinates, that hD + G : Hh (m) → L2 is bijective with inverse E satisfying, E = O(α −1 ) : L2 → Hh (m).

6.4

(6.3.56)

Global Grushin Problem, End of the Proof

As a preparation for the cutting and pasting in the global situation, we establish some microlocal properties for E, E± in Proposition 6.3.2. This will mainly concern the eigenfunctions e0 , f0 . Assume (cf. (6.3.46)) that |ω| ≤ ω,

2

h3

1

ω ≤ h 2 .

1, |x| ( hδ/2 , The symbol ξ + G(x, ω; h) belongs to S(m), m = ξ  + x2 . For 0 < δ δ 2 δ 2 δ we have G ≥ h + |x| /O(1) ≥ h x /O(1) and hence |ξ + G| ≥ h m(x, ξ )/O(1) for all ξ . The same conclusion is valid when |ξ | ( hδ . Thus, |ξ + G| ≥ hδ m(x, ξ ), when |ξ | + |x|2 ( hδ .

(6.4.1)

6.4 Global Grushin Problem, End of the Proof

121

Hence ξ + G(x) is a slightly degenerate elliptic symbol in the region |ξ | + |x|2 ( hδ , and following the standard construction of parametrices for elliptic operators (cf. [41], Ch. 8) we get a symbol J (x, ξ ; h) (depending also on ω) such that α ∂x,ξ J = Oα (1)h−δ(1+|α|) m(x, ξ )−1 , (x, ξ ) ∈ R2 ,

(ξ + G)J, J (ξ + G) ∼ 1 in S(1), when |ξ | + |x|2 ( hδ .

(6.4.2) (6.4.3)

Here  denotes the composition of symbols in the semi-classical Weyl calculus [41], corresponding to composition of h-pseudodifferential operators. Letting J also denote the corresponding h-pseudodifferential operators, this means that (hD + G) ◦ J = 1 + K,

J ◦ (hD + G) = 1 + L,

(6.4.4)

where K and L are h-pseudodifferential operators with symbols K(x, ξ ; h) and L(x, ξ ; h) that satisfy ⎧ ⎨O(h−δ|α| ), on R2 , α α ∂(x,ξ ) K, ∂(x,ξ ) L = ⎩ O(h∞ ), for |ξ | + |x|2 ( hδ .

(6.4.5)

Recall that  = (hD + G)∗ (hD + G), † = (hD + G)(hD + G)∗ , and use the same letters for the symbols of these operators. Both symbols are ≡ |ξ + G|2 mod S(hm2 ). h. We conclude that By (6.3.48), we have t02 |(x, ξ, ω; h) − t02 |, |† (x, ξ, ω; h) − t02 | ≥ h2δ m(x, ξ )2 , when |ξ | + |x|2 ( hδ , (6.4.6) provided that δ > 0 is small enough (< 1/2 suffices here). If δ < 1/4, we can construct  such that symbols R, R α α −2δ(1+|α|)  m(x, ξ )−2 ), (x, ξ ) ∈ R2 , ∂(x,ξ ) R, ∂(x,ξ ) R = Oα (h

 R(  † − t02 ) ∼ 1 ( − t02 )R, R( − t02 ), († − t02 )R, in S(1), in the region |ξ | + |x|2 ( hδ .

(6.4.7)

(6.4.8)

Passing to the corresponding h-pseudodifferential operators, this means that ( − t02 ) ◦ R = 1 + K,

R ◦ ( − t02 ) = 1 + L,

 = 1 + K,  († − t02 ) ◦ R

 ◦ († − t02 ) = 1 + L,  R

(6.4.9)

122

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

  where K, L, K, L are h-pseudodifferential operators with symbols satisfying ⎧ ⎨O(h−2δ(1+|α|) ), on R2 , α α α  α  ∂(x,ξ ) K, ∂(x,ξ ) K, ∂(x,ξ ) L, ∂(x,ξ ) L = ⎩ O(h∞ ), for |ξ | + |x|2 ( hδ .

(6.4.10)

Applying the 2nd and the 4th equations in (6.4.9) to e0 and f0 , respectively, we get e0 = −Le0 , f0 = − Lf0 ,

(6.4.11)

showing that e0 , f0 are microlocally concentrated in a region |ξ | + |x|2 ≤ O(hδ ). This gives a corresponding localization for R± = E∓ in Proposition 6.3.2: E+ = −LE+ , E− = −E−  L∗ .

(6.4.12)

Let χ ∈ C0∞ (R) be a standard cutoff function, equal to 1 on [−1/2, 1/2] and with support in ] − 1, 1[. We claim that E = χ2 Eχ1 + J ((1 − χ2 χ1 ) − [hD + G, χ2 ]Eχ1 ) + O(h∞ ) in L(L2 , Hh (m)), (6.4.13) where J is the h-pseudodifferential operator in (6.4.4) and   χj (x, ξ ) = χ (ξ 2 + x 4 )/(Cj h2δ ) , C1 , C2 ( 1. (In (6.4.13), χj denote the corresponding h-pseudodifferential operators.) In fact, let F = χ2 Eχ1 + J ((1 − χ2 χ1 ) − [hD + G, χ2 ]Eχ1 ). Then, since (hD + G)E + R− E− = 1 and K(1 − χ2 χ1 ), K[hD + G, χ2 ], R− E− − χ2 R− E− χ1 are = O(h∞ ), we have (hD + G)F = 1 − R− E− + O(h∞ ) in L(L2 , L2 ), so (hD + G)F + R− E− = 1 + O(h∞ ) in L(L2 , L2 ).

(6.4.14)

Moreover, R+ F = O(h∞ ) in L(L2 , C).

(6.4.15)

6.4 Global Grushin Problem, End of the Proof

123

Apply E to the left in (6.4.14) and use that E(hD + G) = 1 − E+ R+ , ER− = 0. Then E = (1 − E+ R+ )F + O(h∞ ) in L(L2 , Hh (m)). From (6.4.15) we then infer that E = F + O(h∞ ) and (6.4.13) follows. Let χ j be narrower cutoffs with the same properties as χj so that χ j χj ≡ χj  χj ≡ χ j modulo O(h∞ ) in S(1) and so that (6.4.13) remains valid with χj replaced by χ j : E=χ 2 E χ1 + J ((1 − χ 2 χ 1 ) − [hD + G, χ 2 ]E χ1 ) + O(h∞ ) in L(L2 , Hh (m)). Then modulo O(h∞ ) in L(L2 , Hh (m)), [hD + G, χ2 ]Eχ1 ≡ [hD + G, χ2 ]J (1 − χ 2 χ 1 )χ1 − [hD + G, χ2 ]J [hD + G, χ 2 ]E χ1 χ1 ≡ [hD + G, χ2 ]J (1 − χ 2 χ 1 )χ1 ≡ [hD + G, χ2 ]J χ1 , and (6.4.13) simplifies to E = χ2 Eχ1 + J ((1 − χ2 χ1 ) − [hD + G, χ2 ]J χ1 ) + O(h∞ ) in L(L2 , Hh (m)). (6.4.16) We can now set up the global Grushin problem for P − z in Theorem 6.1.4. Recall that p−1 (z0 ) consists of N points ρ01 , . . . , ρ0N by (6.1.37). For z ∈ neigh (z0 , C), we introduce the new base points ρj (z) as in Remark 6.2.1, so that (6.1.38) reduces to wj = isj . Let κj (depending on z) be an affine canonical transformation, mapping (0, 0) to ρj (z), such that as in the discussion around Remark 6.2.1, p ◦ κj − p(ρj (z)) = eiθj (ρj (z))(ξ + ir j (x, ξ )), or equivalently, p ◦ κj − z = eiθj (ρj (z)) (ξ + ir j (x, ξ ) − ωj ), where r j (x, ξ ) = O(x 2 + ξ 2 ), ωj = isj + O(sj2 ) On the operator level, this means that Uj−1 (P − z)Uj =eiθj (ρj (z)) (Pj − ωj )  j hDx + Gj (x, ωj ; h) + Sj , =A j = eiθj (ρj (z))Aj , A

(6.4.17)

124

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

as in (6.2.12) and where Sj has the properties of “S”, explained after that relation. Here Uj is a unitary metaplectic Fourier integral operator associated to κj (see the notes at the end of this chapter). Most of the work in this section concerns the case when j ∈ N , where we recall that N = {j ∈ {1, . . . , N}; C0 h2/3 ≤ sj ≤ O(1) (h ln(1/ h))2/3 }. For each j , we consider the Grushin problem in Proposition 6.3.2:   j hD + Gj R− Pj = j R+ 0 with inverse  j E j E+ , Ej = j j E− E−+ 

where R+ u = (u|e0 ), R− u− = u− f0 . Recall that E± = (R± )∗ and that E−+ = −tj , where tj fulfills the j -dependent version of (6.3.47). 1 Let ψ ∈ C0∞ (R2 ) be a standard cutoff function, equal to 1 near (0, 0). For 0 < δ0 fixed, we put ψj (x, ξ ; h) = ψ(h−δ0 ((x, ξ ) − ρj )) and also write ψj for the corresponding h-pseudodifferential operator. Our global Grushin problem is then j

j

j

j

j

j

  P − z R− P(z) = : Hh (m) × CN → L2 × CN , R+ 0

j

(6.4.18)

where m now denotes the original order function, associated to P and (R+ u)(j ) = R+ Uj−1 u, j ∈ N , j

R− u− =

 j ∈N

j j R− Uj A u− (j ).

(6.4.19) (6.4.20)

The localization properties of e0 , f0 imply that R− is well defined mod O(h∞ ) despite j , which is only defined microlocally near (0, 0). It is independent of the the presence of A choice of ψj up to operators that are O(h∞ ). Let J (x, ξ ; h) be a parametrix of P − z in the region, j

j



dist ((x, ξ ), {γ1 , . . . , γN }) ≥ hδ ,

γj = γj (τj (z)),

(6.4.21)

6.4 Global Grushin Problem, End of the Proof

where δ0 <  δ

125

1, such that 

α −δ (1+|α|) ∂(x,ξ m(x, ξ )−1 ), ) J = O(h

(P − z)J, J (P − z) ∼ 1 in the region (6.4.21).

(6.4.22) (6.4.23)

As usual, J will also denote the corresponding h-pseudodifferential operator, depending on the context. To construct a right inverse of P(z) amounts to finding a solution (u, u− ) ∈ H (m)×CN of the system ⎧ ⎨(P − z)u + R u = v, − − ⎩R+ u = v+ , for any given (v, v+ ) ∈ L2 × CN . Take first u0 = J (1 −

(6.4.24) N 1

ψj )v, so that

⎧ ⎨(P − z)u = (1 − N ψ )v + O(h∞ v) in L2 , 0 j 1 ⎩R+ u0 = O(h∞ v) in CN . When j ∈ N , we look for a solution uj microlocally concentrated to a small neighborhood of ρj , and u− (j ) ∈ C, so that ⎧ ⎨(P − z)u + R j u (j ) = ψ v, j j − − ⎩(R+ uj ) = v+ (j )δj , up to small errors, where we let δj denote the j th canonical basis vector in CN , so that δj (k) = δj,k , the latter being the Kronecker delta. The concentration of uj to a small neighborhood of ρj will then imply that (R+ uj )(k) = O(h∞ ) for k = j and we try to solve ⎧ j −1 ⎨U A j R− u− (j ) = ψj v, j j (hD + Gj )Uj uj + Uj A j ⎩R U −1 uj = v+ (j ), + j

which formally would follow from ⎧ ⎨(hD + G )U −1 u + R j u (j ) = A −1 U −1 ψj v, j j − − j j j ⎩R j U −1 uj = v+ (j ), +

j

126

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

so we are led to the choice ⎧ j ⎨u = U E j A −1 U −1 ψj v + Uj E+ v+ (j ), j j j j ⎩u− (j ) = E j A −1 U −1 ψj v + E j v+ (j ). −

j

j

−+

j is well-defined and elliptic only in a fixed neighborhood of (0, 0), but Notice here that A −1 U −1 ψj is well-defined modulo O(h∞ ), thanks to the cutoff ψj . the operator A j j When j ∈ {1, 2, . . . , N} \ N =: N c , we know from the discussion at the end of Sect. 6.3 that hDx + Gj : Hh (m) → L2 (R) is bijective, with inverse E j : L2 → Hh (m) of norm O((h2/3 + |sj |)−1 ). The obvious j -dependent version of (6.4.16) also holds. As an approximate right inverse of P(z), we try   E + E , E = − E −+ E 

(6.4.25)

where ⎧   ⎪  = J (1 − N ψj )v + N Uj E j A −1 U −1 ψj v, ⎪Ev 1 1 j j ⎪ ⎪  ⎪ j ⎨E + v+ = j ∈N Uj E+ v+ (j ), j −1 −1 ⎪ − v)(j ) = E− ⎪ (E Aj Uj ψj v, j ∈ N , ⎪ ⎪ ⎪ j ⎩ E−+ = diag (E−+ ).

(6.4.26)

Recall that we work under the assumptions of Theorem 6.1.4, so −hδ0 ≤ sj ≤ O(1)(h ln(1/ h))2/3 , ∀j, where δ0 > 0 is arbitrarily small and fixed. Proposition 6.4.1 We have,  ⎧ 1 − 12 − 4 2/3 −1 ⎪  ⎪ c E = O(1) max maxj ∈N h sj , maxj ∈N (h + |sj |) , ⎪ ⎪ ⎪ ⎪ ⎨ + = O(1) : CN → Hh (m), E ⎪ ⎪ − = O(1) : L2 → CN , ⎪ E ⎪ ⎪ 1 1 ⎪ 3/2 ⎩ E−+ = O(1) maxj ∈N sj4 h 2 e−sj /O(h) : CN → CN .

(6.4.27)

6.4 Global Grushin Problem, End of the Proof

127

Moreover, ⎧ ∞ 2 2 ⎪   ⎪ ⎪(P − z)E + R− E− = 1 + O(h ) : L → L , ⎪ ⎪ ⎨(P − z)E + + R− E −+ = O(h∞ ) : CN → L2 , ⎪  = O(h∞ ) : L2 → CN , ⎪ R+ E ⎪ ⎪ ⎪ ⎩  R+ E+ = 1 + O(h∞ ) : CN → CN .

(6.4.28)

Proof (6.4.27) follows from Proposition 6.3.2, (6.3.54), (6.3.56) (where we recall that  −1 = O(1). α " sj when |sj | ( h2/3 ) and the bounds J = O(h−δ ), Uj , A j The proof of (6.4.28) is just a long calculation with some attention to terms that disappear because of the localization properties. Using (6.4.26) we get modulo O(h∞ ) : L2 → L2 , ≡ 1− (P − z)E

N  1

ψj +

N  −1 U −1 ψj . (P − z)Uj E j A j j

(6.4.29)

1

By (6.4.17), −1 U −1 ψj ≡ Uj A j (hD + Gj )E j A −1 U −1 ψj + Uj Sj E j A −1 U −1 ψj . (P − z)Uj E j A j j j j j j (6.4.30) j A j is a pseudodifferential −1 U −1 ψj ≡ ψ −1 U −1 , where ψ By Egorov’s theorem, A j h j j 

operator of the same class as ψj and whose symbol is supported in an O(hδ )-neighborhood j E j ψ j ≡ ψ j , where of (0, 0). Now by the localization properties of Ej , we see that E j ψ   ψj has the same properties as ψj . Recalling that every term in the h-asymptotic expansion j ≡ 0, and hence the last of Sj vanishes to infinite order at (0, 0), we conclude that Sj E j ψ term in (6.4.30) is ≡ 0. Using also that ⎧ ⎨(hD + G )E j + R j E j = 1, j ∈ N , j − − ⎩(hD + Gj )E j = 1, j ∈ N c we get ⎧ j j −1 −1 ⎨ψ − U A j j j R− E− A j Uj ψj , −1 U −1 ψj ≡ (P − z)Uj E j A j j ⎩ψj , if j ∈ N c .

if j ∈ N ,

(6.4.31)

128

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

On the other hand, using (6.4.26) and (6.4.20), − = R− E

 N

j R− E− A −1 U −1 ψj . Uj A j j j

j

(6.4.32)

Summing (6.4.31) over j and adding (6.4.32), we get N   −1 U −1 ψj + R− E − ≡ (P − z)Uj E j A ψj , j j 1

which together with (6.4.29) gives the first equation in (6.4.28). Next look at + v+ = (P − z)E

 j (P − z)Uj E+ v+ (j ). N

Modulo O(h∞ v+ ) in L2 , we get + v+ ≡ (P − z)E

 N

j j (hD + Gj )E+ Uj A v+ (j ) ≡ −

 N

j j j R− Uj A E−+ v+ (j ).

On the other hand, −+ v+ = R− E

 N

j j j R− Uj A E−+ v+ (j ).

Adding the two equations then gives the second equation in (6.4.28). Next, look at  ) ≡ R+ U −1 Uj E j A −1 U −1 ψj v, R+ Ev(j j j j j

which simplifies to  ) ≡ R+ E j A −1 U −1 ψj v ≡ 0. R+ Ev(j " #$ % j j j

≡0

This gives the third equation in (6.4.28). Finally, we turn to j −1 + v+ )(j ) = R+ (R+ E Uj

 k∈N

k Uk E + v+ (k).

6.4 Global Grushin Problem, End of the Proof

129

k , we get modulo O(h∞ v ): Because of the localization in E+ +

+ v+ )(j ) ≡ R+ U −1 Uj E+ v+ (j ) = R+ E+ v+ (j ) = v+ (j ) (R+ E j j

j

j

j

 

and the fourth equation in (6.4.28) follows.

A similar discussion of the uniqueness of the solutions of (6.4.24) leads to the approximate left inverse of P(z):  E =

  E + E , − E −+ E

(6.4.33)

where ⎧  N ⎪  = (1 − N −1 −1 Ev ⎪ 1 ψj )J v + 1 ψj Uj Ej Aj Uj v, ⎪ ⎪  ⎪ j ⎨E + = N Uj E+ v+ (j ), j −1 −1 ⎪ − v)(j ) = E− ⎪ (E Aj Uj v, j ∈ N , ⎪ ⎪ ⎪ j ⎩ E−+ = diag (E−+ ).

(6.4.34)

j Using the localization properties of E j , E−+ , we can check directly that E = E+ O(h∞ ). This fact also follows from

Proposition 6.4.2 E satisfies (6.4.27). Moreover, ⎧ ⎪  − z) + E + R+ = 1 + O(h∞ ) : Hh (m) → Hh (m), E(P ⎪ ⎪ ⎪ ⎪ ⎨E  (P − z) + E  R = O(h∞ ) : H (m) → CN , −

−+ +

h

⎪  − = O(h∞ ) : CN → Hh (m), ⎪ ER ⎪ ⎪ ⎪ ⎩ E− R− = 1 : CN → CN .

(6.4.35)

We omit the proof, which is merely a variation of the proof of Proposition 6.4.1. An immediate consequence of the two results is Proposition 6.4.3 For h > 0 small enough, P(z) : Hh (m)×CN → L2 ×CN is bijective with a bounded inverse E(z) = P(z)−1 which satisfies,  + O(h∞ ) : L2 × CN → Hh (m) × CN .  + O(h∞ ) = E(z) E(z) = E(z)

(6.4.36)

130

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

We can finally look at the norm of the resolvent, (P − z)−1 = E(z) − E+ (z)E−+ (z)−1 E− (z).

(6.4.37)

Here E(z) can be estimated as in (6.4.27), in view of (6.4.36), and we concentrate on the second term. This term is of rank #N and we pause to consider in general, the norm of a finite rank operator A : L2 → L2 , given by Au =

n 

aj,k (u|fk )ej ,

j,k=1

where f1 , . . . , fn and e1 , . . . , en are two linearly independent families in L2 . We orthonormalize the two families,    −1  f1 . . . fn = f1 . . . fn Gf 2 ,    −1  e n = e 1 . . . e n Ge 2 .  e1 . . .  Here the Gramian Ge is given by ⎛ ⎞ e1∗ ⎜ ∗⎟   ⎜e2 ⎟  ⎟ Ge = (ek |ej ) = ⎜ ⎜ .. ⎟ e1 e2 . . . en , ⎝.⎠ en∗

identifying ej with the operator C  z → zej and letting ej∗ be the adjoint. Gf is defined the same way. A straightforward calculation yields Au =

n 

k ) b e j , j, k (u|f

(6.4.38)

 j , k=1

where 1 1 1 1   2 2 ∗ 2 2 (b j, k ) = Ge ◦ aj,k ◦ (Gf ) = Ge ◦ aj,k ◦ Gf .

(6.4.39)

en and f1 , . . . , fn are orthonormal, it is clear from (6.4.38) that Since the families  e1 , . . . , AL2 →L2 = (b j , k )Cn →Cn ,

6.4 Global Grushin Problem, End of the Proof

131

and (6.4.39) gives 1 1  AL2 →L2 = Ge2 ◦ aj,k ◦ Gf2 Cn →Cn .

(6.4.40)

−1 We apply this to E+ E−+ E− . From (6.4.26) (where we can drop ψj because of the j localization property of E− ) and Proposition 6.4.3 we have in L2 , −1 E− u = E+ E−+

 N

j j j −1 −1 ∞ Uj E+ (E−+ )−1 E− A j Uj u +O(h uL2 ),

#$

"

(6.4.41)

%

=:Au

From the discussion after (6.4.17), we recall that j

j

j

j

E+ v+ (j ) = v+ (j )e0 , (E− v)(j ) = (v|f0 ), and so Au =

 j ∗j )−1 f j )Uj ej . (E−+ )−1 (u|Uj (A 0 0

(6.4.42)

N

To apply (6.4.40), note that the Gramian GU e0 of U1 e01 , . . . , UN e0N is equal to 1 + O(h∞ ) ∗ )−1 f 1 , . . . , UN (A ∗ )−1 f N is of the form D + and that the Gramian GU (A∗ )−1 f0 of U1 (A 1 0 N 0 ∞ O(h ), where D = diag (dj ) with ∗j )−1 f |Uj (A ∗j )−1 f ) = ((A ∗j )−1 f |(A ∗j )−1 f ). dj = (Uj (A 0 0 0 0 j

j

j

j

(6.4.43)

By the complex stationary phase method in the rescaled variables, ∗j )−1 f |(A ∗j )−1 f ) = (1 + O(h/s ))|aj (ρ − (z))|−2 , ((A 0 0 j j j

3/2

j

and since ρj− = ρj (z) + O(sj ) and sj ≤ O(h/sj ) by the upper bound (6.1.43), we can replace ρj− (z) by ρj (z) in the above formula. Here aj denotes the leading symbol in Aj , given in (6.4.17), 1/2

1/2

3/2

p ◦ κj − z = eiθj (γj (z)) aj (ξ + gj ) + sj ,

(6.4.44)

where  sj = O((ρ − ρj (z)∞ ) and ρj (z) ∈ γj still denotes the z-dependent base point, introduced in Remark 6.2.1.

132

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

Combining (6.4.40)–(6.4.43), we get when N = ∅, −1 E+ E−+ E−  = O(h∞ ) + D 1/2 ◦ diag ((E−+ )−1 )   3/2 1 + O(h/sj ) = O(h∞ ) + max . j |a (ρ (z))E j (z)| j j −+ j

(6.4.45)

j

Here we recall that tj := |E−+ (z)| satisfies the j -dependent version of (6.3.47), ⎛

⎞⎞

⎛

tj = ⎝1 + O ⎝

h 3/2

⎠⎠

sj

1  1  Ij 4 h 2 1 j 1 j {qj , q j }(ρ+ ) {q j , qj }(ρ− ) e− h . π 2i 2i

(6.4.46)

From (6.4.44) we see that j

{qj , q j }(ρ± ) =

1

{p, p}(ρ± ) + O(sj∞ ), j

j |aj (ρ± )|2

3/2

j

and again aj (ρ± ) = (1 + O(h/sj )aj (ρj (z)). Then (6.4.46) gives ⎛

⎛

⎞⎞ 1  1  I 4 h h 2 1 j 1 j − hj ⎝ ⎝ ⎠ ⎠ {p, {p, p}(ρ ) p}(ρ ) e . |aj (ρj (z))|tj = 1 + O + − 3/2 π 2i 2i sj (6.4.47) Using this in (6.4.45) gives, when N = ∅, −1 E+ E−+ E−  ⎛

⎛

= max ⎝1 + O ⎝ j ∈N

⎞⎞ h 3/2

sj

⎠⎠

− 1 I π 1  1 4 j 2 j 1 j {p, p}(ρ+ ) {p, p}(ρ− ) eh. h 2i 2i

(6.4.48)

 = E + O(h∞ ) in (6.4.26), we have From (6.4.27) and the fact that E  2 − 14 − 1 −1 2 3 E = O(1) max max sj h , max (h + |sj |) ) . j ∈N

j ∈N c

Combining (6.4.37), (6.4.48), (6.4.49), we get the conclusion in Theorem 6.1.4.

(6.4.49)

6.A Appendix: Quick Review of Almost Holomorphic Functions

6.A

133

Appendix: Quick Review of Almost Holomorphic Functions

Almost analytic functions have been introduced by Hörmander [81], Nirenberg [108] and Dynkin [42]. We prefer the terminology “almost holomorphic functions” to emphasize that these functions are defined in the complex domain. The work [81] has been very important for the author’s works and we refer to the more recent commentary by the author (JS) at the end of the online version of [81], with pointers to more recent works. If ω ⊂ Rn and  ⊂ Cn are open with  ∩ Rn = ω, and if u ∈ C ∞ (ω), there exists a function  u ∈ C ∞ () such that  u|ω = u,

(6.A.1)

∂ z u(z) = O(|z|∞ ), locally uniformly on .

(6.A.2)

Here in the last equation, we write ∂u = ∂ z u = ∂z1  udz1 + · · · + ∂zn  udzn  (∂z1  u, . . . , ∂zn  u) u(z) = O(|z|N ) with ∂zj = (1/2)(∂zj + i∂zj ) and the meaning of the estimate is that ∂ locally uniformly on  for every N ∈ N. We call  u an almost holomorphic (rather than almost analytic) extension of u. The Taylor expansion of  u is uniquely determined by that of u at every real point, so if  u is another almost holomorphic extension of the same function u, then  u − u = O(|z|∞ ) locally uniformly on  (or on the intersection of the domains of the two extensions, when these domains are different). The extension  u can be constructed by means of truncated Taylor expansions and if 0 < r ∈ C(ω), we can choose  u with support in {z ∈ ; |z| < r(z)}. See [81, 102] for further details. In the case n = 1, let γ : [a, b] → C be a smooth injective curve with injective differential, −∞ < a < b < +∞, and let u be a smooth function on γ ( ]a, b[ ). Then we can find a smooth function  u on neigh (γ ( ]a, b[ ), C) such that  u|γ ( ]a,b[ ) = u, ∂ u = O(dist (·, γ ( ]a, b[ ))∞),

(6.A.3)

locally uniformly, on the neighborhood of definition. Again,  u is unique mod ∞ O(dist (·, γ ( ]a, b[ )) ) locally uniformly, and we call  u an almost holomorphic extension. Indeed, if  γ is an almost holomorphic extension of γ , then by the implicit function theorem,  γ is a diffeomorphism: neigh ( ]a, b[ , C) → neigh (γ ( ]a, b[ ), C) and letting γ −1 denote the inverse, we can take   u = u ◦γ ◦ γ −1 ,

(6.A.4)

134

6 Resolvent Estimates Near the Boundary of the Range of the Symbol

where u ◦ γ denotes an almost holomorphic extension of u ◦ γ . This can be generalized to higher dimensions, n ≥ 2, by replacing the range of a curve by a totally real manifold of maximal dimension. (See [102].) Notes The main result of this chapter, Theorem 6.1.4, is a small generalization of a result of Montrieux [18]. In Chap. 10 we will review more general results in all dimensions, which are less precise however. Metaplectic (Fourier integral) operators (in the semi-classical limit) are Fourier integral operators L2 (Rn ) → L2 (Rn ) with quadratic phase functions and constant amplitudes. The phase function is assumed to satisfy standard nondegeneracy assumptions which imply that the operator can be associated to an affine canonical transformation κ : T ∗ Rn → T ∗ Rn . Such an operator U with non-zero amplitude, associated to κ has a bounded inverse U −1 associated to κ −1 . If A = a w (x, hD; h) is the h-Weyl quantization of the symbol a and U is a metaplectic operator as above, then B := U −1 AU is the h-Weyl quantization of the symbol b = a◦κ. This is an exact form of Egorov’s theorem. It suffices that a ∈ S  (T ∗ Rn ). See e.g., [41, Chapter 7, Appendix].

7

The Complex WKB Method

In this chapter we shall study the exponential growth and asymptotic expansions of exact solutions of second-order differential equations in the semi-classical limit. As an application, we establish a Bohr-Sommerfeld quantization condition for Schrödinger operators with real-analytic complex-valued potentials.

7.1

Estimates on an Interval

In this section we derive some basic estimates for differential equations on an interval. Let I = [a, b] be a bounded interval and consider the problem (h∂x − A(x))u(x) = 0, x ∈ I,

(7.1.1)

where A ∈ C ∞ (I ; Mat (n, n)) and we let Mat (n, n) denote the space of complex n × nmatrices. Using the basic result in the theory of linear ODEs about the well-posedness of the Cauchy problem, we can introduce the fundamental matrix E(x, y) ∈ C ∞ (I × I ; Mat (n, n)), determined by (h∂x − A(x))E(x, y) = 0, E(y, y) = 1.

(7.1.2)

Let W (A(x)) denote the numerical range of A(x) as in Definition 2.3.1. Proposition 7.1.1 Let μ+ (A(x)) =

sup λ∈W (A(x))

λ, μ− (A(x)) =

inf

λ∈W (A(x))

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_7

λ

135

136

7 The Complex WKB Method

Then ⎧ ⎨exp( x (μ (A(t))dt/ h), + y E(x, y) ≤ ⎩exp( x (μ− (A(t))dt/ h), y

if x ≥ y, if x ≤ y.

(7.1.3)

Proof If u = u(x) is a solution of (h∂x − A(x))u = 0, we have u(x) = E(x, y)u(y). Moreover, h∂x (u(x)|u(x)) = (A(x)u(x)|u(x)) + (u(x)|A(x)u(x)) = 2(A(x)u(x)|u(x)), so ⎧ ⎨≤ 2μ (A(x))u(x)2 + h∂x u(x)2 ⎩≥ 2μ− (A(x))u(x)2 . Integrating these differential inequalities, we get

2

u(x) ≤

 ⎧ ⎪ ⎪ exp(2 ⎪ ⎨

x

y

⎪ ⎪ ⎪ ⎩exp(2



y

x

μ+ (A(t))dt/ h)u(y)2 ,

if x ≥ y,

μ− (A(t))dt/ h)u(y)2 ,

if x ≤ y,

and (7.1.3) follows, since u(y) can be chosen arbitrarily in Cn .

 

Remark 7.1.2 For x = y we have h∂y E = −h∂x E and hence h∂y E(x, y) + E(x, y)A(y) = 0 when x = y. On the other hand, (h∂x − A(x))(h∂y E(x, y)) = 0, (h∂x − A(x))(E(x, y)A(y)) = 0, so (h∂x − A(x))(h∂y E(x, y) + E(x, y)A(y)) = 0 on I × I. From the uniqueness in the Cauchy problem, we deduce the second differential equation for the fundamental matrix, h∂y E(x, y) + E(x, y)A(y) = 0, x, y ∈ I.

(7.1.4)

7.1 Estimates on an Interval

137

Differentiating (7.1.2), (7.1.4) several times, we see that (h∂x )j (h∂y )k E(x, y) is a linear combination of terms (h∂x )j1 A(x) ◦ · · · ◦ (h∂x )jν A(x) ◦ E(x, y) ◦ (h∂y )k1 A(y) ◦ · · · ◦ (h∂y )kμ A(y), where j , km ≥ 0, ν + j1 + . . . + jν = j, μ + k1 + . . . + kμ = k. It follows that (h∂x )j (h∂y )k E(x, y) ≤ Cj,k × the RHS of (7.1.3). We now assume, in order to fix the ideas, that n = 2. Assume that σ (A(x)) = {λ1 (x), λ2 (x)}, λ1 (x) = λ2 (x), x ∈ I.

(7.1.5)

We then know that A(x) is diagonizable and more precisely that there exists U0 (x) ∈ C ∞ (I ; GL (n)),

(7.1.6)

where GL (n) ⊂ Mat (n, n) is the group of invertible complex n × n matrices, such that U0−1 (x)A(x)U0(x)

  λ1 (x) 0 = =: 0 (x). 0 λ2 (x)

(7.1.7)

Then U0 (x)−1 (h∂x − A(x))U0 (x) = h∂x − 0 (x) + h U0 (x)−1 ∂x (U0 (x)) " #$ % 

=:−B1 (x)

 λ1 (x) + hb11 (x) hb12 (x) = h∂x − . hb21 (x) λ2 (x) + hb22 (x) Naturally, we have the equivalence (h∂x − A(x))(U0(x)u) = 0 ⇔ (h∂x − (0 (x) + hB1 (x)))u = 0.

(7.1.8)

138

7 The Complex WKB Method

If F (x, y; h) is the fundamental matrix for h∂x − (0 (x) + hB1 (x)), then we have the (equivalent) relations, E(x, y; h) = U0 (x)F (x, y; h)U0(y)−1 , F (x, y; h) = U0 (x)−1 E(x, y; h)U0 (y).

(7.1.9)

In addition to (7.1.5), we now assume that λ1 (x) ≥ λ2 (x), x ∈ I.

(7.1.10)

Then, μ+ (0 (x) + hB1 (x)) = λ1 (x) + O(h), μ− (0 (x) + hB1 (x)) = λ2 (x) + O(h), and Proposition 7.1.1 gives   x  ⎧ 1 ⎪ ⎪ exp λ (t)dt + O(|x − y|) , 1 ⎪ ⎨ h y F (x, y; h) ≤    x ⎪ 1 ⎪ ⎪ λ2 (t)dt + O(|x − y|) , ⎩exp h y

if x ≥ y (7.1.11) if x ≤ y

As before, we get (h∂x )j (h∂y )k F (x, y) ≤ Cj,k × the RHS of (7.1.11).

(7.1.12)

Back to h∂x − A(x), we get Theorem 7.1.3 Under the assumptions (7.1.5), (7.1.18), we have  x ⎧ ⎪ −1 ⎪ λ1 (t)dt, ⎪ ⎨exp h y j k (h∂x ) (h∂y ) E(x, y; h) ≤ Cj,k  x ⎪ ⎪ −1 ⎪ λ2 (t)dt, ⎩exp h y

This follows from (7.1.9), (7.1.11), (7.1.12).

if x ≥ y, (7.1.13) if x ≤ y.

7.1 Estimates on an Interval

139

We can eliminate the off-diagonal in (7.1.8) to any order in h by means of   elements c11 c12 ∈ C ∞ (I ; Mat(n, n)) and consider 1+hC(x), additional conjugations. Let C = c21 c22 which is invertible for h small enough, with inverse (1 + hC(x))−1 = 1 − hC(x) + h2 C(x)2 − · · · where the series is convergent but will be viewed as an asymptotic one. Then, (1 + hC(x))−1 (h∂x − (0 (x) + hB1 (x)))(1 + hC(x)) = h∂x − (0 (x) + h(−C(x)0 (x) + 0 (x)C(x) + B1 (x))) + O(h2 ) = h∂x − (0 (x) + h[0 (x), C(x)] + B1 (x)) + O(h2 ), and we see that 

 0 (λ1 − λ2 )c12 [0 , C] = . (λ2 − λ1 )c21 0 1 (x) = U0 (x)(1 + Choose c11 = c22 = 0, cj,k = −bj,k /(λj − λk ), k = j . Then with U hC(x)), we find that 1(x; h) 1 (x; h)−1 (h∂x − A(x))U U   0 λ1 (x) + hb11 (x) = h∂x − + O(h2 ), 0 λ2 (x) + hb22 (x)

(7.1.14)

where the last term has an asymptotic expansion in powers of h. Again we can kill the (2) leading off diagonal entries (of the form h2 bj k (x)) by conjugating with a matrix 1 + h2 D(x), and so on. We get Proposition 7.1.4 Under the assumption (7.1.5), we can find U (x; h) ∼ U0 (x) + hU1 (x) + h2 U2 (x) + · · · ∈ C ∞ (I ; Mat (n, n)) (with the asymptotic sum in the U0 (x)−1 ∈ C ∞ (I ; Mat (n, n)), such that

sense

of

(4.1.3),

U (x; h)−1 (h∂x − A(x))U (x; h) = h∂x − (x; h),

(7.1.15) (4.1.4))

with

(7.1.16)

140

7 The Complex WKB Method

where (x; h) ∼ 0 (x) + h1 (x) + h2 2 (x) + · · · in C ∞ (I ; Mat (n, n)), and each matrix j is diagonal, so   λ1 (x; h) r1,2 (x; h) , rj,k (x; h) ∼ 0 (x; h) = r2,1 (x; h)  λ2 (x; h) 

(7.1.17)

 λj (x; h) ∼ λj (x) + hλj,1 (x) + h2 λj,2 (x) + · · · . Using this result it is easy to find formal asymptotic solutions. The discussion in this section can be applied to the scalar Schrödinger equation (−(h∂x )2 + V (x))v = 0

(7.1.18)

if the potential V is smooth on I . Indeed, introducing 

   u0 (x) v(x) u= = , u1 (x) h∂x v(x) we see that (7.1.18) is equivalent to 

 0 1 (h∂x − A(x))u = 0, where A(x) = . V 0

(7.1.19)

The eigenvalues of A are ±V (x)1/2 and the condition (7.1.5) is equivalent to the fact that V (x) = 0 for all x ∈ I or, in other words that there is no turning point in I at energy 0. (In general, we say that x0 ∈ I is a turning point of V at energy E, if V (x0 ) − E = 0. This turning point is simple if in addition V  (x0 ) = 0.)

7.2

The Schrödinger Equation in the Complex Domain

Let   C be open and simply connected. Let A(z) ∈ Hol (; Mat (2, 2)). As in the case of an interval, the very basic result is that the Cauchy problem is well-posed: Let w ∈ O and let u0 ∈ C2 . Then there is a unique holomorphic solution, u = u(z) ∈ Hol (; C2 ), of the problem (h∂z − A(z))u(z) = 0 in , u(w) = u0 ,

(7.2.1)

7.2 The Schrödinger Equation in the Complex Domain

141

which can be written u(z) = E(z, w)u0 ,

(7.2.2)

where E(z, w) is the fundamental matrix. As in (7.1.5), we now assume that A(z) has distinct eigenvalues: σ (A(z)) = {λ1 (z), λ2 (z)}, where λ1 (z) = λ2 (z), ∀z ∈ .

(7.2.3)

Let E1 (z), E2 (z) ∈ C2 be the corresponding one-dimensional eigenspaces that depend holomorphically on z. Locally, we can find non-vanishing holomorphic sections ej (z) ∈ Ej (z). The choice can be made global if we impose that ∂z e1 (z) ∈ E2 (z), ∂z e2 (z) ∈ E1 (z) everywhere. In fact, this leads to simple differential equations that have global holomorphic solutions: Choose local holomorphic sections ej0 (z) ∈ Ej (z). Then ∂z e10 (z) = a1 (z)e10 (z) + a2 (z)e20 (z) for some holomorphic coefficients. If we put e1 (z) = u1 (z)e10 (z), then the condition that ∂z e1 (z) ∈ E2 (z) is equivalent to the differential equation ∂z u1 + a1 (z)u1 = 0, which locally has a unique non-vanishing solution, if we prescribe u1 (z0 ) in C \ {0} at some point z0 . Since  is simply connected, this leads to a unique non-vanishing holomorphic section e1 in E1 over . The same works for e2 of course. With such a global choice, we let U0 (z) be the invertible matrix with e1 (z) and e2 (z) as the two columns. Then as in (7.1.7) we have   (z) 0 λ 1 =: 0 (z), z ∈ . U0−1 (z)A(z)U0 (z) = 0 λ2 (z)

(7.2.4)

From this we obtain the following analogue of Proposition 7.1.4 Proposition 7.2.1 Under the assumption (7.2.3), we can find U (x; h) ∼ U0 (x) + hU1 (x) + h2 U2 (x) + · · · ∈ Hol (; Mat (n, n))

(7.2.5)

with U0 (x)−1 ∈ Hol (; Mat (n, n)), such that U (z; h)−1 (h∂z − A(z))U (z; h) = h∂z − (z; h), where (z; h) ∼ 0 (z) + h1 (z) + h2 2 (z) + · · · in C ∞ (I ; Mat (n, n)),

(7.2.6)

142

7 The Complex WKB Method

and each matrix j is diagonal, so   λ1 (z; h) r1,2 (z; h) , rj,k (z; h) ∼ 0 (z; h) = λ2 (z; h) r2,1 (z; h)  

(7.2.7)

 λj (z; h) ∼ λj (z) + hλj,1 (z) + h2 λj,2 (z) + · · · . Strictly speaking, for every K  , the inverse of U (x; h) exists for x ∈ K, 0 < h ≤ h(K) for some h(K) > 0 small enough. Corollary 7.2.2 Let φj (z) be holomorphic in  with φj (z) = λj (z). Then there exits a(z; h) ∼ a0 (z) + ha1(z) + · · · in Hol () such that 0 = a0 (z) ∈ N (A(z) − λj (z)) for all z ∈ , and (h∂z − A(z))(a(z; h)eφj (z)/ h ) = r(z; h)eφj (z)/ h , r ∼ 0.

(7.2.8)

j (z; h) ∼ φj (z)+hφj,1 (z)+· · · be a holomorphic primitive of  Proof Let φ λj (z; h). Then  φ (z;h)/ h φ (z)/ h j j e = aj (z; h)e , aj (z; h) = 1 + h aj,1 (z) + · · · , and if ν1 , ν2 is the canonical basis in C2 , we have locally uniformly, 

(h∂z − )(eφj (z)/ h νj ) = O(h∞ )eφj (z)/ h . It then suffices to define a (depending also on j ) by 

a(z; h)eφj (z)/ h = U (z; h)(eφj (z;h)/ hνj ).   By examining directly the equations for a0 and a1 that follow from (7.2.8), we see that a0 (z) = Const ej (z) respectively when j = 1, 2, where ej (z) are the non-vanishing sections of N (A(z) − λj ) that we constructed prior to (7.2.4). Let γ : [a, b]  t → γ (t) ∈  be a smooth curve with γ˙ (t) = 0. If we restrict the equation (h∂z − A(z))u = 0 to γ , we get (h∂t − γ˙ (t)A(γ (t)))u(γ (t)) = 0,

(7.2.9)

from which we deduce the general estimate for the fundamental matrix: E(γ (t), γ (s); h) ≤ O(1) exp for a ≤ s ≤ t ≤ b.

  t 1 max ((γ˙ (τ )λj (γ (τ )))dτ , h s j =1,2

(7.2.10)

7.2 The Schrödinger Equation in the Complex Domain

143

Now assume that (γ˙ (t)λ1 (γ (t))) ≥ (γ˙ (t)λ2 (γ (t))), a ≤ t ≤ b.

(7.2.11)

Then the integral in the exponent in (7.2.10) simplifies to 

t



t

(γ˙ (τ )λ1 (γ (τ )))dτ = 

s

s

d (φ1 (γ (τ )))dτ = (φ1 (γ (t)) − φ1 (γ (s))), dτ

and (7.2.10) becomes, 

1 E(γ (t), γ (s); h) ≤ O(1) exp (φ1 (γ (t)) − φ1 (γ (s))) . h

(7.2.12)

Similarly (still with s ≤ t) 

1 E(γ (s), γ (t); h) ≤ O(1) exp (φ2 (γ (s)) − φ2 (γ (t))) . h

(7.2.13)

Theorem 7.2.3 Under the assumption (7.2.11), let uWKB (z; h) = aWKB (z; h)eφj (z)/ h be an asymptotic solution of (h∂z −A(z))uWKB(z; h) ≈ 0 as in Corollary 7.2.2. Let u(z; h) be the exact solution of (h∂z − A(z))u = 0 in neigh ([a, b], C) such that u(γ (a)) = uWKB (γ (a)) when j = 1, and u(γ (b)) = uWKB (γ (b)) when j = 2. Then u(z; h) − uWKB (z; h) = O(h∞ )eφj (z)/ h with all its derivatives on γ ([a, b]). If we strengthen the assumption (7.2.11) to γ˙ (t)λ1 (γ (t)) > γ˙ (t)λ2 (γ (t)), a ≤ t ≤ b,

(7.2.14)

then u(z; h) − uWKB (z; h) = O(h∞ )eφj (z)/ h in neigh (γ (]a, b]), ) and neigh (γ ([a, b[), ) for j equal to 1 and 2, respectively. Proof The cases j = 1 and j = 2 are basically equivalent and we choose j = 1 in order to fix the ideas. Let u(z; h) be the unique exact solution such that u(γ (a)) = uWKB (γ (a)). Recall that (h∂z − A)uWKB = r(z; h)eφ1 (z)/ h , r ∼ 0,

144

7 The Complex WKB Method

so that  u(z) − uWKB (z) = −

z

E(z, w; h)r(w; h)eφ1 (w)/ h dw

γ (a)

(line integral). If z = γ (t) and we integrate along γ and use (7.2.12), we get the desired conclusion under the assumption (7.2.11). Under the stronger assumption (7.2.14), it suffices to take a smooth family of curves γs : [a, b + ] → , s ∈ neigh (0, R), starting at γ (a), with γ0 |[a,b] = γ so that the   images of the γs fill up a neighborhood of γ ( ]a, b] ). Remark 7.2.4 Assume (7.2.14) and normalize the choice of φ1 , φ2 so that φ1 (γ (a)) = φ2 (γ (a)). Let uWKB (z; h) = aWKB (z; h)eφ1 (z)/ h + bWKB (z; h)eφ2 (z)/ h be the sum of two asymptotic null solutions as in Corollary 7.2.2. Then we have the same conclusion as in Theorem 7.2.3, namely that the exact solution u with the “initial condition” u(γ (a)) = uWKB (γ (a)) satisfies u(z; h) − uWKB (z; h) = O(h∞ )eφ1 (z)/ h in neigh (γ ( ]a, b]), ). Further, notice that uWKB (z; h) = aWKB (z; h)eφ1 (z)/ h + O(h∞ )eφ1 (z)/ h in neigh (γ (]a, b]), ). Now consider the scalar Schrödinger equation (−(h∂)2 + V (z))u = 0,

(7.2.15)

where V is holomorphic in the open simply connected domain . Writing   u  u= h∂u we get the equivalent first-order system (h∂ − A) u = 0,

(7.2.16)

where 

 0 1 A(z) = . V 0

(7.2.17)

7.2 The Schrödinger Equation in the Complex Domain

145

The eigenvalues of A(z) are ±V (z)1/2 , so (7.2.3) is equivalent to the assumption that V (z) = 0 everywhere in , i.e., that there are no turning points in . The earlier discussion can be applied with φ1 = V (z)1/2, φ2 = −V (z)1/2 after fixing a holomorphic branch of the square root of V (z). Notice that this gives an alternative approach to the construction of asymptotic solutions to (7.2.15). Definition 7.2.5 A Stokes line is a curve along which φ is constant. An anti-Stokes line is a curve along which φ is constant. Here φ = φ1 or φ2 . Study Near a Simple Turning Point Let z0 ∈  be a simple turning point (at energy 0) in the sense that V (z0 ) = 0, V  (z0 ) = 0.

(7.2.18)

In order to simplify the notation, assume that z0 = 0. Consider the eikonal equation 1

φ  (z) = V (z) 2

(7.2.19)

in a neighborhood of 0. Clearly, φ(z) will have to be multivalued and in order to understand this better, we pass to the double covering of a pointed neighborhood of 0, by putting z = w2 . Then 1 ∂ ∂ = , ∂z 2w ∂w (w) = V (z) = F (z)z = F (w2 )w2 , φ(z) =  and if we put V φ(w), where F (0) = 0, the eikonal equation becomes φ = F (w2 ) 2 2w2 , ∂w  1

and the right-hand side is an even holomorphic function of w. If we also require that φ(0) =  φ (0) = 0, we see that  φ (w) is an odd holomorphic function of the form 1 1 2 2 3  (0) = F (0) 2 = V  (0) 2 . φ(w) = F (w )w , where F 3

In the original coordinates, we get the double-valued solution φ(z) =

3 2 F (z)z 2 . 3

(7.2.20)

146

7 The Complex WKB Method

Now look for Stokes and anti-Stokes lines that reach 0. On such curves, we have φ = (z)2 z3 = 0. In other words F (z)2 z3 = t 3 for some 0 or φ = 0, i.e., φ 2 = 0: F t ∈ neigh (0, R), and taking the cubic root we get three curves γk , (z) 3 z = e2πik/3t, k ∈ {0, 1, 2}  Z/3Z. F 2

We get the following picture, where we have taken V  (0) > 0 for simplicity. Each curve γk splits into a Stokes curve γk− ending at the turning point and including that point by convention, and an anti-Stokes curve γk+ , which does not include the turning point. Restricting the attention to a small suitably shaped neighborhood W of 0, the three Stokes curves delimit three closed Stokes “sectors” k in that neighborhood. In Fig. 7.1 we also draw some Stokes curves inside each sector. ◦ Let φk be the branch of φ in neigh (0) \ γk− such that φk < 0 in k (and such that φk (0) = 0 since φk is odd). Notice that φk+1 and φk are both well defined in k ∪ k+1 and satisfy φk+1 = −φk there. Let W be a sufficiently small open disc centered at 0. Then (as can be seen by working in the coordinate z3/2 ) each level set {z ∈ W ; φj (z) = const. = 0} is connected and ◦

hence equal to a Stokes line. Let Aj ∈ j , and notice that every point in {z ∈ W \ γj− ; φj (z) > φj (Aj )}

(7.2.21)

can be reached by a curve in W \ γj− , starting at Aj and ending at z, along which φj is strictly increasing. By Theorem 7.2.3, it is clear that we have a holomorphic solution to (−(h∂)2 + V )uj = 0 in W , such that ⎧ ⎨u (z; h) = a (z; h)eφj (z)/ h , j j ⎩aj (z; h) ∼ aj,0 (z) + haj,1 (z) + · · ·

Fig. 7.1 Stokes sectors and Stokes lines near a simple turning point

in the set (7.2.21).

7.2 The Schrödinger Equation in the Complex Domain

147

Here, as in Example 4.1.3 (which extends to the complex case), aj,0 (z) is unique up to a constant factor and we can choose aj,0 (z) = (φj (z))− 2 , 1

where we have not chosen any preferred sign. Further, by Remark 7.2.4, we may arrange so that uj (Aj ; h) = 0. In the following we replace the disc W by W\

1 

{z ∈ j ; φj (z) ≤ φj (Aj )}

−1

and decrease j accordingly. Recall that if u, v are solutions to our homogeneous Schrödinger equation, then the Wronskian Wr (u, v) = (h∂u)v − uh∂v is constant. Applying the asymptotics of u0 and u1 at some point in the interior of 0 ∪ 1 , we see that Wr (u0 , u1 ) has an asymptotic expansion in powers of h: Wr (u0 , u1 ) = 2a0,0a1,0 ∂φ0 + O(h) φ = 2 ( 0 + O(h) φ0 φ1 ( φ0 = 2 ( + O(h). φ1 Similarly, (

φ1 Wr (u1 , u−1 ) = 2 ( + O(h),  φ−1 (  φ−1 Wr (u−1 , u0 ) = 2 ( + O(h). φ0

148

7 The Complex WKB Method

This can be further determined in the following way: Let us fix a branch of (φj )1/2 as above for j = 0, 1, −1 mod 4Z. Then for any two different Stokes sectors, j = k, we have in the interior of j ∪ k that (φj )1/2 = i νj,k (φk )1/2 ,

(7.2.22)

where νj,k ∈ Z/4Z is odd and νj,k = −νk,j . Starting in 0 we make a tour around 0 in the positive direction and write (φ1 )1/2 = i ν1,0 (φ0 )1/2 ,  (φ−1 )1/2 = i ν−1,1 (φ1 )1/2 ,  )1/2 . (φ0 )1/2 = i ν0,−1 (φ−1

This means that if we follow the continuous branch of (φ0 )1/2 around 0 in the positive sense, then after one tour, we get the branch i −(ν0,−1 +ν−1,1 +ν1,0 ) (φ0 )1/2 . But (φ0 )1/2 = V 1/4 for a suitable branch of the fourth root and following this function around 0 once in the positive sense, we get iV 1/4 . Hence we get the cocycle condition ν0,−1 + ν−1,1 + ν1,0 ≡ 1 mod 4Z.

(7.2.23)

We can now specify the signs in the computations of the Wronskians above: ( φj Wr (uj , uk ) = 2 ( + O(h) = 2i νj,k + O(h). φk

(7.2.24)

The space of null solutions is of dimension 2 and any two of u−1 , u0 , u1 are linearly independent, so we have a relation α−1 u−1 + α0 u0 + α1 u1 = 0,

(7.2.25)

where the vector (α−1 , α0 , α1 )t ∈ C3 \ {0} is well defined up to a scalar factor. Applying Wr (uj , ·) to this relation, we get ⎛ ⎞ α−1 ⎜ ⎟ (Wr (uj , uk ))j,k ⎝ α0 ⎠ = 0, α1

(7.2.26)

7.2 The Schrödinger Equation in the Complex Domain

149

or more explicitly, ⎛

⎞⎛ ⎞ 0 a b α−1 ⎜ ⎟⎜ ⎟ ⎝−a 0 c ⎠ ⎝ α0 ⎠ = 0. −b −c 0 α1

(7.2.27)

⎛ ⎞ ⎛ ⎞ c α−1 ⎜ ⎟ ⎜ ⎟ ⎝ α0 ⎠ = ⎝−b⎠ . a α1

(7.2.28)

We can take

Using (7.2.24) we can specify the values of a, b, c and of α−1 , α0 , α1 : a = Wr (u−1 , u0 ) = 2i ν−1,0 + O(h), b = Wr (u−1 , u1 ) = 2i ν−1,1 + O(h),

(7.2.29)

c = Wr (u0 , u1 ) = 2i ν0,1 + O(h), which gives a choice ⎛

⎞ ⎛ ⎞ ⎛ ⎞ α−1 i ν0,1 i ν0,1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ α0 ⎠ = ⎝−i ν−1,1 ⎠ + O(h) = ⎝i ν1,−1 ⎠ + O(h). α1 i ν−1,0 i ν−1,0

(7.2.30)

Bohr-Sommerfeld Quantization Condition for a Potential Well Let V0 be a real-valued and analytic function on neigh ([A, B], R), where −∞ < A < B < +∞. Let E0 ∈ R and assume that there exist A < α0 < β0 < B such that ⎧ ⎨> 0, on [A, α [ ∪ ]β , B], 0 0 V0 − E0 ⎩< 0, on ]α0 , β0 [ .

(7.2.31)

Also assume that α0 , β0 are simple turning points for V0 (x) − E0 : V0 (α0 ) < 0, V0 (β0 ) > 0.

(7.2.32)

Then the situation is stable under small perturbations of the real energy E: For E ∈ neigh (E0 , R) we have simple turning points α(E) < β(E) in ]A, B[ such that V0 −E > 0 on [A, α(E)[ ∪ ]β(E), B] and V0 −E < 0 on ]α(E), β(E)[ . In Fig. 7.2 we draw the Stokes lines of V0 − E in a complex neighborhood of [A, B] that have α or β as an end point.

150

7 The Complex WKB Method

Fig. 7.2 A real potential well with nondegenerate turning points at the ends

Fig. 7.3 The fate of a real potential well after a complex perturbation of the potential and/or the energy

In particular, the real interval between α and β is a Stokes line which connects the two turning points. Let   C be a complex neighborhood of [A, B] to which V0 extends holomorphically. Let V (x) = V0 (x) + W (x), where W is holomorphic in  and |W (x)| < , x ∈ .

(7.2.33)

If > 0 is small enough and E belongs to a small complex neighborhood of E0 , we still have two simple (in general complex) turning points α = α(V , E) and β = β(V , E) close to α0 and β0 , respectively, and in general α and β will not be connected by a Stokes line. The drawing in Fig. 7.3 indicates the three Stokes sectors k near α and the three Stokes sectors Sk near β for k = −1, 0, 1. Note that A ∈ 0 , B ∈ S0 . For each k we have an exact solution uj which is of the form uj = aj (z; h)eφj (z)/ h

◦

in j

with φj (α) = 0 and uj subdominant (i.e., |uj | |uk | for all k = j ) in the interior of j . Similarly, we have the exact solutions vj associated to the sectors Sj of the form vj = bj (z; h)eψj (z)/ h

◦

in S j ,

subdominant in the interior of Sj , ψj (β) = 0. When dealing with vj we think of −z as the new independent variable, rather than z (giving a new Schrödinger operator). Consequently, the leading term in bj becomes bj,0 = (−ψj )−1/4 . In analogy with (7.2.22), we have (−ψj )1/2 = i νj,k (−ψk )1/2 , where we choose the same νj,k .

(7.2.34)

7.2 The Schrödinger Equation in the Complex Domain

151

uj satisfy (7.2.25) with αj as in (7.2.30). vj satisfy the analogous relations with coefficients, that we denote by βj and we have βj = αj + O(h). We may arrange so that u0 (A) = 0, v0 (B) = 0. Also we may arrange so that uj , vj , aj , bj depend holomorphically on E. Now consider the Dirichlet problem (−(h∂)2 + V − E)u = 0, u(A) = u(B) = 0.

(7.2.35)

In other words, we are looking for the spectrum in neigh (E0 , C) of the unbounded operator P = −(h∂x )2 + V : L2 ( ]A, B[ ) → L2 ( ]A, B[ ), with domain D(P ) = {u ∈ Hh2 ( ]A, B[ ); u(A) = u(B) = 0}. It is well known that the spectrum of P is purely discrete and we see that E ∈ σ (P ) ⇐⇒ Wr (u0 , v0 ) = 0.

(7.2.36)

In the construction of the subdominant solutions, we may arrange so that uj = fj (h)v−j for j = ±1.

(7.2.37)

In order to determine the asymptotics of fj , we compare the asymptotic expansions for uj and v−j in j ∩ S−j . x We first look at the exponential factors. In 0 , we have φ0 (x) = α (V (y) − E)1/2 dy with the continuous branch of the square root which is positive for x < α when V , E are real. Thus, 

x

φj (x) = −

(V − E)1/2 dy, j = ±1,

α

where the branch of the square root is given by the continuous extension of the one in 0 to the adjacent sectors (and with a cut along the curve that separates 1 and −1 ). When V , E are real, we get for j = ±1, 

x

φj (x) = −j i

(E − V )1/2 dy, α < x < β,

(7.2.38)

α

where (E − V )1/2 denotes the natural branch of the square root (> 0 on ]α, β[ when V , E are real).

152

7 The Complex WKB Method

Similarly, ψ0 (x) = −

x β

(V (y) − E)1/2 dy in S0 with the natural branch so 

x

ψj (x) =

(V − E)1/2dy in Sj , j = ±1,

β

with the continuous branch, having a cut along the curve separating S1 from S−1 . Hence with the same branch as in (7.2.38), 

x

ψj (x) = j i

(E − V )1/2 dy.

(7.2.39)

β

It follows that  φ1 = ψ−1 − i

β α

 φ−1 = ψ1 + i

(E − V )1/2 dy,

β

(7.2.40) (E − V )

1/2

dy.

α

Next compare the leading amplitudes. From the eikonal equations, we know that φ0 = (V − E)1/2 , −ψ0 = (V − E)1/2 in 0 , S0 respectively, with the natural branches of the square root. Hence, (φ0 )1/2 = (V − E)1/4 , (−ψ0 )1/2 = (V − E)1/4 with the natural “positive” branches of the square and quartic roots. It follows that  1/2 (φ±1 ) = i ν±1,0 (V − E)1/4 = i ν±1,0 e±iπ/4 (E − V )1/4

in ±1 ,

where (E − V )1/4 denotes the branch which is > 0 on ]α, β[ when V , E are real. Thus for j = ±1, a±1,0 = i −ν±1,0 e∓iπ/4 (E − V )−1/4

in ±1 .

(7.2.41)

Similarly, b±1,0 = i −ν±1,0 e∓iπ/4 (E − V )−1/4 in S±1 .

(7.2.42)

7.2 The Schrödinger Equation in the Complex Domain

153

It follows that    i β 1/2 f1 (h) = g1 exp − (E − V ) dy , h α   β  i 1/2 f−1 (h) = g−1 exp (E − V ) dy , h α

(7.2.43)

where, g1 = a1 /b−1 , g−1 = a−1 /b1 have complete asymptotic expansions in powers of h, with leading terms a1,0 = i −ν1,0 +ν−1,0 e−iπ/4−iπ/4 = i −ν1,0 +ν−1,0 −1 , b−1,0 a−1,0 = = i −ν−1,0 +ν1,0 eiπ/4+iπ/4 = i ν1,0 −ν−1,0 +1 . b1,0

g1,0 = g−1,0

(7.2.44)

From (7.2.25), (7.2.30) and the analogous relations for vj with coefficients βj = αj + O(h), we get 1 v1 + β −1 v−1 , α1 u1 +  α−1 u−1 , v0 = β u0 = 

(7.2.45)

j = −βj /β0 ,  j + O(h) = ±1 + O(h).  αj = −αj /α0 , β αj = β

(7.2.46)

From this and (7.2.37), we get 1 Wr (u1 , v1 ) +  −1 Wr (u−1 , v−1 ) Wr (u0 , v0 ) =  α1 β α−1 β 1 f1 Wr (v−1 , v1 ) +  −1 f−1 Wr (v1 , v−1 ) = α1 β α−1 β 1 f1 −  −1 f−1 )Wr (v−1 , v1 ). = ( α1 β α−1 β j = 1 + O(h) has a complete asymptotic expansion in powers of h (as well Here  αj β as similar quantities below) and Wr (v−1 , v1 ) = 2i ν−1,1 + O(h) = 0 by the analogue of (7.2.29). Using also (7.2.43), (7.2.44), we get   i β 1/2 Wr (u0 , v0 ) = 2i ν−1,1 (1 + O(h))i −ν1,0 +ν−1,0 −1 e− h α (E−V ) dy − (1 + O(h))i ν1,0 −ν−1,0 +1 e h i

β α

(E−V )1/2 dy

 .

Here ν1,0 − ν−1,0 + 1 is odd, β

Wr (u0 , v0 ) = 2(1 + O(h))i ν−1,1 +ν1,0 −ν−1,0 +1 e− h α (E−V ) dy  2i  β  1/2 −e h α (E−V ) dy (1 + O(h)) − 1 , i

1/2

154

7 The Complex WKB Method

and using (7.2.23) we get Wr (u0 , v0 ) = −2(1 + O(h))e− h i

β α

(E−V )1/2 dy



 i e h (I (E;h)−πh) − 1 ,

(7.2.47)

where  I (E; h) = I0 (E) + O(h2 ), I0 (E) := 2

β(E)

(E − V (y))1/2dy.

(7.2.48)

α(E)

We have  I0 (E) =

γ 

ζ dz,

γ+ and  γ− , where where  γ ⊂ p−1 (E) is the closed curve which is the concatenation of  ⎧ ⎨ γ+ (t) = ((1 − t)α + tβ, (E − V ((1 − t)α + tβ))1/2), ⎩ γ− (t) = ((1 − t)β + tα, −(E − V ((1 − t)α + tβ)))1/2),

t ∈ [0, 1[ .

When V and E are real,  γ is the real energy curve p = E with the orientation of the Hamiltonian field, and from the Stokes formula it follows that I (E) = volR×R p−1 ( ] − ∞, E]). It is classical (and rather easy to show) that ∂E I (E) = T (E) > 0 is the primitive period of the real energy curve p−1 (E) as a closed Hp -trajectory. In the non-real case, one can still give a sense to and establish the same formula, where now T (E) > 0 and |T (E)| 1. From (7.2.47) we see that the eigenvalues of P near E0 are given by 1

e ih (I (E;h)−πh) = 1,

(7.2.49)

or equivalently by the Bohr-Sommerfeld quantization condition 1 I (E; h) = 2π(k + )h, k ∈ Z. 2

(7.2.50)

From the construction it follows that I (E; h) is a holomorphic function of E in a fixed neighborhood of E0 in C and has a complete asymptotic expansion in powers of h in

7.2 The Schrödinger Equation in the Complex Domain

155

the space of such functions with the two leading terms given in (7.2.48). Moreover, ∂E I (E; h) = T (E) + O(h2 ) = 0, so I (·; h) is biholomorphic from neigh (E0 , C) onto neigh (I (E0 ), C) and (7.2.50) gives a sequence of eigenvalues         1 1 Ek = I −1 2π k + h; h = I0−1 2π k + h + O(h2 ), 2 2 all situated on the curve given by the condition that I (E; h) ∈ R, which is an O(h2 ) deformation of the curve I0 (E) ∈ R. Notice that the reality of I0 (E) is equivalent to the condition that α(E) and β(E) are connected by a Stokes line. It is not hard to show that the eigenvalues Ek are simple, either by deformation of P to the harmonic oscillator, or by appealing to general facts about Grushin problems. The latter argument has been carried out in [101] (see also [23]). Under suitable assumptions on the behaviour of V near ±∞, the spectrum of −(h∂)2 +V as a closed operator in L2 (R) is still given by a Bohr-Sommerfeld quantization condition [23], and for the proof we just need to complement the WKB-arguments above with a completely analogous study near infinity. (See [101] for details.) Notes It is from André Voros’ lecture notes [157] that we have learnt the basic principle of complex WKB analysis, namely to follow the solutions in the directions of growth. See also the monograph of Fedoryuk [44]. There is a large literature on the complex WKB-method, often with lots of sophisticated and deep considerations (like resurgence), far beyond the scope of this book. Also, there are many spectral results for operators in one dimension based on the complex WKB method. See for instance Redparth [115], Shkalikov [127, 128], Nekhaev and Shafarevich [107], and in the special case of PT -symmetry, [23, 101].

8

Review of Classical Non-self-adjoint Spectral Theory

The first section of this chapter deals with Fredholm theory in the spirit of Appendix A in [60], see also an appendix in [105] and [148]. The remaining sections give a brief account of the very beautiful classical theory of non-self-adjoint operators, taken from a section in [136] which is a brief account of parts of the classical book by Gohberg and Krein [48].

8.1

Fredholm Theory Via Grushin Problems

Most of this section follows Appendix A in [60] quite closely. For simplicity we only consider the case of (separable) Hilbert spaces. Let H1 , H2 be two such spaces. Recall that a bounded operator P : H1 → H2 (P ∈ L(H1 , H2 )) is a Fredholm operator if the null space N (P ) is of finite dimension and the range R(P ) is closed and of finite codimension, where codim R(P ) := dim R(P )⊥ . Equivalently, P is (a) Fredholm (operator) if there exists Q ∈ L(H2 , H1 ), such that P Q = 1 + R, QP = 1 + L, where L : H1 → H1 , R : H2 → H2 are compact. If P is Fredholm, we define its index by ind P = dim N (P ) − codim R(P ).

(8.1.1)

Let  ⊂ C be an open connected set (or an open interval in R) and let   z → Pz ∈ L(H1 , H2 )

(8.1.2)

be a continuous family (i.e., continous for the operator norm; uniformly continuous).

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_8

157

158

8 Review of Classical Non-self-adjoint Spectral Theory

Proposition 8.1.1 If Pz0 is Fredholm for some z0 , then there is a neighborhood V ⊂  of z0 such that Pz is Fredholm,

(8.1.3)

ind Pz = ind Pz0 ,

(8.1.4)

for every z ∈ V . Proof Let n0+ = dim N (Pz0 ), n0− = codim R(Pz0 ) and define 0

0

0 0 : H1 → Cn+ , R− : Cn− → H2 , R+

by 0

0 R+ u(j )

= (u|ej ), j =

R− u− =

1, 2, .., n0+ ,

n− 

u− (j )fj ,

1

where e1 , . . . , en0 and f1 , . . . , fn0 are orthonormal bases for N (Pz0 ) and R(Pz0 )⊥ , + − respectively. Put 

0 Pz R− Pz = 0 0 R+

 0

0

: H1 × Cn− → H2 × Cn+ .

(8.1.5)

Using the orthogonal decompositions H1 = N (Pz0 )⊥ ⊕ N (P ), H2 = R(Pz0 ) ⊕ R(P )⊥ , we see that Pz0 is bijective with bounded inverse. By continuity, Pz has the same property for all z in a neighborhood of z0 and the proposition follows from Proposition 8.1.2 below.   Proposition 8.1.2 Let P ∈ L(H1 , H2 ) and let R+ : H1 → Cn+ , R− : Cn− → H2 be bounded linear operators of maximal ranks n+ , n− ∈ N. If  P R− P= R+ 0

 : H1 × Cn− → H2 × Cn+

8.1 Fredholm Theory Via Grushin Problems

159

is bijective with a bounded inverse, then P is Fredholm of index n+ − n− . (When n+ = 0, then R+ , Cn+ are absent as well as the last line in the matrix for P, and similarly when n− = 0.) Proof Let 

 E E+ E= E− E−+

: H2 × Cn+ → H1 × Cn−

be the inverse of P, so that the system ⎧ ⎨P u + R u = v, − − ⎩R+ u = v+ , has the unique solution ⎧ ⎨u = Ev + E v ∈ H , + + 1 ⎩u− = E− v + E−+ v+ ∈ Cn− , for any given v ∈ H2 , v+ ∈ Cn+ . The equation, P u = v can be written P u + R− 0 = v and if we introduce the unknown v+ = R+ u, we get the equivalent system ⎧ ⎨u = Ev + E v , + + ⎩0 = E− v + E−+ v+ . Thus, R(P ) = {v ∈ H2 ; ∃v+ ∈ Cn+ , E− v = −E−+ v+ },

(8.1.6)

N (P ) = {E+ v+ ; E−+ v+ = 0}.

(8.1.7)

The fact that E = P −1 is equivalent to the following two systems of equations: P E + R− E− = 1, P E+ + R− E−+ = 0, R+ E = 0, R+ E+ = 1,

(8.1.8)

EP + E+ R+ = 1, ER− = 0, E− P + E−+ R+ = 0, E− R− = 1.

(8.1.9)

160

8 Review of Classical Non-self-adjoint Spectral Theory

From the last equation, we see that E− is surjective and (8.1.6) shows that R(P ) is closed and that codim R(P ) = codim R(E−+ ). Further, (8.1.7) shows that N (P ) is finitedimensional and dim N (P ) = dim N (E−+ ). Here we also used the injectivity of E+ , provided by the last equation in (8.1.8). Thus P is a Fredholm operator and ind P = dim N (E−+ ) − codim R(E−+ ) = n+ − n− , where the last equality is a general fact for the index of any n− × n+ -matrix.

 

The following result can be proved by straightforward computations (cf. (8.1.8), (8.1.9)): Proposition 8.1.3 Let P , R+ , R− , P be as in Proposition 8.1.2 and assume that P is bijective with a bounded inverse E as in the beginning of the proof of that result. • If P is bijective, then E−+ is bijective (necessarily P is of index 0, so n+ = n− ) and −1 = −R+ P −1 R− . E−+

(8.1.10)

• If E−+ is bijective, then P is bijective and −1 P −1 = E − E+ E−+ E− .

(8.1.11)

See also Proposition 4.1 in [144] for a characterization of the invertibility of P. We next review analytic Fredholm theory. Assume that the family Pz in (8.1.2) is not only continuous, but also holomorphic (for the operator norm topology) and that Pz is Fredholm for every z ∈ . Then we know that ind Pz is constant and we assume that it is equal to 0. Proposition 8.1.4 Assume in addition that Pw is bijective for some w ∈ . Then

:= {z ∈ ; Pz is not bijective} is discrete. If z0 ∈ , then z → Pz−1 has a pole of order N0 < ∞ at z0 : Pz−1 =

−1  −N0

(z − z0 )j Aj + Q(z),

(8.1.12)

8.1 Fredholm Theory Via Grushin Problems

161

where Q(z) ∈ L(H2 , H1 ) is holomorphic in a neighborhood of z0 and the operators A−N0 , . . . , A−1 ∈ L(H2 , H1 ) are of finite rank. Proof If z1 ∈ , we can define 

Pzz1

z1 Pz R− = z1 R+ 0

 : H1 × Cn0 (z1 ) → H2 × Cn0 (z1 )

z1 with R± = R± independent of z, such that Pzz1 is bijective for z in a connected neighborhood Vz1 of z1 in . Let



E

z1

z1 (z) E z1 (z) E+ = z1 z1 E− (z) E−+ (z)



z1 (z) is a holomorphic function of z ∈ Vz1 with values in the be the inverse, so that E−+ n0 (z1 ) × n0 (z1 ) matrices. Now, ∩ Vz1 coincides with the set of zeros of the holomorphic z1 function Vz1  z → det E−+ (z), which is either a discrete set or equal to Vz1 . Covering  with such Vz1 , we conclude that is either discrete or equal to all of . But the latter possibility is excluded by the assumption that w ∈ for some w ∈ . Now, let z0 ∈ and choose Pz , Ez as above with z1 = z0 . Then

Pz−1 = E(z) − E+ (z)E−+ (z)−1 E− (z), z ∈ Vz0 . Here E−+ (z)−1 has a pole at z0 : E−+ (z)−1 =

R−N0 R−1 + hol (z), + ···+ N 0 (z − z0 ) (z − z0 )

1 ≤ N0 < ∞, rank R−j ≤ n0 := n0 (z0 ). Using that E(z), E± (z) are holomorphic we get (8.1.12), where A−N0 , . . . , A−1 can be expressed in terms of R−N0 , . . . , R−1 and (j ) (j ) E+ (z0 ), E− (z0 ), for 0 ≤ j ≤ N0 .   Let P : H → H be a closed densely defined operator with domain D = D(P ). With  as above, we assume that P − z : D → H is Fredholm of index 0 for z ∈  and bijective for some z0 ∈ .

(8.1.13)

If ρ(P ) denotes the resolvent set of P , we know from the above discussion that ρ(P )∩ =  \ , where = σ (P ) ∩  is discrete.

162

8 Review of Classical Non-self-adjoint Spectral Theory

Proposition 8.1.5 Write the Laurent series in (8.1.12) as (z − P )−1 = (z − z0 )−N0 A−N0 + · · · + (z − z0 )−1 A−1 + Q(z) in L(H, D),

(8.1.14)

where Q is holomorphic near z = z0 and A−j are of finite rank. The operator π−1 := A−1 is a projection which commutes with P . This implies that the finite-dimensional space R(π−1 ) is contained in the domains of all powers P k , k ∈ N, and is invariant under P . The restriction of z0 − P to R(π−1 ) is nilpotent. Indeed, A−j = (P − z0 )j −1 π−1 , 1 ≤ j ≤ N0 ,

(8.1.15)

(P − z0 )N0 π−1 = 0.

(8.1.16)

Let γ = γr = ∂D(z0 , r) be the oriented boundary of the disc D(z0 , r) for 0 < r small enough. Then π−1 =

A−j

1 = 2πi

1 2πi





1

(z − P )−1 dz,

(8.1.17)

(z − z0 )j −1 (z − P )−1 dz.

(8.1.18)

γ

γ

Proof Equations (8.1.17), (8.1.18) are standard formulae for the Laurent coefficients, obtained by multiplying (8.1.14) by (z − z0 )j −1 and then integrating along γ . Knowing that π−1 : H → D, we apply P − z0 to the left in (8.1.17) and integrate: (P − z0 )π−1 =

1 2πi



(P − z0 )(z − P )−1 dz. γ

Here the first P in the integral can be replaced by z, since the corresponding difference of integrals is 1 2πi



(P − z)(z − P )−1 dz, γ

which is zero, since the integrand is holomorphic near z0 . Thus, (P − z0 )π−1 =

1 2πi

 γ

(z − z0 )(z − P )−1 dz = A−2 .

8.1 Fredholm Theory Via Grushin Problems

163

By the same argument, we see that the range of π−1 is contained in the domain of P k for all k ∈ N and we get (8.1.16). Of course, if u ∈ D, then π−1 P = P π−1 , so π1 and P commute. 1 and use the resolvent Let us finally recall why π−1 is a projection. Let 0 < r1 < r2 identity, to get with γj = γrj :   2 π−1 =

dz dw 2πi 2πi   1 dz dw 1 dz dw (z − P )−1 − (w − P )−1 w−z 2πi 2πi 2πi 2πi γ2 γ1 w − z

(w − P )−1 (z − P )−1

γ2

γ1

γ2

γ1

  =

=: I + II. Here,   I= γ1

γ2

1 dw w − z 2πi 

II = −



(z − P )−1

(w − P )−1

dz = 2πi



γ2

γ1



(z − P )−1 γ1

1 dz w − z 2πi



dz = π−1 , 2πi

dw = 0. 2πi

2 =π . Hence π−1 −1

 

One can also follow multiplicities through Grushin reductions, and we refer to [105, 148] and many other papers for such discussions. Under the assumptions of the last proposition, the (algebraic) multiplicity of the eigenvalue z0 of P is by definition, m(P , z0 ) = dim R(π−1 ). Similarly, if   P − z R− : D × Cn0 → H × Cn0 P(z) = R+ 0 is bijective for z ∈ neigh (z0 ), with inverse 

 E(z) E+ (z) E(z) = , E− (z) E−+ (z) we let m(E−+ , z0 ) be the multiplicity of z0 as a zero of z → det(E−+ (z)). Proposition 8.1.6 Under the above assumptions, m(P , z0 ) = m(E−+ , z0 ).

(8.1.19)

164

8 Review of Classical Non-self-adjoint Spectral Theory

Proof Anticipating the treatment of traces later in this chapter, we have  m(P , z0 ) = tr π−1 = tr 

γ

E+ (z)E−+ (z)−1 E− (z)

dz 2πi

  dz = tr E+ (z)E−+ (z)−1 E− (z) 2πi γ    dz = tr E−+ (z)−1 E− (z)E+ (z) 2πi γ since E−+ is a finite-rank operator. From the relation ∂z E(z) = −E(z)∂z P(z)E(z), we get 

 E E+ ∂z E(z) = E− E−+

 10 00



E E+ E− E−+

and hence, ∂z E−+ (z) = E− (z)E+ (z). Thus,  γ

  dz tr E−+ (z)−1 ∂z E−+ (z) 2πi

γ

∂z det E−+ (z) dz det E−+ (z) 2πi

m(P , z0 ) =  =

= m(E−+ , z0 ).  

8.2

Singular Values

In the remainder of this chapter we give a brief account of abstract non-self-adjoint theory, following [48] closely. Let H be a separable complex Hilbert space. If A ∈ L(H, H) is compact, we let s1 (A) ≥ s2 (A) ≥ · · ·  0 be the eigenvalues of the compact self-adjoint operator

8.2 Singular Values

165

(A∗ A)1/2. They are called the singular values of A. We notice that sj (A∗ ) = sj (A). In fact, this follows from the intertwining relations: A(A∗ A) = (AA∗ )A, (A∗ A)A∗ = A∗ (AA∗ ). The singular values appear naturally in the polar decomposition: If A ∈ L(H, H), then Au2 = (Au|Au) = (A∗ Au|u) = ((A∗ A)1/2 u|(A∗ A)1/2 u) = (A∗ A)1/2 u2 . Let e1 , e2 , . . . be an orthonormal family of eigenvectors of (A∗ A)1/2 associated to the eigenvalues s1 (A), s2 (A), . . . that are > 0. It is an orthonormal basis for R((A∗ A)1/2 ), and R((A∗ A)1/2)⊥ = N ((A∗ A)1/2 ) = N (A). The operator U : R((A∗ A)1/2)  (A∗ A)1/2 u → Au ∈ R(A) is isometric and bijective. It extends to a unitary operator, which we also denote by U , from R((A∗ A)1/2) to R(A), and to a partial isometry if we put U = 0 on the orthogonal space N (A). We get the polar decomposition A = U (A∗ A)1/2.

(8.2.1)

This leads to the Schmidt decomposition of A Au =



sj (A)(u|ej )fj ,

(8.2.2)

where fj = U ej is also an orthonormal family. Recall the mini-max characterization of the sj ([48], p. 25): sj (A) =

inf

E⊂H; E is a closed subspace of H of codimension ≤j−1

((A∗ A)1/2u|u) . u2 u∈E\{0} sup

(8.2.3)

From that we get the following characterization of the singular values due to Allahverdiev [48, pp. 28, 29]: Theorem 8.2.1 Let A ∈ L(H, H) be compact. Then sn+1 (A) =

min

K∈L(H,H) K of rank ≤n

A − K,

n = 0, 1, .. .

The minimum is achieved by an operator K for which s1 (K) = s1 (A),. . . ,sn (K) = sn (A), sn+1 (K) = 0, s1 (A − K) = sn+1 (A), s2 (A − K) = sn+2 (A), . . ..

166

8 Review of Classical Non-self-adjoint Spectral Theory

Proof If K is of rank ≤ n, then N (K) is of codimension ≤ n and sn+1 (A) ≤

Au (A − K)u = sup ≤ A − K. u u 0 =u∈N (K) 0 =u∈K sup

To obtain the minimizing operator consider the polar decomposition A = U (A∗ A)1/2 and take K = U (A∗ A)1/2Pn , where Pn is the orthogonal projection onto the space spanned by e1 , . . . , en . Then A − K = U (A∗ A)1/2 (1 − Pn ), (A − K)∗ (A − K) = (1 − Pn )(A∗ A)1/2U ∗ U (A∗ A)1/2 (1 − Pn ) = A∗ A(1 − Pn ), and we get sn+1 (A) = A − K as well as the statement about the singular values of K.   The following corollary is due to Ky Fan. Corollary 8.2.2 Let A, B ∈ L(H) be compact. Then for n, m ≥ 1, sm+n−1 (A + B) ≤ sm (A) + sn (B),

(8.2.4)

sm+n−1 (AB) ≤ sm (A)sn (B).

(8.2.5)

Proof Let KA , KB be operators of rank ≤ m − 1 and ≤ n − 1, respectively, such that sm (A) = A − KA , sn (B) = B − KB . Then sm+n−1 (A + B) ≤ A + B − (KA + KB ) ≤ A − KA  + B − KB  = sm (A) + sn (B). Since (A − KA )(B − KB ) = AB − ((A − KA )KB + KA B) , where the last term in the parentheses is of rank ≤ m + n − 2, we have sm+n−1 (AB) ≤ (A − KA )(B − KB ) ≤ A − KA B − KB | ≤ sm (A)sn (B).  

8.2 Singular Values

167

Corollary 8.2.3 |sn (A) − sn (B)| ≤ A − B. Proof Let K be an operator of rank n − 1. Then sn (A) ≤ A − K = B − K + A − B ≤ B − K + A − B. Varying K, we get sn (A) ≤ sn (B) + A − B, and we have the same inequality with A and B exchanged.   We now discuss Weyl inequalities, and start with the following result of Weyl (see [48, pp. 35, 36]): Theorem 8.2.4 Let A ∈ L(H, H) be compact and let λ1 (A), λ2 (A), . . . be the nonvanishing eigenvalues of A arranged in such a way that |λ1 | ≥ |λ2 | ≥ · · · and repeated according to their multiplicity (which by definition is the rank of the associated spectral projection). Then for every n ≥ 1 for which λn (A) is defined, we have |λ1 (A) · · · λn (A)| ≤ s1 (A) · · · sn (A).

(8.2.6)

Proof For n = 1, (8.2.6) just says that |λ1 (A)| ≤ A. Approaching A by a sequence of finite rank operators, we can assume that A is of finite rank and replace H by the finitedimensional space R(A) + (N (A))⊥ , which we denote by H from now on. (Indeed, λj and sj depend continuously on A.) Introduce the space )n

H = H ∧ ···∧ H

generated by n-fold exterior products of vectors in H. product that satisfies

*n

(8.2.7) H is a Hilbert space with a scalar

(u1 ∧ .. ∧ un |v1 ∧ .. ∧ vn ) = det((uj |vk )), uj , vj ∈ H. Further, there is a linear operator ∧n A : the condition

*n

H→

*n

(8.2.8)

H which is uniquely determined by

(∧n A)(u1 ∧ · · · ∧ un ) = Au1 ∧ · · · ∧ Aun , uj ∈ H.

(8.2.9)

Using a basis of generalized eigenvectors, we see that the eigenvalues of ∧n A are the values λj1 · · · λjn , with jν = jμ , for ν = μ. The eigenvalue of greatest modulus is then λ1 · · · λn . On the other hand, the adjoint of ∧n A is ∧n A∗ . We also have

168

8 Review of Classical Non-self-adjoint Spectral Theory

(∧n A)(∧n B) = ∧n (AB). Then (∧n A)∗ (∧n A) = ∧n (A∗ A) and this operator has the eigenvalues sj21 (A) · · · sjn (A)2 , out of which the largest one is (s1 (A) · · · sn (A))2 =  ∧n A2 ≥ |λ1 |2 · · · |λn |2 .  

The proof is complete. In the same spirit we have the inequality of Horn (see [48, p. 48]): n 

sj (AB) ≤

j =1

n 

(sj (A)sj (B)).

(8.2.10)

j =1

Proof As before it suffices to treat the case when H is of finite dimension. The largest eigenvalue of (∧n AB)∗ (∧n AB) = ∧n ((AB)∗ AB) is equal to (s1 (AB) · · · sn (AB))2 . On the other hand, ((∧n AB)∗ (∧n AB)u|u) = (∧n AB)u2 = (∧n A) ◦ (∧n B)u2 ≤  ∧n A2  ∧n B2 u2 ≤ (s1 (A) · · · sn (A))n (s1 (B) · · · sn (B))2 u2 , and taking the supremum over all normalized u, we obtain the required inequality.

 

We next need a convexity inequality, due to Weyl and Hardy, Littlewood, and Pólya [48, p. 37]. Lemma 8.2.5 Let !(x) be a convex function on R, which tends to 0, when x → −∞. Let a1 ≥ · · · ≥ aN , b1 ≥ · · · ≥ bN be real numbers satisfying k 

aj ≤

j =1

k 

bj , 1 ≤ k ≤ N.

j =1

Then, k  j =1

!(aj ) ≤

k  j =1

!(bj ), 1 ≤ k ≤ N.

8.2 Singular Values

169

Proof Approaching ! by a sequence of smooth functions, we can reduce the proof to the case when ! ∈ C ∞ . Then ! ≥ 0, ! (x) → 0, x → −∞.

(8.2.11)

Letting y → −∞ in the identity 

x

!(x) = !(y) +

! (t)dt,

(8.2.12)

y

we get  !(x) =

x −∞

! (t)dt.

0 From the convergence of the last integral, we conclude that y ! (t)dt ≤ C, y ≤ 0, implying that |y|! (y) is a bounded function for y ≤ 0. Also, |x − y|! (y) ≤ !(x) for y ≤ x ≤ 0, so |y|! (y) ≤ 2!(x) when |y| ≥ 2|x|. Hence |y|! (y) → 0 when y → −∞. Integration by parts in (8.2.12) gives !(x) = !(y) + [(t − x)! (t)]xt=y − = !(y) + (x − y)! (y) +



x



x

(t − x)! (t)dt

y

(x − t)! (t)dt.

y

Letting y tend to −∞, we get  !(x) =

x





−∞

(x − t)! (t)dt =

(x − t)+ ! (t)dt.

Hence k 

 !(aj ) =

⎛ ⎝

k 

⎞ (aj − t)+ ⎠ ! (t)dt.

j =1

1

Let k(t) ≤ k be the largest  k ≤ k with ak ≥ t. Then k(t ) k(t ) k k     (aj − t)+ = (aj − t) ≤ (bj − t) ≤ (bj − t)+ . j =1

Hence

k

1 !(aj )

j =1

≤

k

1 !(bk ).

j =1

j =1

 

170

8 Review of Classical Non-self-adjoint Spectral Theory

As a consequence we get the following result of H. Weyl [48, pp. 39, 40]: Theorem 8.2.6 Let A : H → H be a compact operator, and let f (x) ≥ 0 be a continuous function on [0, ∞[ with f (0) = 0 such that f (et ) is convex. Let λj and sj be the eigenvalues and singular values of A, arranged as |λ1 | ≥ |λ2 | ≥ · · · , s1 ≥ s2 ≥ · · · . Then for every k ≥ 1, k 

f (|λj |) ≤

j =1

k 

f (sj ).

(8.2.13)

j =1

Proof We know from Theorem 8.2.4 that k 

log |λj | ≤

j =1

k 

log sj ,

j =1

and it suffices to apply the preceding convexity lemma.

 

Corollary 8.2.7 For every p > 0, we have n 

|λj (A)|p ≤

j =1

n 

sj (A)p .

j =1

For every r > 0, we have n 

(1 + r|λj (A)|) ≤

j =1

n 

(1 + rsj (A)).

j =1

Let A, B be compact operators. With !(t) = et , aj = log sj (AB), bj log(sj (A)sj (B)), we get from Horn’s inequality (8.2.10) and Lemma 8.2.5: Corollary 8.2.8

n

j =1 sj (AB)

≤

=

n

1 sj (A)sj (B).

Let C∞ ⊂ L(H) be the subspace of compact operators. The following Lemma is due to Ky Fan [48, p. 47]. Lemma 8.2.9 Let A ∈ C∞ . Then for every 1 ≤ n ∈ N, we have n  j =1

sj (A) = max

n  j =1

(U Aφj |φj ),

8.3 Schatten–von Neumann Classes

171

where the maximum is taken over the set of all unitary operators U and all orthonormal systems φ1 , . . . , φn . For the next result, see [48, p. 48]: Corollary 8.2.10 If A, B ∈ S∞ , then n 

sj (A + B) ≤

j =1

8.3

n 

sj (A) +

j =1

n 

sj (B).

j =1

Schatten–von Neumann Classes

We are now ready to discuss the Schatten–von Neumann classes (still following [48]). Definition 8.3.1 For 1 ≤ p ≤ ∞, we put Cp = {A ∈ C∞ ; (sj (A))∞ 1 p < ∞}, where (sj (A))∞ 1 p

⎧ ⎨∞ s (A)p 1/p , if p < ∞, 1 j = ⎩sup sj (A), if p = ∞.

Theorem 8.3.2 Cp is a closed two-sided ideal in L(H, H) equipped with the norm ACp = (sj (A))∞ 1 p . If p1 ≤ p2 , then Cp1 ⊂ Cp2 . The space of finite rank operators is dense in Cp for every p. We will only recall the proof of the fact that  · Cp satisfies the triangle inequality. Let A, B ∈ Cp , and put ξj = sj (A + B), ηj = sj (A) + sj (B). According to Corollary 8.2.10,   we have n1 ξj ≤ n1 ηj , ∀n, and hence A + BC1 ≤ AC1 + BC1 , by letting n tend to ∞. The case p = ∞ follows from the triangle inequality for the ordinary operator norm.

172

8 Review of Classical Non-self-adjoint Spectral Theory

It remains to treat the case 1 < p < ∞. Both sequences ξj and ηj are decreasing. It suffices to show that ξ p ≤ ηp .

(8.3.1)

We have ξ p = sup ξ, ζ , ζ ∈q , ζ q =1

by Hölder’s inequality, where q ∈ [1, +∞[ is the conjugate index, given by p−1 +q −1 = 1 and ·|· denotes the real scalar product on 2 ({1, 2, . . .}). We also know that the supremum p/q is attained by a ζ = ζ 0 of the form ζj0 = (const. > 0)ξj , and in particular ζj0 is a j decreasing sequence. We use partial summation with "j = 1 ξk , "0 = 0: ξ |ζ 0 (n) : =

n 

ξj ζj0 =

n 

j =1

=

n 

("j − "j −1 )ζj0

j =1

"j ζj0 −

j =1

n−1  j =1

"j ζj0+1 = "n ζn0 +

n−1  j =1

"j (ζj0 − ζj0+1 )

⎛ ⎞ j n n−1    =⎝ ξk ⎠ ζn0 + ξk (ζj0 − ζj0+1 ) #$ % " j =1 j =1 k=1 ≥0

The last expression is ≤ the same expression with ξ replaced by η, and running the same calculation backwards, the latter expression is equal to η|ζ 0 (n) . Hence ξ |ζ 0 (n) ≤ η|ζ 0  ≤ ηp . Letting n → ∞, we get (8.3.1), and this completes the proof of the triangle inequality for the Cp -norms.  We notice that A = s1 (A) ≤ ACp . The space C1 is the space of nuclear or traceclass operators, and C2 is the space of Hilbert–Schmidt operators. We have the following Hölder type result: Theorem 8.3.3 Let p, q ∈ [1, ∞] be conjugate indices; p−1 + q −1 = 1. If A ∈ Cp , B ∈ Cq , then AB ∈ C1 and ABC1 ≤ ACp BCq . Proof By Corollary 8.2.8, we know that n  j =1

sj (AB) ≤

n  j =1

sj (A)sj (B),

8.4 Traces and Determinants

173

and letting n tend to ∞, we get from the usual Hölder inequality: ∞  j =1

sj (AB) ≤

∞ 

sj (A)sj (B) ≤ s· (A)p s· (B)q = ACq BCq .

j =1

  Remark 8.3.4 When p = ∞, the assumption A ∈ C∞ can be weakened to A ∈ L(H, H), with AC∞ replaced by AL(H,H) . Idem for B when q = ∞.

8.4

Traces and Determinants

We next discuss the trace of a nuclear operator. If A ∈ L(H, H) is of finite rank, we choose a finite-dimensional subspace H ⊂ H such that N (A)⊥ , R(A) ⊂ H . We then define the trace of A, tr A as the trace tr A|H of the restriction of A to H . We can check that this does not depend on the choice of H and that tr A is the sum of the finitely many non-vanishing eigenvalues of A (each counted with its algebraic multiplicity). We see that A → tr A

(8.4.1)

is a linear functional on the space of finite rank operators. Moreover, by Corollary 8.2.7, |tr A| ≤



|λj (A)| ≤ AC1 .

(8.4.2)

We can then extend (8.4.1) to a continuous linear functional on C1 , and we still have |tr A| ≤ AC1 .

(8.4.3)

In the case of finite rank operators, we also have the cyclicity of the trace tr AB = tr BA.

(8.4.4)

Let now A ∈ Cp , B ∈ Cq , where p, q ∈ [1, ∞] are conjugate indices, and choose Aν , Bν , ν = 1, 2, . . . of finite rank, so that A − Aν Cp → 0, B − Bν Cq → 0. Then AB − Aν Bν C1 = (A − Aν )B + Aν (B − Bν )C1 ≤ A − Aν Cp BCq + Aν Cp B − Bν Cq → 0, ν → ∞.

174

8 Review of Classical Non-self-adjoint Spectral Theory

Using this also for BA and the cyclicity of the trace (8.4.4) for finite rank operators, we obtain (8.4.4) also in the case A ∈ Cp , B ∈ Cq , where p, q ∈ [1, ∞] are conjugate indices. When A ∈ L(H, H) we still have (8.4.4). Indeed, if B is of finite rank, let A1 , A2 , . . . be a sequence of finite-rank operators converging to A weakly. Then, tr (AB) ← tr (Aν B) = tr (BAν ) → tr (BA), ν → ∞. If A is bounded and B ∈ C1 , let Bν ∈ C1 be a sequence of operators converging to B in C1 . Then tr (AB) ← tr (ABν ) = tr (Bν A) → tr (BA), ν → ∞, and the claim follows. Remark 8.4.1 One simple way of extending most of the theory to the case of operators A : H1 → H2 , where H1 , H2 are two different Hilbert spaces, is the following. Consider the corresponding operator 

 0 0 : H1 ⊕ H2 → H1 ⊕ H2 , A0

(8.4.5)

and say that A belongs to Cp if the operator in (8.4.5) does. The cyclicity of the trace still holds in this setting, namely if A : H1 → H2 and B : H2 → H1 belong to Cp and Cq , respectively, where p and q are conjugate indices. We next discuss determinants of trace-class perturbations of the identity operator. Let first A ∈ L(H, H) be of finite rank and chose a finite dimensional Hilbert space as above. Then we define det(1 − A) = det((1 − A)|H ) =

 (1 − λj (A)),

(8.4.6)

j

where λj (A) denote the non-zero eigenvalues, repeated according to their multiplicity. We remark that | det(1 − A)| ≤

   (1 + |λj (A)|) ≤ (1 + sj (A)) ≤ e j sj (A) , j

j

where the second inequality follows from Corollary 8.2.7. We want to extend the definition to the case when A ∈ C1 . Let first I be a compact interval and let I  t → At be a C 1 family of finite-rank operators, with

8.4 Traces and Determinants

175

N (At )⊥ , R(At ) ⊂ H , for some finite-dimensional subspace H which is independent of t. We first assume that 1 − At is invertible for all t ∈ I , or in other words, that 1 − λj (At ) = 0 for all t and j . Then det(1 − At ) = 0, and by a classical formula, ∂ log det(1 − At ) = −tr ∂t

     ∂ −1 ∂ −1 (At ) = −tr At (1 − At ) . (1 − At ) ∂t ∂t

Hence,     ∂ det(1 − A )   ∂  ∂ t   ∂t −1   = log det(1 − A At C1 . )    t  ≤ (1 − At )   det(1 − At )  ∂t ∂t In particular, if I = [0, 1] and At = tA1 + (1 − t)A0 , we get | log det(1 − A1 ) − log det(1 − A0 )| ≤ sup (1 − (tA1 + (1 − t)A0 ))−1 A1 − A0 C1 . 0≤t ≤1

(8.4.7) Now let A ∈ C1 . If 1 − A is not invertible, we put det(1 − A) = 0. Assume then that 1 − A is invertible. Let Aν be a sequence of finite-rank operators which converges to A in C1 . For ν, μ large enough, we have (1 − Aν )−1  ≤ C0 for some fixed constant C0 , and more generally (1 − (tAν + (1 − t)A0 ))−1  ≤ C0 . Then | log det(1 − Aν ) − log det(1 − Aμ )| ≤ C0 Aν − Aμ C1 , and consequently limν→∞ log det(1 − Aν ) exists. We then put det(1 − A) = exp



 lim log det(1 − Aν ) .

ν→∞

(8.4.8)

Notice that | det(1 − A)| ≤

 (1 + sj (A)) ≤ eAC1 .

(8.4.9)

j

Using approximation by finite-rank operators, we also see that det((1 − A)(1 − B)) = det(1 − A) det(1 − B).

(8.4.10)

By the same argument, we can extend (8.4.7) to general trace-class operators for which 1 − (tA1 + (1 − t)A0 ) is invertible. We now add a complex variable z ∈ C and consider the function z → det(1 − zA). If A is of finite rank, then this is an entire function of z. If A ∈ C1 , let Aν → A be a

176

8 Review of Classical Non-self-adjoint Spectral Theory

sequence of finite-rank operators. Then | det(1 − zAν )| ≤ e|z|Aν C1 . If zλj (A) = 1, ∀j , ! then det(1 − zAν ) → det(1 − zA) with locally uniform convergence in C \ j {1/λj }, and it follows that det(1 − zA) is a holomorphic function on this set, that verifies | det(1 − zA)| ≤ eAC1 |z| .

(8.4.11)

Therefore, det(1 − zAν ) converges to an entire function f (z) locally uniformly on C. If z = 1/λj (A), where λj (A) is of multiplicity m, then exactly m eigenvalues of Aν will converge to λj (A), while the others will stay away from a neighborhood of this point (when ν is large enough). Considering the argument variation (Rouché), we conclude that f (z) vanishes to the order m at 1/λj (A), and in particular we have f (z) = det(1 − zA) also at that point. In conclusion, we have Proposition 8.4.2 Let A ∈ C1 . Then DA (z) := det(1 − zA) is an entire function whose zeros, counted with multiplicity, coincide with the values 1/λ1 (A), 1/λ2 (A), . . . counted with the multiplicities of λ1 (1), λ2 (A), . . . Observe that DA (0) = 1

(8.4.12)

Also observe that DA (z) is of subexponential growth, in the sense that for every > 0 there exists a constant C > 0 such that |DA (z)| ≤ C e |z| .

(8.4.13)

In fact, by a limiting argument, we have |DA (z)| ≤

∞  j =1

(1 + |z|sj (A)) ≤

N 

(1 + |z|sj (A))e

∞

N+1 sj (A))|z|

.

j =1

Here the prefactor is of polynomial growth for every fixed N and for a given > 0, we can always choose N > 0 so large that the exponent is ≤ |z|. + The next observation is that W (z) := ∞ 1 (1 − zλj (A)) is also an entire function of subexponential growth. This follows by the same argument, if we recall that the series ∞ 1 |λj (A)| is convergent. We now use a special case of a theorem of Hadamard (see [4, Ch. 3, Section 3.2]): Since DA (z) and W (z) have the same zeros (counted with multiplicity), we have DA (z) = W (z)eg(z) ,

8.4 Traces and Determinants

177

where g(z) is an entire function. It is also clear that we can choose g with g(0) = 0. The function  g(z) := log |DA (z)| −

∞ 

log |1 − λj z|

(8.4.14)

j =1

is harmonic. Let R ≥ 2. For |z| ≤ R/2, we have   g(z) =

|w|=R

PR (z, w) g(w)|dw|,

(8.4.15)

where PR (z, w) = R −1 P1 (z/R, w/R) is the Poisson kernel for the disc of radius R and |dw| denotes the length element on the boundary of this disc. It is easy to see that R C 1 ≤ PR (z, w) ≤ , when |z| ≤ , |w| = R, CR R 2

(8.4.16)

where C > 0 is independent of R. Using the subexponential growth of DA (z), we get  |w|=R

PR (z, w) log |DA (w)||dw| ≤

C

RR ≤ C R, R

(8.4.17)

for R ≥ R large enough. On the other hand,    ∞ ∞     1   PR (z, w) log |1 − λj w||dw| ≤ | log |1 − λj w|||dw|   |w|=R  R |w|=R 1

1

(8.4.18) If |λj |R ≤ 12 , we write | log |1 − λj w|| ≤ C|λj |R and the sum over the corresponding j in (8.4.18) can be bounded by   C 2π|λj |RR = 2πCR |λj | = o(R), R → ∞. R 1 1

|λj |≤ 2R

(8.4.19)

|λj |≤ 2R

If 12 ≤ |λj |R ≤ T , where T ( 1 is independent of R, then by straightforward estimates, 1 R

 |w|=R

| log |1 − λj w|||dw| ≤ CT .

178

8 Review of Classical Non-self-adjoint Spectral Theory

Let us estimate the number of λj in this case: 



1 ≤ 2R

1 2 ≤|λj |R≤T

|λj | = oT (R).

1 T 2R ≤|λj |≤ R

Hence  1 2 ≤|λj |R≤T

1 R

 |w|=R

| log |1 − λj w|||dw| = oT (R), R → ∞.

(8.4.20)

It remains to consider the case |λj |R ≥ T . Here log |1 − λj R| ∼ log(|λj |R). Hence, with constants C and Cδ that are independent of T :  |λj |R≥T

≤ Cδ

1 R





|w|=R

| log |1 − λj w|||dw| ≤ C

|λj |δ R δ = Cδ R δ

|λj |≥ TR





log(|λj |R)

|λj |R≥T

|λj |δ .

T |λj |≥ R

Put δ = 1/p, 1 < p < ∞, and let q be the conjugate index. Then, if N denotes the number of λj with |λj | ≥ T /R, we get from Hölder’s inequality ⎛ 

⎞1

p

⎜  ⎟ |λj | ≤ N ⎝ |λj |⎠ .

|λj |≥ TR

1 p

1 q

(8.4.21)

|λj |≥ TR

 Here NT /R ≤ |λj | ≤ C, so N ≤ CR/T and the expression (8.4.21) is bounded by CR 1/q /T 1/q . Hence 1 1 +  1  CR p q CR | log |1 − λj w|||dw| ≤ = 1. 1 R q |w|=R T Tq |λj |≥ T

(8.4.22)

R

Combining the three cases, we find   ∞    CR   PR (z, w)( log |1 − λj w|)|dw| ≤ oT (R) + 1 = o(R), R → ∞   |w|=R  Tq 1 (8.4.23)

8.4 Traces and Determinants

179

Combining this with (8.4.14), (8.4.15) and (8.4.17), we get  g(z) ≤ o(R), on |z| ≤

R . 2

Now we can apply Harnack’s inequality (see e.g., [4, Section 6.3]) to the function  g − o(R), which is ≤ 0 on the disc |z| ≤ R/2 and ≥ −o(R) at 0, and conclude that  g ≥ −o(R) on the disc |z| ≤

R . 4

Since g is harmonic, it follows from the last two estimates that  g = 0. Hence g is constant and since we have chosen g with g(0) = 0, we get g(z) = 0 for all z ∈ C. Thus, we have established Theorem 8.4.3 Let A : H → H be a trace-class operator with exactly N non-vanishing eigenvalues λ1 (A), λ2 (A), . . ., 0 ≤ N ≤ ∞, repeated according to multiplicity, 0 ≤ N ≤ +∞. Then DA (z) = det(1 − zA) satisfies DA (z) =

N 

(1 − λj z), z ∈ C,

(8.4.24)

j =1

where the product is defined to be = 1 when N = 0. From this we get the important Lidskii’s theorem as a corollary (see [48, p. 101]): Corollary 8.4.4 If A ∈ C1 we have tr A =

N 

λj (A),

(8.4.25)

j =1

where λj (A) are the non-vanishing eigenvalues as in Theorem 8.4.3. Proof We know the result when A is of finite rank. In this case, we also know that  DA ∂ log det(1 − zA) = −tr ((1 − zA)−1 A), = DA ∂z

away from the zeros of z → det(1 − zA). In particular,  (0) DA = −tr A, DA (0)

when A is of finite rank.

(8.4.26)

180

8 Review of Classical Non-self-adjoint Spectral Theory

When A ∈ C1 , let Aν be a sequence of finite-rank operators converging to A in the C1  (z) → D  (z), when ν → ∞, uniformly for z in a norm. Then DAν (z) → DA (z), DA A ν neighborhood of 0. Since DA (z) = 0 in such a neighborhood, we have also  (0) DA ν

DAν (0)

 (0) DA . DA (0)

→

By (8.4.26), we know that the right-hand side of the last relation is equal to −tr Aν , and we also know that this quantity converges to −tr A. Consequently (8.4.26) remains valid for general trace-class operators. In view of Theorem 8.4.3, we know on the other hand that  (0)  DA =− λj (A), DA (0) N

j =1

 

and the corollary follows. The last proof also shows that  (z)  λj (A) DA ∂ = log DA (z) = −tr (1 − zA)−1 A = − , DA (z) ∂z 1 − zλj (A) N

(8.4.27)

j =1

for all z with 1 − zλj (A) = 0, ∀j . We end this section by recalling Jensen’s formula and the standard application to getting bounds on the number of zeros of holomorphic functions. Let f (z) be a holomorphic function on the open disc D(0, R) with a continuous extension to the corresponding closed disc. Assume that f (0) = 0 and f (z) = 0 for |z| = R. Assume first that f (z) has no zeros at all. Then log |f (z)| =  log f (z) is a harmonic function in the open disc which is continuous up to the boundary, and the mean value property of harmonic functions tells us that 1 log |f (0)| = 2π



2π

log |f (Reiθ )|dθ.

0

We now allow f to vanish and let z1 , . . . , zN be the zeros repeated according to their multiplicity. Since f is not allowed to vanish at 0 or at the boundary of D(0, R), we have 0 < |zj | < R. Then F (z) := f (z)

N  R 2 − zj z R(z − zj )

j =1

8.4 Traces and Determinants

181

is holomorphic in the open disc, continuous up to the boundary and has no zeros in the closed disc of radius R. Moreover, |F (z)| = |f (z)| when |z| = R, so according to the preceding paragraph, we have log |F (0)| =

1 2π



2π

log |f (Reiθ )|dθ.

0

Expanding the left-hand side, we get Jensen’s formula: log |f (0)| +

N  j =1

1 R = log |zj | 2π



2π

log |f (Reiθ )|dθ.

(8.4.28)

0

A standard application of this formula is to notice that if N(R/2) is the number of zeros zj of f with |zj | ≤ R/2, then we get N(R/2) log 2 ≤

1 2π



2π

log |f (Reiθ )|dθ − log |f (0)|.

(8.4.29)

0

Notes The presentation in Sect. 8.1 is very close to the one in Appendix A in [60]; further developments can be found in [105] and [148]. Sections 8.2–8.4 give a brief account of the classical theory of non-self-adjoint operators, following closely the monograph [48] (as in the lecture notes [136]). The proof of Lidskii’s theorem (Corollary 8.4.4) is slightly simplified in comparison to [48].

Part II Some General Results

9

Quasi-Modes in Higher Dimension

Recall that if a(x, ξ ) and b(x, ξ ) are two C 1 -functions defined on some domain in R2n x,ξ , 0 then we can define the Poisson bracket to be the C -function on the same domain given by {a, b} = aξ · bx − ax · bξ = Ha (b). Here Ha = aξ ·∂x −ax ·∂ξ denotes the Hamilton vector field of a. The following result is due to Zworski [161], who obtained it via a semi-classical reduction from the above mentioned result of Hörmander. A direct proof was given in [40] and here we give a variant. We will assume some familiarity with symplectic geometry. (See e.g. [51, Ch. 5, 9] for real symplectic geometry, and [102] for corresponding notions in the complex domain.) Theorem 9.0.1 Let P (x, hDx ) =

 |α|≤m

aα (x)(hDx )α , Dx =

1 ∂ , i ∂x

(9.0.1)

 have smooth coefficients in the open set  ⊂ Rn . Put p(x, ξ ) = |α|≤m aα (x)ξ α . Assume z = p(x0 , ξ0 ) with 1i {p, p}(x0 , ξ0 ) > 0. Then there exists u = uh ∈ C0∞ (), with u = 1, (P − z)u = O(h∞ ), when h → 0. Moreover, u is concentrated to x0 in the sense that if W ⊂  is any fixed neighborhood of x0 , then uL2 (\W ) = O(h∞ ).

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_9

185

186

9 Quasi-Modes in Higher Dimension

More generally, the result remains valid if aα depend on h in such a way that aα = aα (x; h) ∼

∞ 

aα,k (x)hk in C ∞ (),

0

now with p(x, ξ ) =



|α|≤m aα,0 (x)ξ

α.

In the case when the coefficients are all analytic near x0 we can replace “h∞ ” by for some C > 0”. See [74, 132]. This implies that if P has an extension from C0∞ () to a closed densely defined ) → L2 ( ),   ⊃ , and the resolvent (P − z)−1 exists for the operator P : L2 ( value z above, then its norm is greater than any negative power of h when h → 0 (and even exponentially large in the analytic case). In the case n ≥ 2, it was noticed in [104] that if z = p(ρ) and p, p are independent at ρ, then 1i {p, p} times the natural (non-vanishing) Liouville differential form (of maximal degree 2n − 2) on p−1 (z) is equal to a constant times the restriction of the closed form σ n−1 to p−1 (z). Here “e−1/Ch

σ =

n 

dξj ∧ dxj

1

is the symplectic 2-form on T ∗ . If  is a compact connected component of p−1 (z) on which dp and dp are pointwise independent, it follows that the average of 1i {p, p} over  with respect to the Liouville form has to vanish. Hence, if there is a point on  where the Poisson bracket is = 0, then there is also point where it is positive. In the case n = 1 we have a similar phenomenon, explained in the remarks after Proposition 4.2.6. Example 9.0.2 Take P = −h2  + V (x), p(x, ξ ) = ξ 2 + V (x), 1i {p, p} = −4ξ · V  (x). Proof of Theorem 9.0.1 We will first treat the case when p is analytic in a neighborhood of (x0 , ξ0 ) and use the same notation for analytic functions defined in a real neighborhood of some point, and their holomorphic extensions to a complex neighborhood of the same point. If φ is analytic near x0 such that φ  (x0 ) = ξ0 ∈ Rn , and φ  (x0 ) > 0,

(9.0.2)

then we can define the complex Lagrangian manifold φ := {(x, φ  (x)); x ∈ neigh (x0 , Cn )}.

(9.0.3)

9 Quasi-Modes in Higher Dimension

187

As observed by Hörmander [82], the positivity assumption (9.0.2) can be formulated equivalently by saying that 1 σ (t, (t)) > 0, 0 = t ∈ T(x0 ,ξ0 ) (φ ). i

(9.0.4)

Here: • T(x0 ,ξ0 ) (φ ) denotes the tangent space (bundle) of φ as a smooth real submanifold of C2n . The corresponding tangent vectors t = tx · ∂x + t x · ∂x + tξ · ∂ξ + t ξ · ∂ξ split into a “holomorphic” component thol = tx · ∂x + tξ · ∂ξ and an “antiholomorphic” one, tahol = thol = t x · ∂x + t ξ · ∂ξ . Here, ∂x = (∂x1 , . . . , ∂xn ), ∂xj = (1/2)(∂xj + i −1 ∂xj ), ∂x = (∂x 1 , . . . , ∂x n ), ∂x j = ∂xj . •  : T(x0 ,ξ0 ) Cn → T(x0 ,ξ0 ) Cn is the unique antilinear map which is equal to the identity on T(x0 ,ξ0 ) R2n .  • σ denotes the symplectic 2-form n1 dξj ∧ dxj on C2n (now a differential form of type (2, 0)), here viewed as a bilinear form on the complexified tangent space of the cotangent space at (x0 , ξ0 ). In other terms, we have the identification and identity σ (t, s) = σ |t ∧ s = σ |thol ∧ shol  and more explicitly, σ (t, s) = tξ · sx − tx · sξ , when s is represented similarly to t above. In practice we identify tangent vectors with their holomorphic parts, and define zt = zthol + ztahol . • Writing thol = tx · ∂x + tξ · ∂ξ as above, we have (t)hol = t x · ∂x + t ξ · ∂ξ . From now on we identify tangent vectors and vector fields in the complex domain with their holomorphic parts, and by (very small) abuse of notation we write t = t. • More generally, let  ⊂ C2n be a complex Lagrangian manifold, i.e., a complex manifold such that σ | = 0, of maximal complex dimension with this property: dim () = n. Assume that (x0 , ξ0 ) ∈  and that (9.0.4) holds with φ replaced by . Then T(x0 ,ξ0 )  cannot contain any non-zero tangent vector (with holomorphic part equal to) tξ · ∂ξ , and it follows that  = φ with φ as above. If t = tx · ∂x + tξ · ∂ξ is (the holomorphic part of) a tangent vector to φ at (x0 , ξ0 ), then tξ = φ  (x0 )tx and 1 1  σ, t ∧ t = (φ  tx · t x − φ tx · t x ) = (φ  tx · t x ), 2i 2i so φ  (x0 ) > 0.

188

9 Quasi-Modes in Higher Dimension

Let z, p, (x0 , ξ0 ) be as in the theorem. Then we observe that   1 1 1 σ (Hp , H p ) = σ (Hp , (Hp )) = {p, p} > 0. i i i Moreover, the real set := p−1 (z) is a smooth symplectic manifold near (x0 , ξ0 ). Indeed, the orthogonal space to Tx0 ,ξ0 with respect to the real symplectic form is generated by Hp , Hp and is transverse to Tx0 ,ξ0 , so the restriction to of the symplectic 2-form is nondegenerate near (x0 , ξ0 ). Using the Darboux theorem, we can identify it locally with R2(n−1) and hence find a Lagrangian submanifold  in its compexification, passing through (x0 , ξ0 ), that satisfies the positivity condition (9.0.4). Viewing the complexification of as a submanifold of C2n , we can define  = {exp sHp (ρ); s ∈ neigh (0, C), ρ ∈ neigh ((x0 , ξ0 ),  )}. Using that Hp is symplectically orthogonal to the tangent space of

, it is then quite easy to verify that  is a complex Lagrangian manifold, contained in the complex characteristic hypersurface {ρ ∈ neigh ((x0 , ξ0 ), C2n ); p(ρ) = 0} and satisfying the positivity condition (9.0.4). Hence,  is of the form φ for an analytic function φ as in (9.0.2), (9.0.3), which also fulfills the eikonal equation p(x, φ  (x)) − z = 0.

(9.0.5)

We normalize φ by requiring that φ(x0 ) = 0. Then the function eiφ(x)/ h is rapidly decreasing with all its derivatives away from any neighborhood of x0 . By a complex version of the standard WKB-construction we can construct an elliptic symbol a(x; h) ∼ a0 (x) + ha1 (x) + · · · , by solving the suitable transport equations to infinite order at x0 , such that if χ ∈ C0∞ (neigh (x0 , Rn )) is equal to 1 near x0 , then u(x; h) = χ(x)h−n/4 a(x; h)eiφ(x)/ h has the required properties. Now, we drop the analyticity assumption and assume merely that p is smooth, as in the statement of the theorem. Let p(1) be the 1st order Taylor polynomial of p at (x0 , ξ0 ), so that p(x, ξ ) = p(1) (x, ξ ) + O((x − x0 , ξ − ξ0 )2 ), (x, ξ ) → (x0 , ξ0 ). Then the arguments above apply to p(1) and noticing that (1) := (p(1) )−1 (z) is affine linear, we can find  in the complexification of (1) as above, also affine. Then (1) , defined similarly to  above, is an affine linear Lagrangian space of the form (2) = φ (2) , where φ (2) is a 2nd order polynomial, with φ(x0 ) = 0, ∂x φ (2) (x0 ) = ξ0 , ∂x2 φ (2) > 0, p(1) (x, ∂x φ (2) (x)) − z = 0, so p(x, ∂x φ (2) (x)) − z = r, where r = O((x − x0 )2 ).

9 Quasi-Modes in Higher Dimension

189

Look for ψ (3) = O((x − x0 )3 ) such that p(x, ∂x (φ (2) + ψ (3) )) − z = O((x − x0 )3 ). This leads to a linear transport-type equation, pξ (x, ∂x φ (2) ) · ∂x ψ (3) = −r + O((x − x0 )3 ), which is easy to solve, using that pξ (x0 , ξ0 ) = 0. Iterating the construction, we can find ψ (j ) = O((x − x0 )j ), j = 4, 5, . . . such that if φ = φ (2) + ψ (3) + ψ (4) + · · · in the sense of formal Taylor series, then in the same sense, p(x, ∂x φ) = 0. Again we can find a WKB solution u = uh with the required properties, noticing that   eiφ(x)/ hf (x; h) = O(h∞ ) in L2 (neigh (x0 , Rn )) if f = O(|x − x0 |∞ + h∞ ). Notes The background is the same as in Sect. 4.2. Davies [33] showed that for the onedimensional Schrödinger operator we can construct quasimodes for values of the spectral parameter that may be quite far from the spectrum of the operator. Zworski [161] observed that this result can be viewed as a special case of a more general result of Hörmander [78, 79] in the context of linear PDE. Pravda-Starov [111] has generalized Theorem 9.0.1 by adapting a more refined quasimode construction of R. Moyer (in two dimensions) and Hörmander [83] for adjoints of operators that do not satisfy the Nirenberg-Trèves condition (#) for local solvability.

Resolvent Estimates Near the Boundary of the Range of the Symbol

10.1

10

Introduction and Outline

In this chapter, which closely follows [140], we study bounds on the resolvent of a non-self-adjoint h-pseudodifferential operator P with (semi-classical) principal symbol p when h → 0, when the spectral parameter is in a neighborhood of certain points on the boundary of the range of p. In Chap. 6 we have already described a very precise result of W. Bordeaux Montrieux in dimension 1. Here we consider a more general situation; the dimension can be arbitrary and we allow for more degenerate behaviour. The results will not be quite as precise as in the one-dimensional case. In [40] we obtained resolvent estimates at certain boundary points in the following two cases: (1) under a non-trapping condition, (2) under a stronger “subellipticity condition”. The case (1) was studied in [40] with general and simple arguments related to the propagation of regularity; the treatment in case (2) was based on Hörmander’s work on subellipticity for operators of principal type [83]. In this case the resolvent extends and has temperate growth in 1/ h in discs of radius O(h ln 1/ h), centered at the appropriate boundary points. In case (2) we have an extension up to distance O(hk/(k+1) ), where the integer k ≥ 2 is determined by a condition of “subellipticity type”. In this chapter we concentrate on points of type (2) and obtain resolvent estimates by studying an associated semigroup as a Fourier integral operator with complex phase in the spirit of Maslov [99], Kucherenko [89], Melin and Sjöstrand [103]. (See also Menikoff and Sjöstrand [106], Matte [100].) It turned out to be more convenient to use Bargmann-

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_10

191

192

10 Resolvent Estimates Near the Boundary of the Range of the Symbol

FBI transforms in the spirit of [132] and [60]. The semigroup method leads to a stronger result: The resolvent can be extended to a disc of radius O((h ln 1/ h)k/(k+1)) around the appropriate boundary points, as in dimension 1 when k = 2 in the result of Bordeaux Montrieux [16], cf. Theorem 6.1.4. In that case Bordeaux Montrieux also constructed quasimodes for values of the spectral parameter that are close to the boundary points. We next state the results. Let X denote either Rn or a compact smooth manifold of dimension n. In the first case, let m ∈ C ∞ (R2n ; [1, +∞[ ) be an order function (cf. [41, Def. 7.4], Sect. 5.1) so that for some C0 , N0 > 0, m(ρ) ≤ C0 ρ − μN0 m(μ), ρ, μ ∈ R2n ,

(10.1.1)

where ρ − μ = (1 + |ρ − μ|2 )1/2. Let P = P (x, ξ ; h) ∈ S(m), in the sense that P is smooth in x, ξ and satisfies α P (x, ξ ; h)| ≤ Cα m(x, ξ ), (x, ξ ) ∈ R2n , α ∈ N2n , |∂x,ξ

(10.1.2)

where Cα is independent of h. We also assume that P (x, ξ ; h) ∼ p0 (x, ξ ) + hp1 (x, ξ ) + · · · , in S(m),

(10.1.3)

and write p = p0 for the principal symbol. We adopt the ellipticity assumption ∃w ∈ C, C > 0, such that |p(ρ) − w| ≥ m(ρ)/C, ∀ρ ∈ R2n .

(10.1.4)

As in (5.1.3), let P = P w (x, hDx ; h) = Op(P (x, hξ ; h))

(10.1.5)

be the Weyl quantization of the symbol P (x, hξ ; h) that we can view as a closed unbounded operator on L2 (Rn ). m (T ∗ X) (the classical Hörmander In the second (compact manifold) case, we let P ∈ S1,0 symbol space) of order m > 0, meaning that |∂xα ∂ξ P (x, ξ ; h)| ≤ Cα,β ξ m−|β| , (x, ξ ) ∈ T ∗ X, β

(10.1.6)

where Cα,β are independent of h. Similarly to (10.1.3), we assume that there exist pj ∈ m−j S1,0 (T ∗ X) such that P (x, ξ ; h) −

N−1  0

m−N hj pj (x, ξ ) ∈ hN S1,0 (T ∗ X), N = 1, 2, . . . ,

(10.1.7)

10.1 Introduction and Outline

193

and we quantize the symbol P (x, hξ ; h) in a standard (non-unique) way, by doing it for various local coordinates and gluing the quantizations together by means of a partition of unity. When m > 0 we make the ellipticity assumption ∃C > 0, such that |p(x, ξ )| ≥

ξ m , |ξ | ≥ C. C

(10.1.8)

Let (p) = p(T ∗ X) (the closure of p(T ∗ X)) and let ∞ (p) be the set of accumulation points of p(ρj ) for all sequences ρj ∈ T ∗ X, j = 1, 2, 3, . . . that tend to infinity. The main result of this chapter is taken from [40, 140]: Theorem 10.1.1 We adopt the general assumptions above. Let z0 ∈ ∂ (p) \ ∞ (p) and assume that dp = 0 at every point of p−1 (z0 ). Then for every such point ρ there exists θ ∈ R (unique up to a multiple of π) such that d(e−iθ (p − z0 )) is real at ρ. We write θ = θ (ρ). Consider the following two cases: (1) For every ρ ∈ p−1 (z0 ), the maximal integral curve of H(e−iθ (ρ) p) through the point ρ is not contained in p−1 (z0 ). (2) There exists an integer k ≥ 1 such that for every ρ ∈ p−1 (z0 ), there exists j ∈ {1, 2, . . . , k} such that j

Hp (p)(ρ)/(j !) = 0. Here Hp = pξ · ∂x − px · ∂ξ is the Hamiltonian field, viewed as a differential operator. In case (1), there exists a constant C0 > 0 such that for every constant C1 > 0 there is a constant C2 > 0 such that the resolvent (z−P )−1 is well-defined for |z−z0 | < C1 h ln h1 , h < C12 , and satisfies the estimate (z − P )

−1

  C0 C0 exp |z − z0 | . ≤ h h

(10.1.9)

In case (2), there exists a constant C0 > 0 such that for every constant C1 > 0 there is a constant C2 > 0 such that the resolvent (z − P )−1 is well-defined for |z − z0 | < C1 (h ln h1 )k/(k+1), h < C12 and satisfies the estimate (z − P )−1  ≤

C0 k

h k+1

 exp

k+1 C0 |z − z0 | k h

 .

(10.1.10)

In [40] the bound O(1/ h) in (10.1.9) was obtained for |z − z0 | ≤ h/O(1), as a well as a less precise polynomial bound for |z − z0 | < C1 h ln h1 . The condition in case (2) was

194

10 Resolvent Estimates Near the Boundary of the Range of the Symbol

formulated a little differently, but the two formulations lead to the same microlocal models and are therefore equivalent. Let us now consider a special situation of interest for evolution equations, namely the case when z0 ∈ iR,

(10.1.11)

p(ρ) ≥ 0 in neigh (p−1 (z0 ), T ∗ X).

(10.1.12)

Theorem 10.1.2 We adopt the general assumptions above. Let z0 ∈ ∂ (p) \ ∞ (p) and assume (10.1.11), (10.1.12). Also assume that dp = 0 on p−1 (z0 ), so that dp = 0, dp = 0 on that set. Consider the two cases: (1) For every ρ ∈ p−1 (z0 ), the maximal integral curve of Hp through the point ρ contains a point where p > 0. j (2) There exists an integer k ≥ 1 such that for every ρ ∈ p−1 (z0 ), we have Hp p(ρ) = 0 for some j ∈ {1, 2, . . . , k}. Then, in case (1), there exists a constant C0 > 0 such that for every constant C1 > 0 there is a constant C2 > 0 such that the resolvent (z − P )−1 is well-defined for |(z − z0 )|
0 there is a constant C2 > 0 such that the resolvent (z − P )−1 is well-defined for |(z − z0 )|
0, neighborhood of (x0 , y0 ) ∈ Cn × Rn , and −φy (x0 , y0 ) = η0 ∈ Rn , φy,y 0 0  (x , y ) = 0. Let κ be the associated canonical transformation. Then, T is det φx,y 0 0 T bounded L2 → H!0 := Hol () ∩ L2 (, e−2!0 / h L(dx)) and has (microlocally) a bounded inverse, where  is a small complex neighborhood of x0 in Cn . The weight !0 0 is smooth and strictly pluri-subharmonic. If !0 := {(x, 2i ∂! ∂x ); x ∈ neigh (x0 )}, then (locally) !0 = κT (T ∗ X). The operator P = T P T −1 can be defined locally modulo O(h∞ ) (cf. [91]) as a bounded operator from H! → H! when the weight ! is smooth and satisfies ! − !0 = O(hδ ) for some δ > 0. (In the analytic framework it suffices that ! − !0 is small.)  is given by Egorov’s theorem applies in this situation, so the principal symbol p  of P p|! ≥ 0. This p  ◦ κT = p. Thus (under the assumptions of Theorem 10.1.2), we have  

0

in turn can be used to see that for 0 ≤ t ≤ hδ , we have e−t P / h = O(1): H!0 → H!t , where !t ≤ !0 is determined by the real Hamilton–Jacobi problem 2 ∂!t ∂!t +  p (x, ) = 0, !t =0 = !0 . ∂t i ∂x

(10.1.21)

Now the bound (10.1.16) follows from the estimate !t ≤ !0 −

t k+1 , C

(10.1.22)

where C > 0. In [140] we obtained this estimate by studying certain I-Lagrangian deformations of the real phase space prior to the FBI-transformation. Here we shall use a shorter and more direct approach. In Sect. 10.4 we discuss some examples. There are many works that develop similar ideas and results, see [11,68–70,76,77,153, 154].

10.2

Evolution Equations on the Transform Side

Let P(x, ξ ; h) be a smooth symbol defined in neigh ((x0 , ξ0 ), !0 ), with an asymptotic expansion P(x, ξ ; h) ∼ p (x, ξ ) + h p1 (x, ξ ) + . . . in C ∞ (neigh ((x0 , ξ0 ), !0 )).

10.2 Evolution Equations on the Transform Side

197

Let P also denote an almost holomorphic extension (see e.g., [103] and Appendix 6.A) to a complex neighborhood of (x0 , ξ0 ): (x, ξ ; h) ∼ p P (x, ξ ) + h p1 (x, ξ ) + · · ·

in C ∞ (neigh ((x0 , ξ0 ), C2n )),

where p , p j are smooth extensions such that ∂p , ∂ p j = O(dist ((x, ξ ), !0 )∞ ). Then, as shown in [91] and later in [104], if u = uh is holomorphic in a neighborhood V of x0 and belonging to H!0 (V ) in the sense that uL2 (V ,e−2!0 /h L(dx)) is finite and of u = P(x, hDx ; h)u can be defined temperate growth in 1/ h when h tends to zero, then P in any smaller neighborhood W  V by the formula Pu(x) =

1 (2πh)n



((x + y)/2, θ ; h)u(y)dydθ, e h (x−y)·θ P i

(10.2.1)

(x)

0 x+y where (x) is a good contour (in the sense of [132]) of the form θ = 2i ∂! ∂x ( 2 ) + i ∞  C1 (x − y), |x − y| ≤ 1/C2 , C1 , C2 > 0. Then ∂ P is negligible, i.e., of norm O(h ): 2 H!0 (V ) → L!0 (W ), and modulo such negligible operators, P is independent of the choice of a good contour. By solving a ∂-problem (assuming, as we may, that our neighborhoods are pseudoconvex) we can always correct P with a negligible operator such that (after an arbitrarily small decrease of W ) P = O(1) : H!0 (V ) → H!0 (W ). Also, if ! = !0 + O(h ln h1 ) in C 2 , then P = O(h−N0 ) : H! (V ) → H! (W ), for some N0 . By means of Stokes’ formula, we can show that P will change only by a negligible term if we replace !0 by ! in the definition of (x), and then it follows that P = O(1) : H! (V ) → H! (W ). Recall [104] (see also [121]) that the identity operator H!0 (V ) → H!0 (W ) is, up to a negligible operator (i.e., an operator of norm O(h∞ )) of the form

I u(x) = h−n



2

2

e h #0 (x,y) a(x, y; h)u(y)e− h !0 (y)dydy,

(10.2.2)

where #0 (x, y), a(x, y; h) are almost holomorphic on the antidiagonal y = x with #0 (x, x) = !0 (x), a(x, y; h) ∼ a0 (x, y) + ha1(x, y) + · · · , a0 (x, x) = 0. More generally a pseudodifferential operator like P takes the form Pu(x) = h−n



2

2

e h #0 (x,y) q(x, y; h)u(y)e− h !0 (y)dydy

  2 ∂!0 q0 (x, x) = p (x) a0 (x, x),  x, i ∂x

(10.2.3)

198

10 Resolvent Estimates Near the Boundary of the Range of the Symbol

and where q0 denotes the first term in the asymptotic expansion of the symbol q. In this discussion, !0 can be replaced by any other smooth exponent ! which is O(hδ ) close to !0 in C ∞ and we make the corresponding replacement of #0 . A well-known consequence of the strict pluri-subharmonicity of ! is that 2#(x, y) − !(x) − !(y) " −|x − y|2 ,

(10.2.4)

so the uniform boundedness H! → H! follows from the domination of the modulus of the kernel of e−!/ h ◦ P ◦ e!/ h by a Gaussian convolution kernel. Now, we study the evolution problem (t) = 0, U (0) = 1, (h∂t + P)U

(10.2.5)

where t is restricted to the interval [0, hδ ] for some arbitrarily small but fixed δ > 0. We review the approximate solution of this problem by a geometrical optics construction: (t) of the form Look for U (t)u(x) = h−n U



2

e h #t (x,y) at (x, y; h)u(y)e−2!0(y)/ h dydy,

(10.2.6)

where #t , at depend smoothly on all the variables and #t =0 = #0 , at =0 = a0 in (10.2.3), (0) = 1 up to a negligible operator. so U (t) is the Fourier integral operator Formally U (t)u(x) = h−n U



2

e h (#t (x,θ)−#0 (y,θ))at (x, θ ; h)u(y)dydθ,

(10.2.7)

where we choose the integration contour θ = y. Writing 2#t (x, θ ) = iφt (x, θ ) leads to more standard notation and we impose the eikonal equation i∂t φ + p (x, φx (x, θ )) = 0.

(10.2.8)

We cannot hope to solve the eikonal equation exactly, but we can do so to infinite order at t = 0, x = y = θ . If we put φt (·,θ) = {(x, φx (t, x, θ ))},

(10.2.9)

  1 (φ0 (·,θ) ), φt (·,θ) = exp t H p 

(10.2.10)

then

i

10.2 Evolution Equations on the Transform Side

199

p = Hp + Hp denotes the real vector field to infinite order at t = 0, θ = x. Here H 1 . (As in Chap. 9 we sometimes drop associated to the (1,0)-field Hp, and similarly for H  ip the distinction between a vector field of type (1,0) and the corresponding real vector field  = 0, we have  ν = ν + ν.) At a point where ∂ p p = H σ = H σ , H  p  p

σ σ i H p = −H p = H p,

(10.2.11)

where the other fields are the Hamiltonian fields of  p,  p with respect to the real symplectic forms σ and σ , respectively. See [104, 132]. Thus (10.2.10) can be written as −σ φt (·,θ) = exp(tH p )(φ0 (·,θ) ).

(10.2.12)

A complex Lagrangian manifold is also I-Lagrangian (i.e., a Lagrangian manifold for σ ) and (10.2.12) can be viewed as a relation between I-Lagrangian manifolds. It defines the I-Lagrangian manifold φt (·,θ) unambiguously, once we have fixed an almost holomorphic extension of p . Locally, a smooth I-Lagrangian manifold , for which the x-space projection   (x, ξ ) → x ∈ Cn is a local diffeomorphism, takes the form  = ! , where ! is real and smooth and ! := {(x,

2 ∂! ); x ∈ },  ⊂ Cn open. i ∂x

With a slight abuse of notation, we can therefore identify the C-Lagrangian manifold 2 ∂(−φ0 ) 0 φ0 with the I-Lagrangian manifold −φ0 , since ∂φ , at all points where ∂x = i ∂x ∂ x φ0 = 0. Relation (10.2.4) shows that !0 (x) + !0 (θ ) − (−φ0 (x, θ )) " |x − θ |2 .

(10.2.13)

Define −σ !t = exp(tH p )(!0 ),

(10.2.14)

and fix the t-dependent constant in this definition of !t by imposing the real Hamilton– Jacobi equation, p(x, ∂t !t + 

2 ∂!t ) = 0, !t =0 = !0 . i ∂x

(10.2.15)

The real part of (10.2.8) gives a similar equation for −φt , p(x, ∂t (−φ) + 

2 ∂ (−φ)) = 0. i ∂x

(10.2.16)

200

10 Resolvent Estimates Near the Boundary of the Range of the Symbol

Relation (10.2.13) implies that !0 and −φ0 (·,θ) intersect transversally at −σ 2 ∂!0 (x0 (θ ), ξ0 (θ )), where (xt (θ ), ξt (θ )) := exp(tH p )(θ , i ∂x (θ )). Equations (10.2.14) and (10.2.12) then show that !t and −φt (·,θ) intersect transversally at (xt (θ , ξt (θ )). Then (10.2.13), (10.2.15), (10.2.16) imply that !t (x) − (−φt (x, θ )) = O(|x − xt (θ)|2 ), and again by (10.2.13) and the transversal intersection, we get !t (x) + !0 (θ ) − (−φt (x, θ )) " |x − xt (θ )|2 .

(10.2.17)

Determining at by solving a sequence of transport equations, we arrive at the following result: (t) constructed above is O(1) : H!0 (V ) → Proposition 10.2.1 The operator U H!t (W ) (W  V being small pseudoconvex neighborhoods of a fixed point x0 ) uniformly for 0 ≤ t ≤ hδ and it solves the problem (10.2.5) up to negligible terms. This local statement makes sense, since by (10.2.17) we have 2#t (x, y) − !t (x) − !0 (y) " −|x − xt (y)|2 .

(10.2.18)

Using (10.2.5), we get up to negligible errors that (t)] = 0, [P, U (0)] = 0, (h∂t + P)[P, U and examining this evolution problem, we conclude that the Fourier integral operator (t)] is negligible. In particular, [P, U (t) + U (t)P = 0, 0 ≤ t ≤ hδ , h∂t U

(10.2.19)

up to negligible errors. Let us briefly recall an alternative approach, leading to the same weights !t (cf. [104]). Consider formally 



(e−t P / h u|e−t P / h u)H!t = (ut |ut )H!t , u ∈ H!0 , and look for !t such that the time derivative of this expression vanishes to leading order. We get  0 ≈ h∂t

ut ut e−2!t / h L(dx)    ∂!t = − (Put |ut )H!t + (ut |Put )H!t + 2 (x)|u|2 e−2!t / h L(dx) . ∂t

10.2 Evolution Equations on the Transform Side

201

Here (Put |ut )H!t =



( p |!t + O(h))|ut |2 e−2!t / h L(dx),

and similarly for (ut |Put )H!t , so we would like to have  0≈

(2

∂!t + 2 p|!t + O(h))|ut |2 e−2!t / h L(dx). ∂t

We choose !t to be the solution of (10.2.15). Then the preceding discussion again  shows that e−t P / h = O(1) : H!0 → H!t . −σ  p is constant along the integral curves of H p . Therefore, the second term in (10.2.15) is ≥ 0, so !t ≤ !0 , t ≥ 0,

(10.2.20)

 p |! ≥ 0.

(10.2.21)

under the assumption that

0

Recall that we confine our discussion to the interval 0 ≤ t ≤ hδ . Write !t = !0 − #t ,         2 2 2 2  p  x, ∂x !t = p  x, ∂x !0 − p ξ x, ∂x !0 · ∂x #t + O (∂x #t )2 . i i i i Using this is in (10.2.15), we get         2 2 2  −∂t #t +  p x, ∂x !0 −  p ξ x, ∂x !0 · ∂x #t + O (∂x #t )2 = 0. i i i Define a =  p, b =  p on !0 and extend these functions to a neighborhood by means of almost holomorphic extension, so that p  = a + ib also in a neighborhood of !0 . On !0 , we have 1 ∂t #t + 2



  2  2  aξ + ibξ ) · ∂x + aξ + ibξ ) · ∂x i i



  #t + O (∂x #t )2 = a. (10.2.22)

202

10 Resolvent Estimates Near the Boundary of the Range of the Symbol

  If μ = n1 μj ∂xj , let  μ = μ + μ = (μj ∂xj + μj ∂xj ) be the associated real vector field with the same action as a differential operator on holomorphic functions. Then we get 2    ∂t #t + b ξ · ∂x #t − iaξ · ∂x #t + O((∂x #t ) ) = a, #0 = 0,

(10.2.23)

 where bξ , aξ and a are evaluated on !0 . Here we observe that ν := b ξ · ∂x is the x-space projection of the real Hamiltonian field of b| ! with respect to the real symplectic form 0 σ |! . 0 ν#t = a, where  ν is some real Notice first that (10.2.23) can be written as ∂t #t +  t-dependent vector field on !0 (locally identified with Cnx ). It follows again that #t ≥ 0. Then by the standard inequality ∇f = O(f 1/2 ) for non-negative C 2 functions, we have (∂#t )2 = O(#t ),

aξ · ∂x #t = O(|aξ ||∂x #t |) = O(1)a 1/2#t

1/2

= O(1)( a + −1 #t ),

uniformly with respect to > 0. (Because of the almost holomorphy, we have |aξ | = O(∇a| ! ).) Using this in (10.2.23), we get 0

∂t #t + ν#t + O(1 + −1 )#t = (1 + O( ))a.

(10.2.24)

This is a differential inequality along the integral curves of ν = H p on !0 and fixing

> 0 small enough, we get (t, ρ), #t (exp tH p (ρ)) " J for ρ ∈ neigh ((x0 , ξ0 ), !0 ), 0 < t J(t, ρ) =

(10.2.25)

1, where 

t 0

 p (exp sH p (ρ))ds.

(10.2.26)

Notice that J(t, κT (ρ)) =



t 0

p(exp sHp (ρ))ds =: J (t, ρ),

∞

(10.2.27)

0 ≤ J ∈ C (neigh ((0, ρ0 ), [0, +∞[×R )). 2n

Now, introduce the following assumption corresponding to the case (2) in Theorem 10.1.2, ⎧ ⎨= 0, j Hp (p)(ρ0 ) ⎩> 0,

if j ≤ k − 1, if j = k,

(10.2.28)

10.2 Evolution Equations on the Transform Side

203

where k has to be even since p ≥ 0. We will work in a sufficiently small neighborhood of ρ0 . Then ⎧ ⎨= 0, j +1 j ∂t J (0, ρ0 ) = Hp (p)(ρ0 ) ⎩> 0,

if j ≤ k − 1, if j = k,

(10.2.29)

Proposition 10.2.2 Let 0 ≤ J (t, ρ) ∈ C ∞ (neigh (0, ρ0 ), [0, +∞[ ×R2n ) satisfy (10.2.29) for some k ≥ 1. Then there is a constant C > 0 such that J (t, ρ) ≥

t k+1 , C

(t, ρ) ∈ neigh ((0, ρ0 ), [0, +∞[ ×R2n ).

(10.2.30)

Proof Assume that (10.2.30) does not hold. Then there is a sequence (tν , ρν ) ∈ [0, +∞[ ×R2n converging to (0, ρ0 ) such that J (tν , ρν ) tνk+1

→ 0,

and since J (t, ρ) is an increasing function of t, we get sup

0≤t ≤tν

J (t, ρν ) tνk+1

→ 0.

Introduce the Taylor expansion, J (t, ρν ) = aν(0) + aν(1)t + · · · + aν(k+1)t k+1 + O(t k+2 ), and define uν (s) =

J (tν s, ρν ) tνk+1

, 0 ≤ s ≤ 1.

Then, by (10.2.31), sup uν (s) → 0, ν → ∞. 0≤s≤1

On the other hand, uν (s) =

aν(0) t k+1 "ν

+

aν(1) s + · · · + aν(k+1)s k+1 + O(tν s k+2 ), tνk #$ % =:pν (s)

(10.2.31)

204

10 Resolvent Estimates Near the Boundary of the Range of the Symbol

so sup |pν (s)| → 0, ν → ∞. 0≤s≤1

The corresponding coefficients of pν have to tend to 0, and, in particular, aν(k+1) =

1 (∂ k+1 J (0, ρν ) → 0 (k + 1)! t  

which is in contradiction with (10.2.29). From (10.2.26), (10.2.27), (10.2.30), we get Proposition 10.2.3 Under the assumption (10.2.28), we have !t (x) ≤ !0 (x) −

10.3

t k+1 , (t, x) ∈ neigh ((0, x0 ), [0, ∞[ ×Cn ). C

(10.2.32)

Resolvent Estimates

Let P be an h-pseudodifferential operator satisfying the general assumptions of the introduction. Let z0 ∈ (∂ (p)) \ ∞ (p). We will treat the case (2) of Theorem 10.1.2, so that z0 ∈ iR,

(10.3.1)

p(ρ) ≥ 0 in neigh (p−1 (z0 ), T ∗ X),

(10.3.2)

∀ρ ∈ p−1 (z0 ), ∃j ≤ k, such that Hp p(ρ) > 0. j

(10.3.3)

Proposition 10.3.1 ∃C0 > 0 such that ∀C1 > 0, ∃C2 > 0 such that for z, h as in (10.1.14) and u ∈ C0∞ (X) it holds that k

|z|u ≤ C0 (z − P )u, when z ≤ −h k+1 ,   k+1 k C0 k k+1 (z)+ (z − P )u, h u ≤ C0 exp h  k  k 1 k+1 k+1 when z ≥ −h ≤ z ≤ O(1) h ln . h

(10.3.4)

10.3 Resolvent Estimates

205

Proof The required estimate is easy to obtain microlocally in the region where P − z0 is elliptic, in the sense that if χ ∈ C0∞ (T ∗ X) and supp χ ∩ p−1 (z0 ) = ∅ and writing χ also for a corresponding h-quantization, then by standard calculus [41], we have χu ≤ O(1)(z − P )u + O(h∞ )u for small values of |z − z0 |. It then suffices to show the following statement: For every ρ0 ∈ p−1 (z0 ), there exists χ ∈ C0∞ (T ∗ X), equal to 1 near ρ0 , such that for z, h as in (10.1.14) and letting χ also denote a corresponding h-pseudodifferential operator, we have k

|z|χu ≤ C0 (z − P )u + CN hN u, when z ≤ −h k+1 ,   k+1 k C0 k k+1 (z)+ (z − P )u + CN hN u, h χu ≤ C0 exp h  k  k 1 k+1 k+1 when − h ≤ z ≤ O(1) h ln , h

(10.3.5)

where N ∈ N is arbitrary. When z ≤ −hk/(k+1) this is an easy consequence of the semi-classical sharp Gårding inequality (see for instance [41]), so from now on we assume that z ≥ −hk/(k+1) . Let T : L2 → H!0 (neigh (x0 , Cn )) be an FBI transform as in Sect. 10.1, with (y0 , η0 ) = ρ0 . We then have a conjugated operator (cf. [132]) P as in (10.2.1), with  = p ◦ κT−1 near (x0 , ξ0 ), with p  (x0 , ξ0 ) = κT (y0 , η0 ) ∈ !0 , and !0 = κT (T ∗ X), p  the principal symbol of P , such that T P u − PT uH!0 (V ) ≤ O(h∞ )uL2 ,

(10.3.6)

where V is a small neighborhood of x0 . When supp χ is small enough, we also have χu ≤ O(1)T uH!0 (V ) + O(h∞ )uL2 .

(10.3.7)

It suffices to show that k − k+1

uH!0 (V1 ) ≤ h



k+1 C0 (z)+k C0 exp h



(P − z)uH!0 (V2 )

(10.3.8)

∞

+ O(h )uH!0 (V3 ) , u ∈ H!0 (V3 ), where V1  V2  V3 are neighborhoods of x0 , given by (x0 , ξ0 ) = κT (ρ0 ) ∈ !0 . (t) : H!0 (V2 ) → H!t (V1 ), we see that From Proposition 10.2.3 and the fact that U (t)uH! (V1 ) ≤ Ce−t U 0

k+1 /(Ch)

uH!0 (V2 ) .

(10.3.9)

206

10 Resolvent Estimates Near the Boundary of the Range of the Symbol

Choose δ > 0 small enough so that δ(k + 1) < 1 and put 

 = 1 R(z) h

hδ

(t)dt. ehU tz

(10.3.10)

0

 is an approximate left inverse to P − z, we study the norm of Before verifying that R this operator in H!0 . We have in L(H!0 (V2 ), H!0 (V1 )):    1 t k+1  e U (t) ≤ C exp tz − . h C tz h

The right-hand side is O(h∞ ) for t O(1) (h ln(1/ h))k/(k+1). We get C  R(z) ≤ h

 0

+∞

(10.3.11)

= hδ , since δ(k + 1) < 1 and z ≤

 1  k+2    1 C k+1 t k+1 C k+1 exp I z , tz − dt = k k h C h k+1 h k+1

(10.3.12)

where  I (s) =

∞

est −t

k+1

(10.3.13)

dt.

0

Lemma 10.3.2 We have I (s) = O(1), when |s| ≤ 1,

(10.3.14)

O(1) , when s ≤ −1, |s|

(10.3.15)

I (s) =  I (s) ≤ O(1)s

− k−1 2k

exp



k (k + 1)

k+1 k

s

k+1 k

, when s ≥ 1.

(10.3.16)

Proof The first two estimates are straightforward, so we concentrate on the last one, where we may also assume that s ( 1. A computation shows that the exponent fs (t) = st − t k+1 on [0, +∞[ has a unique critical point t = t (s) = (s/(k +1))1/ k , which is a nondegenerate maximum, since 1

fs (t (s)) = −k(k + 1) k s

k−1 k

,

10.3 Resolvent Estimates

207

with critical value fs (t (s)) =

k (k + 1)

k+1 k

s

k+1 k

.

Now fs (t) = −(k + 1)kt k−1  fs (t (s)), for

t (s) ≤ t < +∞, 2

∞ k+1 so t (s)/2 est −t dt satisfies the required upper bound. Here we write a  b when a, b ≥ 0 and a ≤ O(b). On the other hand, fs (t (s)) − fs (t) ≥

k+1 k

s

C

, for 0 ≤ t ≤

t (s) , s ( 1, 2

so 

t (s) 2

 e

st −t k+1

1 k

dt ≤ O(1)s exp fs (t (s)) −

0

s

k+1 k

C

 ,  

and (10.3.16) follows. Applying the lemma to (10.3.12), we get Proposition 10.3.3 We have C

 R(z) ≤ h  R(z) ≤

C , |z|

k k+1

,

k

if |z| ≤ O(1)h k+1 , k

if − 1

  k+1 C (z) k  R(z) ≤ k exp Ck , h h k+1

z ≤ −h k+1 ,

h

k k+1

  k 1 k+1 ≤ z ≤ C h ln . h

(10.3.17)

(10.3.18)

(10.3.19)

From the beginning of the proof of Lemma 10.3.2, or more directly from (10.3.11), we see that (t) ≤ C exp e h U tz

k+1 Ck (z)+k , h

208

10 Resolvent Estimates Near the Boundary of the Range of the Symbol k

which is bounded by some negative power of h, since z ≤ O(1)(h ln h1 ) k+1 . Working locally, we then see that modulo a negligible operator,  P − z) ≡ 1 R(z)( h



hδ

(t)dt ≡ 1, e h (−h∂t − z)U tz

0

(t) where the last equivalence follows from an integration by parts and the fact that et z/ hU δ is negligible for t = h . Combining this with Proposition 10.3.3, we get (10.3.8), and then (10.3.5) by means of (10.3.6), which completes the proof of Proposition 10.3.1.   We can now complete the Proof of Theorem 10.1.2 Using standard pseudodifferential machinery (see for instance [41]) we first notice that P has discrete spectrum in a neighborhood of z0 and that P − z is a Fredholm operator of index 0 from D(P ) to L2 when z varies in a small neighborhood of z0 . On the other hand, Proposition 10.3.1 implies that P −z is injective and hence bijective k for z ≤ O(h ln h1 ) k+1 and we also get the corresponding bounds on the resolvent.   As for the proof of Theorem 10.1.1, we refer to [140]: It is based on a factorization with the help of Malgrange’s preparation theorem which allows for a reduction to a situation close to that of Theorem 10.1.2.

10.4

Examples

Consider P = −h2  + iV (x), V ∈ C ∞ (X; R),

(10.4.1)

where X is either a smooth compact manifold of dimension n or X = Rn . In the second case we assume that p = ξ 2 + iV (x) belongs to a symbol space S(m), where m ≥ 1 is an order function. If V ∈ Cb∞ (R2 ), then we can take m = 1 + ξ 2 , and if ∂ α V (x) = O((1 + |x|)2 ) for all α ∈ Nn and satisfies the ellipticity condition |V (x)| ≥ C −1 |x|2 for |x| ≥ C, for some constant C > 0, then we can take m = 1 + ξ 2 + x 2 . We have (p) = [0, ∞[ +iV (X). When X is compact ∞ (p) is empty, and when X = Rn , we have ∞ (p) = [0, ∞[ +i ∞ (V ), where ∞ (V ) is the set of accumulation points at infinity of V . Let z0 = x0 + iy0 ∈ ∂ (p) \ ∞ (p). • In the case x0 = 0 we see that Theorem 10.1.2 (2) is applicable with k = 2, provided that y0 is not a critical value of V .

10.4 Examples

209

• When x0 > 0, then y0 is either the maximum or the minimum value of V . Assume that V −1 (y0 ) is finite and that each element of that set is a non-degenerate maximum or minimum. Then Theorem 10.1.2 (2) is applicable to ±iP with k = 2. By allowing a more complicated behaviour of V near its extreme points, we can produce examples where the same result applies with k > 2. Next, consider the non-self-adjoint harmonic oscillator Q=−

d2 + iy 2 dy 2

(10.4.2)

on the real line, studied by Boulton [22] and Davies [34]. Introduce a large spectral √ parameter E = iλ + μ, where λ ( 1 and |μ| λ. The change of variables y = λx d2 2 permits us to identify Q with Q = λP , where P = −h2 dx 2 + ix and h = 1/λ → 0. μ Hence Q − E = λ(P − (1 + i λ )) and Theorem 10.1.2 (2) is applicable with k = 2. We conclude that (Q − E)−1 is well-defined and of polynomial growth in λ (which can be specified further), respectively O(λ−1 ) when 2 μ ≤ C1 (λ−1 ln λ) 3 λ

and

μ ≤ C1 λ

respectively,

for any fixed C1 > 0, i.e., when 1

2

μ ≤ C1 λ 3 (ln λ) 3

and

1

μ ≤ C1 λ 3

respectively.

(10.4.3)

Actually, we are here in dimension 1 with k = 2, so Theorem 6.1.4 gives an even more precise result. We refer to [137] for some comments about the Kramers–Fokker–Planck operator 1 P = hy · ∂x − V  (x) · h∂y + (y − h∂y ) · (y + h∂y ) 2

(10.4.4)

on R2n = Rnx × Rny , where V is smooth and real-valued. Notes This chapter gives bounds on the resolvent of an h-pseudodifferential operator P near the boundary of the range of the semi-classical principal symbol. We follow [140] quite closely; the results are the same but one essential step of the proof has been simplified. They partially extend those of [40]. They are more general than those in Chap. 6: the dimension can be arbitrary and we allow a higher order of degeneration. In the case {p, {p, p}} = 0 it would be interesting to know if one can get the same precision as in the one-dimensional case in Chap. 6.

From Resolvent Estimates to Semigroup Bounds

11.1

11

Introduction

In Chap. 10 we saw a concrete example of how to get resolvent bounds from semigroup bounds. Naturally, one can go in the opposite direction and in this chapter we discuss some abstract results of that type, including the Hille–Yoshida and Gearhardt–Prüss–Hwang– Greiner theorems. As for the latter, we also give a result of Helffer and the author [62] that provides a more precise bound on the semigroup. We refer to [62] for the discussion of some examples.

11.2

General Results

We start by recalling some general results and here we follow [43]. Let B be a complex Banach space. Definition 11.2.1 A map [0, +∞[  t → T (y) ∈ L(B, B) is a strongly continuous semigroup if ⎧ ⎨T (t + s) = T (t)T (s), t, s ≥ 0, (11.2.1) ⎩T (0) = 1, and the orbit maps [0, +∞[  t → T (t)x ∈ B

(11.2.2)

are continuous for every x ∈ B. © Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_11

211

212

11 From Resolvent Estimates to Semigroup Bounds

Example 11.2.2 Let A ∈ Mat (n, n) (the space of complex n × n matrices) and put T (t) = exp(tA) : Cn → Cn . This is a uniformly continuous semigroup: T (t) − T (s) → 0, t → s for every s ≥ 0. Example 11.2.3 Translation semigroups: Let B = Lp ([0, +∞[ ), 1 ≤ p < +∞ and define T (t) : B → B by T (t)u(x) = u(x + t). Then T (t), 0 ≤ t < +∞ is a strongly continuous semigroup which is not uniformly continuous. Example 11.2.4 Let B = L2 (Rn ) and let T (t)u = U (t, x) ∈ C 1 ([0, +∞[) ; H 0 (Rn ), be the solution of the heat equation ∂t U (t, x) = x U (t, x), U (0, x) = u(x). Then T (t) is a strongly continuous semigroup and also a contraction semigroup in the sense that T (t) ≤ 1 for all t ≥ 0. Let (T (t))0≤t ω we have λ ∈ ρ(A) and (λ − A)−1  ≤

1 . λ − ω

Example 11.2.12 Let −A = −(h∂x )2 + ix 2 : L2 → L2 with domain B 2 defined in the beginning of Sect. 2.5. The numerical range of A is contained in the third quadrant and in particular in the half-plane λ < 0, so we know that (λ − A)−1  ≤ 1/λ when λ > 0. Consequently, A generates a contraction semigroup. The following theorem of Feller, Miyadera and Phillips (see e.g. [43, Theorem II.3.8]) characterizes generators of general strongly continuous semigroups: Theorem 11.2.13 Let ω ∈ R, M ≥ 1 and let A : B ⊃ D(A) → B be a linear operator. The following properties are equivalent: (a) A generates a strongly continuous semigroup T (t), t ≥ 0 which satisfies (P(M, ω)). (b) A is closed, densely defined. For every λ > ω we have λ ∈ ρ(A) and (λ − A)−n  ≤

M , ∀n ∈ N \ {0}. λ−ω

(c) A is closed, densely defined. For every λ ∈ C with λ > ω we have λ ∈ ρ(A) and (λ − A)−n  ≤

M , ∀n ∈ N \ {0}. λ − ω

A drawback with the Hille–Yoshida theorem is that it is essentially limited to the special case of contraction semigroups. The Feller–Miyadera–Phillips theorem gives a general characterization of generators of strongly continuous semigroups, but it may be less clear how to verify the conditions of that theorem in concrete applications.

11.3

The Gearhardt–Prüss–Hwang–Greiner Theorem

In this section we first recall the Gearhardt–Prüss–Hwang–Greiner theorem (see [43], Theorem V.I.11, [152], Theorem 19.1 as well as [158], [37]) and then give a variant with explicit bounds on the norm of the semigroup due to Helffer and the author [62], also published in [57]. The GPHG-theorem reads: Theorem 11.3.1 (a) Assume that B = H is a Hilbert space and that (z − A)−1  is uniformly bounded in the half-plane z ≥ ω. Then there exists a constant M > 0 such that P(M, ω) holds.

11.3 The Gearhardt–Prüss–Hwang–Greiner Theorem

215

(b) If P(M, ω) holds, then for every α > ω, (z − A)−1  is uniformly bounded in the half-plane z ≥ α. The part (b) follows from the more precise statement in Theorem 11.2.9. An idea in the proof is to use that the resolvent and the inhomogeneous equation (∂t − A)u = w in exponentially weighted spaces are related via Fourier–Laplace transform and we can use Plancherel’s formula. This is why we need to work in a Hilbert space. Variants of this simple idea have also been used in more concrete situations. See [25, 46, 66, 122]. Before stating the more explicit version, we make some simple remarks. We can improve slightly the conclusion of (a). If the assumption in (a) is true for some ω, then it is automatically true for some ω < ω: Lemma 11.3.2 If for some r(ω) > 0, (z − A)−1  ≤ ω ∈ ]ω − r(ω), ω] we have (z − A)−1  ≤

1 r(ω)

for z > ω, then for every

1 , z > ω . r(ω) − (ω − ω )

Proof If z ∈ C and z > ω , we take a sequence of numbers z0 ∈ C such that z0 = z and z0  ω, and the lemma follows from Proposition 2.1.6.   Remark 11.3.3 Let ω1 = inf{ω ∈ R; {z ∈ C; z > ω} ⊂ ρ(A) and sup (z − A)−1  < ∞}. z>ω

For ω > ω1 , we define r(ω) by 1 = sup (z − A)−1 . r(ω) z>ω Then r(ω) is an increasing function of ω; for every ω ∈ ]ω1 , ∞[, we have ω − r(ω) ≥ ω1 and for ω ∈ [ω − r(ω), ω] we have r(ω ) ≥ r(ω) − (ω − ω ). We may state all this more elegantly by saying that r is a Lipschitz function on ]ω1 , +∞[ satisfying 0≤

dr ≤ 1. dω

Moreover, if ω1 > −∞, then r(ω) → 0 when ω  ω1 .

216

11 From Resolvent Estimates to Semigroup Bounds

Remark 11.3.4 By Theorem 11.2.9, we already know that (z − A)−1  is uniformly bounded in the half-plane z ≥ β if β > ω0 , where ω0 is the growth bound for (T (t))0≤t 0 by 1 = sup (z − A)−1 . r(ω) z≥ω Let m(t) ≥ T (t) be a continuous positive function. Then for all t, a, a > 0, such that t = a + a , we have T (t) ≤

eωt r(ω) m1 e−ω· L2 ([0,a])  m1 e−ω· L2 ([0,a])

.

(11.3.2)

Here the norms are always the natural ones obtained from H, L2 , thus for instance T (t) = T (t)L(H,H) , if u is a function on R with values in C or in H, u denotes the natural L2 norm; when the norm is taken over a subset J of R, this is indicated with a “L2 (J )”. In (11.3.2) we also have the natural norm in the exponentially weighted space a instead of a; f e−ω· L2 ([0,a]) = eω· f (·)L2 ([0,a]). e−ω· L2 ([0, a]), and similarly with  Notice that we only need the bound m(t) for small t, say 0 ≤ t ≤ 1 and that we have (11.3.2) with m(t) replaced by m(t)1[0,1] (t) + ∞1]1,+∞[ . Thus from the given bound m(t) for small times and the resolvent bound, we get a global bound m (t) on T (t). We can then replace m(t) by min{m(t), m (t)} and reapply the theorem. It is an interesting

11.3 The Gearhardt–Prüss–Hwang–Greiner Theorem

217

problem to understand what would be the optimal bound that we can get from such an iteration. Some steps in that direction were taken in [62]. The following variant of the main result, obtained with Helffer in [62], could be useful in problems of return to equilibrium. Theorem 11.3.6 We make the assumptions of Theorem 11.3.5, so that (11.3.2) holds. Let  ω < ω and assume that A has no spectrum on the line z =  ω and that the spectrum of A in the half-plane z >  ω is compact (and included in the strip  ω < z < ω). Assume that ω } \ U , where U is any neighborhood (z − A)−1  is uniformly bounded on {z ∈ C; z ≥  ω}, and define r( ω) by of σ+ (A) := {z ∈ σ (A); z >  1 = sup (z − A)−1 . r( ω) z= ω Then for every t > 0, T (t) = T (t)+ + R(t) = T (t)+ + T (t)(1 − + ), where for all a, a > 0 with a +  a = t, R(t) ≤

ωt e

r( ω)1/me−ω· L2 ([0,a])1/me−ω· L2 ([0,a])

I − + .

(11.3.3)

Here + denotes the spectral projection associated to σ+ (A): 1 + = 2πi



(z − A)−1 dz, ∂V

where V is any compact neighborhood of σ+ (A) with C 1 boundary, disjoint from σ (A) \ σ+ (A). We refer to [62] for the proofs of the last two theorems, which are based on the study of the inhomogeneous equation (∂t − A)u = w on R

(11.3.4)

and exponentially weighted L2 spaces on the real line, as well as Laplace transforms. Notes In this chapter we have collected some results about semigroup bounds that follow from estimates on the resolvent in certain half-planes.

Counting Zeros of Holomorphic Functions

12.1

12

Introduction

In this chapter we will generalize Proposition 3.4.6 of Hager about counting the zeros of holomorphic functions of exponential growth. In [56] we obtained such a generalization, by weakening the regularity assumptions on the functions φ. However, due to some logarithmic losses, we were not quite able to recover Hager’s original result, and we still had a fixed domain  with smooth boundary. In many spectral problems, the domain should be allowed to depend on h and other parameters, it could be a long thin rectangle, and the boundary regularity should be relaxed. In this chapter, based on [141] we revisit systematically the proof of the counting proposition in [56] and obtain a general and quite natural result allowing an h-dependent exponent φ to be merely continuous and the h-dependent domain  to have Lipschitz boundary. The result generalizes the two earlier ones. By allowing suitable small changes of the points zj we also get rid of the logarithmic losses. Compared with the results in [141] we relax a subharmonicity assumption about the exponent in the exponential bounds. We next formulate the results. Let   C be an open set and γ = ∂ be the boundary of . Let r : γ →]0, ∞[ be a Lipschitz function of Lipschitz modulus ≤ 1/2: |r(x) − r(y)| ≤

1 |x − y|, x, y ∈ γ . 2

(12.1.1)

We further assume that γ is Lipschitz in the following precise sense, where r enters: There exists a constant C0 such that for every x ∈ γ there exist new affine coordinates y2 ) of the form  y = U (y − x), y ∈ C  R2 being the old coordinates, where  y = ( y1 ,  U = Ux is orthogonal, such that the intersection of  and the rectangle Rx := {y ∈

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_12

219

220

12 Counting Zeros of Holomorphic Functions

C; | y1 | < r(x), | y2 | < C0 r(x)} takes the form {y ∈ Rx ;  y2 > fx ( y1 ), | y1 | < r(x)},

(12.1.2)

where fx ( y1 ) is Lipschitz on [−r(x), r(x)], with Lipschitz modulus ≤ C0 (i.e., |fx (t) − fy (s)| ≤ C0 |t − s|). Notice that our assumption (12.1.2) remains valid if we decrease r. It will be convenient to extend the function r(x) to all of C, by putting 1 r(x) = inf (r(y) + |x − y|). y∈γ 2

(12.1.3)

The extended function is also Lipschitz with modulus ≤ 12 : 1 |x − y|, x, y ∈ C. 2

|r(x) − r(y)| ≤ Notice that r(x) ≥

1 dist (x, γ ), 2

(12.1.4)

where dist (x, y) = infy∈γ |x − y| denotes the distance from x to γ , and that |y − x| ≤ r(x) 2⇒

3r(x) r(x) ≤ r(y) ≤ . 2 2

(12.1.5)

For convenience, we shall also assume that  is simply connected. Before stating the general version of zero counting result, we review some facts about Radon measures. If  is open in Rn for some n ≥ 1, we say that μ is a (Radon) measure on  if it is a distribution on  of order 0. We will only consider real-valued measures i.e., μ, ψ ∈ R for every ψ ∈ C0 (). Thus for every K   there exists CK > 0 such that |μ, ψ| ≤ CK ψL∞ ,

(12.1.6)

for all ψ ∈ C0 () with support in K. If we have can find a constant C = CK which is independent of K, then we say that μ is a finite (Radon) measure. Notice that if μ is a measure, then its restriction to any open set    is finite. When μ is finite we let |μ|() denote the smallest K-independent constant in (12.1.6). We say that μ is positive and write μ ≥ 0 if μ, ψ ≥ 0 for all 0 ≤ ψ ∈ C0 (). We have the Jordan decomposition μ = μ+ −μ− of μ into the difference of two positive Radon measures which is minimal in the sense that if μ = ν+ − ν− with ν± ≥ 0, then μ± ≤ ν± . See [45, Corollary 2]. We write |μ| = μ+ + μ− . When ν ≥ 0 is a finite Radon measure we can define ν() = limν, ψj , where C0 (; [0, 1])  ψj # 1 , j → ∞. This is in agreement with the earlier definition

12.1 Introduction

221

of |μ|(). Also when μ is a finite Radon measure on , we define μ() =

lim

C0 (;[0,1])ψj #1

μ, ψj .

(12.1.7)

The main result of this chapter is: Theorem 12.1.1 Let   C be open, simply connected and have Lipschitz boundary γ with an associated Lipschitz weight r as in (12.1.1)–(12.1.3). Put ! γαr = x∈γ D(x, αr(x)) for any constant α > 0. Let zj0 ∈ γ , j ∈ Z/NZ be distributed  along the boundary in the positively oriented sense such that1 r(zj0 )/4 ≤ |zj0+1 − zj0 | ≤ r(zj0 )/2. Then there exists a positive constant C10 , depending only on the constant C0 in the assumption around (12.1.2), such that for every constant C1 ≥ C10 , there exists a constant C2 > 0 such that we have the following for any zj ∈ D(zj0 , r(zj0 )/(2C1 )): Let 0 < h ≤ 1 and let φ be a continuous function defined on some open neighborhood of the closure of  γr such that μ := φL(dz) is a finite Radon measure there, and denote by the same symbol a distribution extension to  ∪  γr . If u is a holomorphic function on ∪ γr satisfying h ln |u| ≤ φ(z) on  γr , h ln |u(zj )| ≥ φ(zj ) − j , for j = 1, 2, . . . , N,

(12.1.8) (12.1.9)

where j > 0, then the number of zeros of u in  satisfies    −1  #(u (0) ∩ ) − 1 μ() ≤   2πh ⎛ ⎞     N   |w − zj |  C2 ⎝  μ+ (dw) ⎠ . ln

j + |μ|( γr ) + r(z )  h r(zj )  D(zj , 4Cj ) j =1

(12.1.10)

1

Here μ() and |μ|( γr ) are well defined as in (12.1.7) (with smooth test functions equal to 1 away from a neighborhood of  γr in the case of μ). μ+ and |μ| are defined from the Jordan decomposition.

1 Here “4” can be replaced by any fixed constant >2.

222

12 Counting Zeros of Holomorphic Functions

   |w−z |  By observing that the average of ln r(zj )j  with respect to the Lebesgue measure r(z0 )

L(dzj ) over D(zj0 , 2Cj1 ) is O(1), we can get rid of the logarithmic terms in Theozj , and we get: rem 12.1.1, by making a suitable choice of zj =  Theorem 12.1.2 Let   C be simply connected and have Lipschitz boundary γ with an associated Lipschitz weight r as in (12.1.1)–(12.1.3). Let zj0 ∈ γ , j ∈ Z/NZ, be distributed along the boundary in the positively oriented sense such that (see Footnote 1) r(zj0 )/4 ≤ |zj0+1 − zj0 | ≤ r(zj0 )/2. Then there exists a positive constant C10 , depending only on the constant C0 in the assumption around (12.1.2), such that for every constant C1 ≥ C10 , there exists a constant C2 > 0 such that we have the following: Let 0 < h ≤ 1 and let φ be a continuous function defined on some neighborhood of the closure of  γr such that μ := φL(dz) is a finite Radon measure there and denote by the same symbol a distribution extension to  ∪  γr . Then there exist  zj ∈ D(zj0 , that if u is a holomorphic function on  ∪  γr satisfying h ln |u| ≤ φ(z) on  γr ,

r(zj0 ) 2C1 )

such

(12.1.11)

and zj ) − j , j = 1, 2, . . . , N, h ln |u( zj )| ≥ φ( instead of (12.1.9), then       −1  #(u (0) ∩ ) − 1 μ() ≤ C2 |μ|( γr ) +

j .   2πh h Of course, if we already know (in the subharmonic case) that       |w − zj |  ln  μ+ (dw) = O(1)μ+ D(zj , r(zj ) ) , r(z )  r(zj )  4C1 D(zj , 4Cj )

(12.1.12)

(12.1.13)

(12.1.14)

1

then we can keep  zj = zj in (12.1.10) and get (12.1.13). This is the case if we assume that μ+ is equivalent to the Lebesgue measure L(dω) in the following sense: μ+ (dw) r(z ) μ+ (D(zj , 4Cj1 ))

"

L(dw) r(z ) L(D(zj , 4Cj1 ))

on D(zj ,

r(zj ) ), 4C1

uniformly for j = 1, 2, . . . , N. Then we get,

(12.1.15)

12.2 Thin Neighborhoods of the Boundary and Weighted Estimates

223

Theorem 12.1.3 Under the conditions of Theorem 12.1.1 as well as the assumption (12.1.14) or the stronger assumption (12.1.15), conditions (12.1.8) and (12.1.9), imply (12.1.13). In particular, we recover Proposition 3.4.6, where , φ are independent of h, γ of class √ C ∞ and φ ∈ C 2 (neigh (γ )). Then |μ| ≤ O(1)L(dz) and it suffices to choose r = ,

j = and to notice that we can replace φ by φ + . The counting proposition in [56] can also be recovered. (In the subharmonic case we have μ = μ+ = |μ| near  γr .) A simple application will be given in Sect. 12.4. Other (for us) more important applications will be given in Chaps. 13, 14, 17. The outline of the chapter is the following: In Sect. 12.2 we consider thin neighborhoods of the boundary where the width is variable and determined by the function r. We verify that we can find such neighborhoods with smooth boundary and estimate the derivatives of the function that defines the boundary. Then we develop some exponentially weighted estimates for the Laplacian in such domains in the spirit of what can be done for the Schrödinger equation [59] and in a large number of works in thin domains, see for instance [21,49]. From that we also deduce pointwise estimates on the corresponding Green kernel. In Sect. 12.3 we prove the main results by following the general strategy of the proof of the corresponding result in [56] and carry out the averaging argument that leads to the elimination of the logarithms. In Sect. 12.4 we consider as a simple illustration the zeros of sums of exponentials of holomorphic functions. These results can also be obtained with more direct methods, cf. [15, 36, 72]. This chapter is a slightly improved version of [141], where a last section—not included here—establishes a connection with classical results on zeros of entire functions.

12.2

Thin Neighborhoods of the Boundary and Weighted Estimates

Let , γ = ∂, r be as in the introduction. We can find a smooth function  r (x) such that 1 r(x) ≤  r(x) ≤ r(x), C

|∇ r(x)| ≤

1 , 2

∂ α r(x) = O( r 1−|α| ),

(12.2.1)

where C > 0 is a universal constant. Indeed, let 0 ≤ χ ∈ C0∞ (D(0, 1)) satisfy  χ(z)L(dz) = 1 and put, for > 0 small enough and C1 > 0 large enough,  r(x) =

C1−1



 χ

 x−y ( r(y))−2 )r(y)L(dy).

r(y)

224

12 Counting Zeros of Holomorphic Functions

From now on, we replace r(x) by  r(x) and drop the tilde. Then (12.1.1), (12.1.2), (12.1.5) remain valid and (12.1.4) remains valid in the weakened form r(x) ≥

1 dist (x, γ ), C

(12.2.2)

where C > 0 is a new constant. Consider the signed distance to γ : ⎧ ⎨dist (x, γ ), if x ∈  g(x) = ⎩−dist (x, γ ), if x ∈ C \ 

(12.2.3)

Possibly after multiplying r by a small positive constant, we deduce from (12.1.2) that for every x ∈ γ there exists a normalized constant real vector field ν = νx (namely ∂ y1 , cf. (12.1.2)) such that ν(g) ≥ In the set

! x∈γ

1 in Rx , C

C = (1 + C02 )−1/2 .

(12.2.4)

Rx , we consider the regularized function  g (x) =

1 χ ( r(x))2



 x−y g(y)L(dy),

r(x)

(12.2.5)

 where 0 ≤ χ ∈ C0∞ (D(0, 1)), χ(x)L(dx) = 1. Here > 0 is small and we notice that r(x) " r(y), g(y) = O(r(y)), when χ((x − y)/( r(x)) = 0. It follows that g (x) = O(r(x)) and more precisely, since g is Lipschitz, that g (x) − g(x) = O( r(x)).

(12.2.6)

  −∇r(x) x − y g(y) χ L(dy)

r(x) r(x)

2 r(x)2    g(y) 1 x−y  x−y (−∇r(x)) L(dy), χ + · 2 2

r(x)

r(x)

r(x) r(x)

(12.2.7)

Differentiating (12.2.5), we get 

∇x g (x) = (∇x g) + 2

where (∇x g) is defined as in (12.2.5) with g replaced by ∇x g. It follows that ∇x g (x) − (∇g) (x) = O(1)

|g(y)| . y∈D(x, r(x)) r(x) sup

(12.2.8)

12.2 Thin Neighborhoods of the Boundary and Weighted Estimates

225

In particular, ∇x g = O(1) and with ν = νx : ν(g )(y) ≥

1 , when y ∈ Rx , and 2C

sup

|z−y|≤ r(y)

|g(z)|

r(y).

(12.2.9)

Differentiating (12.2.7) further, we get ∂ α g (x) = Oα (( r(x))1−|α|), |α| ≥ 1.

(12.2.10)

Let C > 0 be large enough but independent of . Put γ ,C r = {x ∈ 



Ry ; |g (x)| < C r(x)}.

(12.2.11)

y∈γ

If C > 0 is sufficiently large, then in the coordinates associated to (12.1.2),  γ ,C r takes the form fx− ( y1 ) <  y2 < fx+ ( y1 ), | y1 | < r(x),

(12.2.12)

where fx± are smooth on [−r(x), r(x)] and satisfy k ± 1−k ∂ ), k ≥ 1, y1 fx = Ok (( r(x))

(12.2.13)

0 < fx+ − fx , fx − fx− " C r(x).

(12.2.14)

Later, we will fix > 0 small enough and write γr =  γ ,C r and more generally, γαr =  γ ,C αr .

(12.2.15)

We shall next establish an exponentially weighted estimate for the Dirichlet Laplacian in γr : Proposition 12.2.1 Let C > 0 be sufficiently large and > 0 sufficiently small in the definition of γr in (12.2.15). Then there exists a new constant Cnew > 0 such that if ψ ∈ C 2 (γ r ) and |ψx | ≤

1 Cnew r

,

(12.2.16)

we have eψ Du +

1 1 ψ  e u ≤ Cnew reψ u, u ∈ (H01 ∩ H 2 )(γr ), Cnew r

(12.2.17)

226

12 Counting Zeros of Holomorphic Functions

where w denotes the L2 norm when the function w is scalar and we write (v|w) =

 

vj (x)wj (x)L(dx), v =



(v|v),

for Cn -valued functions with components in L2 . H01 and H 2 are the standard Sobolev spaces.2 Proof Let ψ ∈ C 2 (γ r ; R) and put −ψ = eψ ◦ (−) ◦ e−ψ = Dx2 − (ψx )2 + i(ψx ◦ Dx + Dx ◦ ψx ), where the last term is formally anti-self-adjoint. Then for every u ∈ (H 2 ∩ H01 )(γr ), (−ψ u|u) = Dx u2 − ((ψx )2 u|u).

(12.2.18)

We need an a priori estimate for Dx . Let v : γ r → Rn be sufficiently smooth. We sometimes consider v as a vector field. Then for u ∈ (H 2 ∩ H01 )(γr ): (Du|uv) − (uv|Du) = i(div (v)u|u). Assume div (v) > 0. If v = ∇w, then div (v) = w, and our assumption is then that w is strictly subharmonic. Then  div (v)|u|2 dx ≤ 2uvDu ≤ Du2 + uv2 , which we write  (div (v) − |v|2 )|u|2 dx ≤ Du2 . Using this in (12.2.18), we get 1 Du2 + 2

 

 1 (div (v) − |v|2 ) − (ψx )2 |u|2 dx 2

1 1 1 1 ≤  (−ψ )uku ≤  (−ψ )u2 + ku2 , k 2 k 2

2 In the following we simply write C instead of C new , following the general rule that constants change from time to time.

12.2 Thin Neighborhoods of the Boundary and Weighted Estimates

227

where k is any positive continuous function on γ r . We write this as 1 Du2 + 2

 

 1 1 1 2 2  2 (div (v) − |v| − k ) − (ψx ) |u|2 dx ≤  (−ψ )u2 . 2 2 k (12.2.19)

We shall see that we can choose v so that div (v) ≥ r −2 ,

|v| ≤ O(r −1 ).

(12.2.20)

After replacing v by C −1 v for a sufficiently large constant C, we achieve that div (v) − |v|2 " r −2 .

(12.2.21)

Before continuing, let us establish (12.2.20). Let g = g be the function in the definition of γr =  γ ,C r in (12.2.11), so that C −1 ≤ |∇g| ≤ 1 (with the new C independent of , C in (12.2.11)), ∂ α g = O (r(x)1−|α| ). Put v = ∇(eλg/r ), where λ > 0 will be sufficiently large. Notice that ∇

g  r

=

g∇r ∇g − 2 , r r

where    ∇g  1    r " r uniformly with respect to , and    g∇r  1 g1    r 2  = O(1) r r = O( ) r , in γr , so if we fix > 0 sufficiently small, then   g  1   " . ∇ r r We have λg

v = e r λ∇

g  r

,

λ |v| " eλO( ) , r

(12.2.22)

228

12 Counting Zeros of Holomorphic Functions

so the second part of (12.2.20) holds for every fixed value of λ. Further, div (v) = e

λg r

     g  g 2 2 . λ ∇  + λ r r

Here,   g 2 1   ∇  " 2, r r



g  r



1 =O 2 r

 ,

so if we fix λ large enough, we also get the first part of (12.2.20). If we choose k = (Cr)−1 for a sufficiently large constant C, we get from (12.2.21) and (12.2.16) that 1 (div (v) − |v|2 − k 2 ) − (ψx )2 " r −2 . 2 Thus, with a new sufficiently large constant C, (12.2.19) yields Du2 +

1 C

 γr

1 2 |u| dx ≤ Cr(−ψ )u2 , r2

(12.2.23)

1 1  u ≤ Cr(−ψ )u. C r

(12.2.24)

which we can also write as Du + Thus, Deψ u +

1 1 ψ  e u ≤ reψ u, C r

and by (12.2.16), eψ Du ≤ Deψ u + O(1/r)eψ u ≤ O(1)reψ u, so we get (12.2.17) with a new constant C.

 

If   C has smooth boundary, let G , P denote the Green and the Poisson kernels of , so that the Dirichlet problem, u = v, u|∂ = f,

u, v ∈ C ∞ (), f ∈ C ∞ (∂),

12.2 Thin Neighborhoods of the Boundary and Weighted Estimates

229

has the unique solution 



u(x) =

G (x, y)v(y)L(dy) + 

P (x, y)f (y)|dy|. ∂

Recall that −G ≥ 0, P ≥ 0. It is also clear that − G (x, y) ≤ C −

1 ln |x − y|, 2π

(12.2.25)

where C > 0 only depends on the diameter of . Indeed, let −G0 (x, y) denote the righthand side of (12.2.25) and choose C > 0 large enough so that −G0 ≥ 0 on  × . Then on the operator level, G v = G0 v − P (G0 v|∂ ), so that  G (x, y) = G0 (x, y) −

P (x, z)G0 (z, y)|dz|, ∂

and hence G ≥ G0 , −G ≤ −G0 . The same argument (replacing G0 by G  with  ⊃ ) shows that −G is an increasing function of :  1 ⊂ 2 2⇒ −G1 ≤ −G2 on 1 × 1 . We will also use the elementary scaling property: G (x/t, y/t) = Gt  (x, y), x, y ∈ t, t > 0.

(12.2.26)

Proposition 12.2.2 Under the same assumptions as in Proposition 12.2.1 there exists a (new) constant C > 0 such that we have − Gγr (x, y) ≤ C −

− Gγr (x, y) ≤ C exp(−

|x − y| r(y) 1 ln , when |x − y| ≤ , 2π r(y) C

1 C



πγ (x) πγ (y)

r(y) 1 |dt|), when |x − y| ≥ , r(t) C

(12.2.27)

(12.2.28)

where it is understood that the integral is evaluated along γ from πγ (y) ∈ γ to πγ (x) ∈ γ , where πγ (y), πγ (x) denote points in γ with |x − πγ (x)| = dist (x, γ ), |y − πγ (y)| = dist (y, γ ), and we choose these two points (when they are not uniquely defined) and the intermediate segment in such a way that the integral is as small as possible.

230

12 Counting Zeros of Holomorphic Functions

Proof Let y ∈ γr , and put t = r(y). Then we can find   C uniformly bounded (with respect to y) whose boundary is uniformly bounded in the C ∞ sense,3 such that γr coincides with y + t =: y in D(y, 2r(y)/C), y ⊂ D(y, 4r(y)/C) and r " r(y) in that disc. In view of (12.2.25) and (12.2.26) we see that −Gy (x, y) satisfies the upper bound in (12.2.27). r(y) ·−y Let χ = χ( x−y r(y) ) be a standard cutoff equal to 1 on D(y, C ) with supp χ( r(y) ) ⊂ D(y, 2r(y)/C), and write the identity:  Gγr (·, y) = χ

·−y r(y)



 Gy (·, y) − Gγr [, χ

 ·−y ]Gy (·, y). r(y)

(12.2.29)

Using that the non-negative function −Gy satisfies (12.2.27), we see that the L2 -norm of Gy (·, y) over the cutoff region (i.e., the support of the x-gradient of the cutoff) is O(r(y)). Gy is harmonic with boundary value 0 in a neighborhood of supp ∇χ. From standard estimates for elliptic boundary value problems, we conclude after scaling that the L2 -norm of ∇x Gy (x, y) over the same region is O(1). It follows that [, χ(

·−y 1 )]Gy (·, y) = O( ), r(y) r(y)

and hence, by applying (12.2.17) with ψ = 0 to u = Gγr [, χ(

·−y )]Gy (·, y), r(y)

we get 1 ·−y Gγ [, χ( )]Gy (·, y) = O(1), in L2 (γr ). r r r(y) ·−y ·−y )] the function Gγr [, χ( r(y) )]Gy (·, y) is harmonic on γr Away from supp [, χ( r(y) with boundary value zero and, appealing as above to apriori estimates for elliptic boundary ·−y value problems and scaling, we conclude that inside the region where χ( r(y) ) = 1, it is O(1). From (12.2.29) we then get the estimate (12.2.27). To get (12.2.28), we apply (12.2.17) and (12.2.16) to the second term in (12.2.29) with ψ(x) = ψ(x, y) equal to the distance from y to x with respect to the Lithner–  > 0 large enough. Then (12.2.16) holds, so (12.2.17)  −2 |dx|2 with C Agmon distance (Cr)

3 That is, given by an equation f (z) = 0, where f belongs to a bounded family of smooth real

functions with the property that f (z) = 0 2⇒ |∇f (z)| ≥ 1/O(1) uniformly for all f in the family.

12.2 Thin Neighborhoods of the Boundary and Weighted Estimates

231

applies with u = Gγr [, χ((· − y)/r(y))]#y (·, y), u = [, χ((· − y)/r(y))]#y (·, y). Noticing that ψ(x) ≥ O(1)−1



πγ (x) πγ (y)

we get for |x − y| ≥ r(y)/O(1):  r

−1

exp O(1)

−1



πγ (·) πγ (y)

1 |dt|, r(t)

 1 |dt| Gγr (·,y) ≤ O(1). r(t)

Using again that |u| ≤ O(r(y))u in a domain y , where u is harmonic, near a part of the boundary where u = 0, we get (12.2.28)   We will also need a lower bound on Gγr on suitable subsets of γr . For > 0 fixed and sufficiently small, we say that M  γr is an elementary piece of γr if • M ⊂ γ(1− 1 )r , cf. (12.2.15), C r(x) 1 ≤ ≤ C, x, y ∈ M, • C r(y)  where M  belongs to a bounded set of • there exists y ∈ M such that M = y + r(y)M, relatively compact subsets of C with smooth boundary. In the following, it will be tacitly understood that we choose our elementary pieces with  some uniform control (C fixed and uniform control on the M). Proposition 12.2.3 If M is an elementary piece in γr , then − Gγr (x, y) " 1 + | ln

|x − y| |, x, y ∈ M. r(y)

(12.2.30)

Proof We just outline the argument. First, by using arguments from the proof of Proposition 12.2.2 (without any exponential weights), we see that − Gγr (x, y) " − ln

|x − y| |x − y| , when x, y ∈ M, r(y) r(y)

1.

(12.2.31)

 then from Next, if M  is a slightly larger elementary piece of the form y + (1 + C1 )r(y)M, 1  r(y)), Harnack’s inequality for the positive harmonic function −Gγr (·, y) on M \D(y, 2C 1 we see that −Gγr (x, y) " 1 in M \ D(y, C r(y)), which together with (12.2.31) gives (12.2.30).  

232

12.3

12 Counting Zeros of Holomorphic Functions

Distribution of Zeros

Let φ be a real C 2 function defined in some neighborhood of γr . Put μ = μφ = φL(dz).

(12.3.1)

Let u be a holomorphic function defined in a neighborhood of  ∪ γr . We assume that h ln |u(z)| ≤ φ(z), z ∈ γr .

(12.3.2)

Lemma 12.3.1 Let z0 ∈ M, where M is an elementary piece, such that h ln |u(z0 )| ≥ φ(z0 ) − , 0 <

1.

(12.3.3)

   −Gγr (z0 , w)μ(dw) .

+

(12.3.4)

Then the number of zeros of u in M is ≤

C h

γr

Here C does not depend on φ, . Since φ is of class C 2 , we have μ(dw) = φ(w)L(dw), φ ∈ C 0 and 

 γr

Gγr (z0 , w)μ(dw) = −φ(z0 ) +

∂γr

Pγr (z0 , w)φ(w)|dw|,

(12.3.5)

where P is the Poisson kernel. Proof of Lemma 12.3.1 Let nu (dz) =



2πδ(z − zj ),

where zj are the zeros of u counted with their multiplicity. We may assume that no zj is situated on ∂γr . Then, since  ln |u| = nu , h ln |u(z)| = 

 Gγr (z, w)hnu (dw) +

γr

Pγr (z, w)h ln |u(w)||dw| ∂γr





(12.3.6)

Gγr (z, w)hnu (dw) +

≤ γr

 =

γr

Pγr (z, w)φ(w)|dw| ∂γr

Gγr (z, w)hnu (dw) + φ(z) −

 γr

Gγr (z, w)μ(dw).

12.3 Distribution of Zeros

233

Putting z = z0 in (12.3.6) and using (12.3.3), we get 

 −Gγr (z0 , w)hnu (dw) ≤ + γr

−Gγr (z0 , w)μ(dw). γr

Now by (12.2.30), −Gγr (z0 , w) ≥

1 , w ∈ M, C  

and we get (12.3.4).

Notice that this argument is basically the same as when using Jensen’s formula to estimate the number of zeros of a holomorphic function in a disc and was used in the proof of Proposition 3.4.6. Let zj0 , zj be as in Theorem 12.1.1 and keep the assumption that φ is of class C 2 near γ r . We may arrange so that  γr/C1 ⊂ γr ⊂  γr . In particular, the assumptions of Theorem 12.1.1 imply (12.3.2). Now we sharpen the assumption (12.3.3) and assume, as in Theorem 12.1.1, that h ln |u(zj )| ≥ φ(zj ) − j .

(12.3.7)

Let Mj ⊂ γr be elementary pieces such that r(zj ) when k = j, C  1 γ r ⊂ Mj ,  r = (1 − )r,  C

zj ∈ Mj , dist (zj , Mk ) ≥

(12.3.8)

j

 ( 1. Recall that γr =  γ ,C r , where C, are now fixed (cf. (12.2.11)), and that where C γ ,αC r . γαr =  According to Lemma 12.3.1, we have #(u

−1

C3 (0) ∩ Mj ) ≤ h

  

j + −Gγr (zj , w)μ(dw) .

(12.3.9)

γr

Consider the harmonic functions on γ r ,   #(z) = h ln |u(z)| + γ r

 −Gγr (z, w)nu (dw) ,

(12.3.10)

 !(z) = φ(z) + γ r

−Gγr (z, w)μ(dw).

(12.3.11)

234

12 Counting Zeros of Holomorphic Functions

Then !(z) = φ(z) on ∂γ r . Similarly, #(z) ≥ h ln |u(z)| on γ r , with equality on ∂γ r . Consider the harmonic function H (z) = !(z) − #(z), z ∈ γ r .

(12.3.12)

Then on ∂γ r , we have, by (12.3.2), that H (z) = φ(z) − h ln |u(z)| ≥ 0, so by the maximum principle, H (z) ≥ 0, on γ r .

(12.3.13)

By (12.3.7), we have H (zj ) = !(zj ) − #(zj ) = φ(zj ) − h ln |u(zj )|   + −Gγr (zj , w)μ(dw) − γ r



≤ j + γ r

γ r

−Gγr (zj , w)hnu (dw)

(12.3.14)

−Gγr (zj , w)μ(dw).

Harnack’s inequality4 implies that  H (z) ≤ O(1)( j +

−Gγr (zj , w)μ(dw)) on Mj ∩ γ r ,  r = (1 −

1 ) r.  C

(12.3.15)

Now assume that u extends to a holomorphic function in a neighborhood of  ∪ γr . We want to evaluate the number of zeros of u in . Using (12.3.9), we first have #(u−1 (0) ∩ γ r ) ≤

  N  C −Gγr (zj , w)μ(dw) .

j + h γr

(12.3.16)

j =1

4 Which says that if   C is a connected open set with smooth boundary and K ⊂  a compact

subset, then there exists a constant C = C,K > 0 such that u(z1 ) ≤ Cu(z2 ) for all z1 , z2 ∈ K and every non-negative harmonic function u on , uniformly if  varies in a family of uniformly bounded subsets of C and dist (K, ∂) ≥ 1/O(1) uniformly, see e.g., [4, Section 6.3].

12.3 Distribution of Zeros

235

Let χ ∈ C0∞ ( ∪ γ r ; [0, 1]) be equal to 1 on . Of course, χ will have to depend on r, but we may assume that for all k ∈ N, ∇ k χ = O(r −k ). We are interested in   χ(z)hnu (dz) =

(12.3.17)

h ln |u(z)|χ(z)L(dz).

(12.3.18)

γ r

On γ r we have  h ln |u(z)| = #(z) − γ r

−Gγr (z, w)hnu (dw) 

= !(z) − H (z) − γ r

 = φ(z) + γ r

−Gγr (z, w)hnu (dw) 

−Gγr (z, w)μ(dw) − H (z) −

γ r

−Gγr (z, w)hnu (dw)

=: φ(z) + R(z), (12.3.19) where the last equality defines R(z). Inserting this in (12.3.18), we get 

 χ(z)hnu (dz) =

 χ(z)μ(dz) +

(12.3.20)

R(z)χ(z)L(dz).

(Here we also used some extension of φ to  with μ = φ.) The task is now to estimate R(z) and the corresponding integral in (12.3.20). Put μj = μ+ (Mj ∩ γ r ),

(12.3.21)

where μ = μ+ − μ− is the Radon decomposition of μ, here with μ± = m± L(dz), 0 ≤ m± ∈ C 0 , m+ m− = 0. Using the exponential decay property (12.2.28) (equally valid for Gγr ) we get for z ∈ Mj ∩ γ r , dist (z, ∂Mj ) ≥ r(zj )/O(1):    

γ r

   −Gγr (z, w)μ(dw) ≤

Mj ∩γ r

−Gγr (z, w)μ+ (dw) + O(1)



μk e

− C1 |j −k| 0

,

k =j

(12.3.22)

236

12 Counting Zeros of Holomorphic Functions

where |j − k| denotes the natural distance from j to k in Z/NZ and C0 > 0. Similarly, from (12.3.15) we get ⎛ H (z) ≤ O(1) ⎝ j +

 Mj ∩γ r

−Gγr (zj , w)μ+ (dw) +



⎞ e

− C1 0

|j −k|

μk ⎠ ,

(12.3.23)

k =j

for z ∈ Mj ∩ γ r . This gives the following estimate on the contribution from the first two terms in R(z) to the last integral in (12.3.20), where we also use that χ = O(r −2 ): 

 γ r

γ r

= O(1)

 −Gγr (z, w)μ(dw) − H (z) χ(z)L(dz)



 ( j +

Mj ∩γ r

j

+ O(1)

−Gγr (zj , w)μ+ (dw)) +



e

− C1 |j −k| 0

μk )

k =j

 Mj ∩γ r Mj ∩γ r

j



(12.3.24)

−Gγr (z, w)μ+ (dw)|χ(z)|L(dz).

Here by (12.2.30),  Mj ∩γ r

−Gγr (z, w)|χ(z)|L(dz) = O(1),

(12.3.25)

so (12.3.24) leads to 



 γ r

γ r

−Gγr (z, w)μ(dw) − H (z) χ(z)L(dz)

⎛

= O(1) ⎝μ+ (γ r ) +

 j

j +

⎞

 j

Mj ∩γ r

(12.3.26)

−Gγr (zj , w)μ+ (dw)⎠ .

The contribution from the last term in R(z) (in (12.3.19)) to the last integral in (12.3.20) is 

 z∈γ r w∈γ r

Gγr (z, w)hnu (dw)χ(z)L(dz).

(12.3.27)

Here, by using an estimate similar to (12.3.22) with μ(dw) replaced by L(dz), together with (12.3.25), we get  z∈γ r

Gγr (z, w)(χ)(z)L(dz) = O(1),

12.3 Distribution of Zeros

237

so the expression (12.3.27) is by (12.3.16) equal to O(h)#(u−1 (0) ∩ γ r )   N   = O(1) −Gγr (zj , w)μ+ (dw)

j + γr

j =1

⎛

= O(1) ⎝μ+ (γr ) +

N 



⎞



j + Mj

j =1

(12.3.28)

−Gγr (zj , w)μ+ (dw) ⎠ .

This is quite similar to (12.3.26). Using Proposition 12.2.2, we have  Mj ∩γ r

−Gγr (zj , w)μ+ (dw) ≤



O(1)

|w−zj |≤

      − w| |z j  μ+ (dw) + μ+ (Mj ∩ γ r ) ln  r(z ) 

r(zj ) C

j

and similarly for the last integral in (12.3.28). Using all this in (12.3.20), we get 

 χ(z)hnu (dz) = ⎛

+ O(1) ⎝μ+ (γr ) +

χ(z)μ(dz)  j





j +

|w−zj |≤r(zj )/C

⎞      − w| |z j ln  μ+ (dw) ⎠ .  r(z ) 

(12.3.29)

j

Now we observe that     −1  #(u (0) ∩ ) − 1 χ(z)hnu (dz) ≤ #(u−1 (0) ∩ γ r ),  2πh which can be estimated by means of (12.3.28). Combining this with (12.3.29) and the fact that      μ() − χ(z)μ(dz) ≤ |μ|(γr ), |μ| = μ+ + μ− ,   we get 1 μ()| ≤ |#(u−1 (0) ∩ ) − 2πh ⎛   O(1) ⎝|μ|(γr ) + ( j + r(z ) h |w−zj |≤ Cj j

⎞      − w| |z j  μ+ (dw))⎠ . ln  r(zj ) 

(12.3.30)

238

12 Counting Zeros of Holomorphic Functions

This completes the proof of Theorem 12.1.1 in the case when φ is of class C 2 near γ r . Now we remove the C 2 assumption and take φ continuous as in Theorem 12.1.1 As there, we let μ = φL(dz). Let 0 ≤ χ ∈ C0∞ (D(0, 1)) be a radial function with  1. Then φ := χ ∗ φ χ(z)L(dz) = 1, and put χ = −2 χ(·/ ) for 0 <

(convolution of χ and φ) and μ := χ ∗ μ = (φ )L(dz), are well defined in slightly shrinked domains, and hence the C 2 special case of the theorem is applicable. If u is as in the theorem, we have (cf. (12.1.8) and (12.1.9)): γr , h ln |u| ≤ φ (z) + min j on  h ln |u(zj )| ≥ φ (zj ) − 2 j , j = 1, 2, . . . , N, when > 0 is small enough, since φ → φ uniformly, when → 0. We can then apply the C 2 -version of the theorem, and replace r with r/2 in the conclusion (12.1.10), and get 1 μ ()| ≤ 2πh        N   |w − zj |  C2  γr/2) + .

j + |μ |( μ ,+ (dw) ln r(z )  h r(zj )  D(zj , 8Cj )

|#(u−1 (0) ∩ ) −

1

(12.3.31)

1

Since χ is radial and hence even, we have  μ () =

1 ∗ χ (z)μ(dz)

when is small enough. Also,  μ() = lim

j →∞

χj (z)μ(dz),

where 0 ≤ χj ∈ C0∞ () is an increasing sequence of functions, equal to 1 away from a small neighborhood of γ , that converges to 1 . It follows that |μ() − μ ()| ≤ 2|μ|( γr ),

(12.3.32)

when is small enough. Let us next look at the right-hand side in (12.3.31). From the Jordan decomposition μ = μ+ − μ− , we get μ = χ ∗ μ+ − χ ∗ μ− , χ ∗ μ± ≥ 0, in a slightly shrinked neighborhood of γ r . By the minimality of the Jordan decomposition, we have μ ,± ≤ χ ∗ μ± and hence |μ | ≤ χ ∗ |μ|. When is small enough, we get |μ |( γr/2) ≤ |μ|( γr )|.

12.3 Distribution of Zeros

239

The integral in the general term in the sum in (12.3.31) is     |w − zj |   ≤  (χ ∗ μ+ )(dw) ln rj D(zj ,rj /(8C1 ))       |w − zj |   = χ ∗ 1D(zj ,rj /(8C1 )) (w) ln  μ+ (dw) rj     | · −zj |   ≤ χ ∗  ln  (w)μ+ (dw), rj D(zj ,rj /(4C1 )) 

(12.3.33)

where rj := r(zj ) and we assume that is small enough. Notice that here we work in a region where ln(|w − zj |/rj ) < 0. By the spherical mean-value property for harmonic functions, we have χ ∗ ln(| · −zj |/rj )(w) = ln |w − zj |/rj when |w − zj | ≥ . Comparing w ln(|w − zj |/rj ) = 2πδzj , w (χ ∗ ln(| · −zj |/rj )) = 2πχ (· − zj ), in log-radial coordinates, t = ln | · −zj |, we see that the equality above extends to an inequality inside D(zj , )5 : 0 ≥ χ ∗ ln(| · −zj |/rj )(w) ≥ ln(|w − zj |/rj ), so the last integral in (12.3.33) can be estimated from above by  D(zj ,rj /(4C1 ))

| ln(|w − zj |/rj )|μ+ (dw).

(12.3.34)

Combining (12.3.32)–(12.3.34) with (12.3.31), we get (12.1.7) which completes the proof of Theorem 12.1.1.  

5 This also follows more directly from the spherical mean value inequality for subharmonic functions.

240

12 Counting Zeros of Holomorphic Functions

We next discuss when the contribution from the logarithmic integrals in (12.1.10) can be eliminated or simplified. Let r, C1 , zj0 be as in Theorem 12.1.1. We get 

 D zj0 ,

 ≤

r(z0 j) 2C1



D zj0 ,





r(z0 ) j 2C1

    |w − z|  L(dz)   ln    μ+ (dw)   r(z)  r(zj0 ) r(z) D z, 4C 0 1 L D zj , 2C1





 D zj0 ,





≤ O(1)μ+ D

zj0 ,

r(z0 ) j C1

    L(dz) |w − z|   μ+ (dw)    r(zj0 ) r(z)  0 L D zj , 2C1

 ln

r(zj0 )



C1

,

where we changed the order of integrations in the last step and also used that r(z) " r(zj0 ) in D(zj0 ,

r(zj0 ) C1 ).

We conclude that the mean value of  D

zj0 ,

r(zj0 ) 2C1



  z →

r(z) D(z, 4C )

     ln |w − z|  μ+ (dw)  r(z) 

1

  0 r(z0 ) 0 r(zj ) is O(1)μ+ D(zj , C1 ) . Thus we can find  zj ∈ D(zj0 , 2Cj1 ) such that  D( zj ,

r( zj ) 4C1 )

    zj |  ln |w − μ+ (dw) = O(1)μ+ (D(zj0 , r(zj0 )/C1 )).  r( z ) 

(12.3.35)

j

 

This establishes Theorem 12.1.2.

Remark 12.3.2 Under the assumptions leading to (12.3.35), the set j (C) of all  z ∈ 0 0 D(zj , r(zj )/(2C1 )) such that  D( zj ,

r( zj ) 4C1 )

   zj |   ln |w − μ+ (dw) ≥ Cμ+ (D(zj0 , r(zj0 )/C1 )),  r( z )  j

satisfies L(j (C)) L(D(zj0 , r(zj0 /(2C1 ))

→ 0, C → +∞.

12.4 Application to Sums of Exponential Functions

241

This makes it possible to find  zj ∈ D(zj0 , r(zj0 )/(2C1 )) simultaneously satisfying several (finally many) estimates of the form (12.3.35), for instance if we have several measures j μ+ , 1 ≤ N < ∞.

12.4

Application to Sums of Exponential Functions

Consider the function u(z; h) =

N 

eφj (z)/ h ,

(12.4.1)

1

where N is finite and φj are holomorphic in the open set  ⊂ C and independent of h for simplicity. Put ψj (z) = φj (z),

(12.4.2)

let    have C ∞ boundary γ and assume ∀x ∈ γ , #(x) := max ψj (x) is attained j

(12.4.3)

for at most two different values of j. If x ∈ γ , #(x) = ψj (x) = ψk (x), j = k, then ν(x, ∂x )(ψj (x) − ψk (x)) = 0,

(12.4.4)

where ν denotes the normalized tangent vector field (say positively oriented) to γ . We shall see that Theorem 12.1.2 allows us to determine the number of zeros of u in  up to O(1). This result can be further strengthened by using direct arguments (see for instance [72]), but the purpose of this section is simply to illustrate the results above. We also notice that the results will be valid if u is holomorphic in , but with the representation (12.4.1) and the φj defined only in a neighborhood of γ . Define # in  or more generally in a neighborhood of γ in  as in (12.4.3). Define the curves γj,k ⊂  near γ by γj,k = {x; #(x) = ψj (x) = ψk (x)}, j = k. We shall establish the following result (without any claim of novelty, see [72] as well as [15, 36]). For a closely related old result on entire functions, see [93], Chapter VI, Section 3, Theorem 9, attributed to Pfluger [110].

242

12 Counting Zeros of Holomorphic Functions

Proposition 12.4.1 We have      −1 #(u (0) ∩ ) − 1 #(z)L(dz) = O(1).  2πh 

(12.4.5)

Here, in the case when ψj and # are defined only in a neighborhood of γ , we take any distribution extension of # to a neighborhood of . Notice that near γ the function # is subharmonic and # is supported by the union of the curves γj,k . On each such curve, ∂ ∂ (ψj −ψk )|dz|, where n is the unit normal to γj,k , oriented so that ∂n (ψj −ψk ) > # = ∂n 0. This can be proved with the help of Green’s formula, cf. the end of the proof of the main result in [146]. We shall prove Proposition 12.4.1 by means of Theorem 12.1.2. Put !(z) = h ln

N 

 e

, z ∈ neigh (γ ),

ψj (z)/ h

(12.4.6)

1

so that !(z) = hf ( ψh1 , .., ψhN ), where  f (x) = ln

N 

 e

xj

(12.4.7)

.

1

If we define θj = exj /



exk , then θj > 0, θ1 + · · · + θN = 1, and ∂xj f (x) = θj ,

(12.4.8)

f  (x) = diag (θj ) − (θj θk )j,k .

(12.4.9)

For y ∈ RN , we have f  (x)y|y =



θj yj2 −



2 θj yj

,

which is ≥0, since the function t → t 2 is convex. Hence f is convex.  We apply this to !(z), now with θj = eψj (z)/ h / k eψk / h , and get ∂z !(z) = ∂z ∂z !(z) =



θj ∂z ψj ,

∂z =

1 1 (∂z + ∂z ) 2 i

 2   1  1    f ∂z ψ|∂z ψ = θj |∂z ψj |2 −  θj ∂z ψj  . h h

(12.4.10)

(12.4.11)

12.4 Application to Sums of Exponential Functions

243

In the last calculation, we also used that ψj are harmonic. It follows that ! is subharmonic near γ . Also notice that !(z) = O(h−1 ),

(12.4.12)

and that this estimate can be considerably improved away from the union of the γj,k : Assume for instance that in some open relatively compact subset, #(z) = ψ1 ≥ maxj =1 ψj + δ, where δ > 0. Then ! = h ln(eψ1 / h (1 + O(e−δ/ h ))) = ψ1 + O(he−δ/ h ).

(12.4.13)

θ1 = 1 + O(e−δ/ h ), θj = O(e−δ/ h ) for j = 1,

(12.4.14)

Further,

so ∂z ! = ∂z ψ1 + O(e−δ/ h ),

(12.4.15)

f  (eψ1 / h , . . . , eψN / h ) = O(e−δ/ h ), 1 ∂z ∂z ! = O( e−δ/ h ). h

(12.4.16)

We will always be able to express the final result in terms of the simpler function #: Lemma 12.4.2 We have 

 !L(dz) − 

#L(dz) = O(h).

(12.4.17)



Proof Using Green’s formula, the left-hand side of (12.4.17) can be written   γ

∂! ∂# − ∂n ∂n

 |dz|,

where n is the suitably oriented normal direction. It then suffices to apply (12.4.15), with ! ψ1 replaced by #, in the region where d(z) := dist (z, j =k γj,k ) ( h and use that the gradients of !, # are O(1).   We next notice that h ln |u(z; h)| ≤ !(z)

(12.4.18)

244

12 Counting Zeros of Holomorphic Functions

in a neighborhood of γ . On the other hand, for z near γ , d(z) ( h, we have h ln |u(z; h)| ≥ !(z) − O(h)e−d(z)/(Ch).

(12.4.19) −d(z0 )/(Ch)

j We can now apply Theorem 12.1.2 with r = Const. h, d(zj0 ) ≥ Ch, j = O(he  φ = !. In view of (12.4.12), (12.4.16), we see that μ( γr ) = O(h), j = O(h), so

#(u−1 (0) ∩ ) −

1 2πh

),

 !L(dz) = O(1), 

and we obtain Proposition 12.4.1 from Lemma 12.4.2.

 

Notes There has been a considerable activity in the study of the zero set of random holomorphic functions where the Edelman Kostlan formula has similarities with the above results (and the earlier ones by Hager and others mentioned above) and where further results have been obtained. See Sodin [149] Sodin and Tsirelson [150], Zrebiec [160], Shiffman et al. [125]. The two last papers deal with holomorphic functions of several variables and it would be of interest to see if our results also have extensions to that case. Our results are similar in spirit to classical results on zeros of entire functions, see Levin [93]. We have followed closely the paper [141] where φ was assumed to be subharmonic, which is certainly the main case of interest. In some applications however, the subharmonicity condition seems to be too restrictive and in this chapter we have made the effort to remove it.

Perturbations of Jordan Blocks

13.1

13

Introduction

In this chapter we shall study the spectrum of a random perturbation of the large Jordan block A0 , introduced in Sect. 2.4: ⎛ 0 ⎜ ⎜0 ⎜ ⎜0 A0 = ⎜ ⎜. ⎜ ⎜ ⎝0 0

10 01 00 . . 00 00

⎞ 0 ... 0 ⎟ 0 . . . 0⎟ ⎟ 1 . . . 0⎟ ⎟ : CN → C N . . . . . .⎟ ⎟ ⎟ 0 . . . 1⎠ 0 ... 0

• Zworski [162] noticed that for every z ∈ D(0, 1), there are associated exponentially accurate quasimodes when N → ∞. Hence the open unit disc is a region of spectral instability. • We have spectral stability (a good resolvent estimate) in C \ D(0, 1), since A0  = 1. • σ (A0 ) = {0}. Thus, if Aδ = A0 + δQ is a small (random) perturbation of A0 we expect the eigenvalues to move inside a small neighborhood of D(0, 1). In the special case when Qu = (u|e1 )eN , N where (ej )N 1 is the canonical basis in C , we have seen in Sect. 2.4 that the eigenvalues of Aδ are of the form δ 1/N e2πik/N , k ∈ Z/NZ,

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_13

245

246

13 Perturbations of Jordan Blocks

so if we fix 0 < δ 1 and let N → ∞, the spectrum “will converge to a uniform distribution on S 1 ”. Davies and Hager [38] studied random perturbations of A0 . They showed that with probability close to 1, most of the eigenvalues are close to a circle: Theorem 13.1.1 Let A = A0 + δQ, Q = (qj,k (ω)), where qj,k are independent random variables ∼ NC (0, 1). If 0 < δ ≤ N −7 , R = δ 1/N , σ > 0, then with probability ≥ 1 − 2N −2 , we have σ (Aδ ) ⊂ D(0, RN 3/N ) and #(σ (Aδ ) ∩ D(0, Re−σ )) ≤

4 2 + ln N. σ σ

The angular distribution was not treated in [38], except for more special perturbations, and the main purpose of this chapter is to do so following the general strategy of Hager’s thesis that we have already followed in Chap. 3. A result by Guionnet et al. [52] implies that when δ is bounded from above by N −κ−1/2 for some κ > 0 and from below by some negative power of N, then 1 N



δ(z − μ) → the uniform measure on S 1 ,

μ∈σ (Aδ )

weakly in probability. The main focus of the chapter is to get probabilistic statements about the distribution of eigenvalues near the critical radius R = δ 1/N when Q = (qj,k ), with qj,k ∼ NC (0, 1) independent. Then we know from Proposition 3.4.1 that QHS ≤ C1 N with probability 2 ≥ 1 − e−N . We distinguish between the cases of small and of larger perturbations. • Small perturbations: 5

e−CN ≤ δ ≤ N − 2 − 0 ,

(13.1.1)

where C, 0 > 0 are fixed. • Larger perturbations: 5

N 0 − 2 ≤ δ

3

N−2 .

The intermediate case can undoubtedly be treated along the lines of the case of larger perturbations. As an outgrowth of this chapter, Vogel and the author studied in [147] the expectation density of eigenvalues inside the critical disc, adapting some methods from Vogel [155] devoted to Hager’s operator, somewhat in the spirit of various works on zeros of random polynomials. This is a new and probably very rich area, but for the present book we finally

13.2 Main Results

247

preferred to limit the perspective to eigenvalue counting with probability. More recent results on large Toeplitz matrices can be found in [145, 146].

13.2

Main Results

Small Perturbations We now state the results and take Q = (qj,k ), with qj,k ∼ NC (0, 1) independent. Then we know from Proposition 3.4.1 (cf. Proposition 3.4.2) that QHS ≤ 2 C1 N with probability ≥ 1 − e−N . We distinguish between the cases of small and of larger perturbations. We first consider the case of small perturbations: 5

e−CN ≤ δ ≤ N − 2 − 0 ,

(13.2.1)

where C, 0 > 0 are fixed. Define a “spectral radius” R = Rσ > 0 to be the unique solution in ]0, N/(N + 1)[ of m(R) = 2C1 Nδ, where m(R) := R N (1 − R).

(13.2.2)

In the beginning of Sect. 13.8, we will see that m is increasing on [0,N/(N+1)], decreasing on [N/(N+1),1] with a maximum mmax = m(N/(N + 1)) = (1 + o(1))/(eN), N → ∞. Moreover, as we shall see in (13.8.3), e N (ln δ+ln N+O(1)) ≤ Rσ ≤ e N (ln δ+2 ln N+O(1)) ≤ e− N (( 0 + 2 ) ln N+O(1)) , 1

1

1

1

(13.2.3)

and with probability ≥ 1 − e−N , 2

σ (Aδ ) ⊂ D(0, Rσ ).

(13.2.4)

To see the last inclusion, we notice that by (2.4.4), (z − A0 )−1  ≤

1 1 − |z|N ≤ , N |z| (1 − |z|) m(|z|)

0 < |z| ≤

N , N +1

and that (z − A0 )−1  ≤

 m

1 N N+1

,

N ≤ |z| ≤ 1, N +1

so when Q ≤ C1 N we get for 0 < |z| ≤ N/(N + 1), δQ(z − A0 )−1  ≤ C1 Nδ/m(|z|),

248

13 Perturbations of Jordan Blocks

which is ≤1/2 for |z| > Rσ , since m is increasing on [0, N/(N + 1)]. The same holds for |z| ≥ N/(N + 1) by the maximum principle, and writing z − Aδ = (z − A0 )(1 + (z − A0 )−1 δQ) we get (13.2.4). Theorem 13.2.1 Let 0 ≤ θ1 < θ2 ≤ 2π, θ = θ2 − θ1 ,  = ]r− , r+ [ ei]θ1 ,θ2 [ , 0 < 1 < 1, 1 1 ≤ r− ≤ Rσ − , O(1) N With probability ≥ 1 − O(N)e−N

1

and Rσ +

1 ≤ r+ ≤ (1 + Rσ )/2. N

− e−N , we have 2

     r− #(σ (Aδ ) ∩ ) − θ N 1 −   2π N(1 − r− )    1 δN 2 N 1 −1 − 0 + +N ≤ O(θ )N Rσ − r− (1 − Rσ )2     N  N  Rσ − O(1) N 1 ln N 1 − O(1) +θ + O(1)N +θ . N r− Rσ

(13.2.5)

In view of (13.2.4), the choice of r+ ≥ Rσ is of no real interest (and the stronger lower bound for r+ , is merely a pointer to the proof, where we shall take r+ = (1 + Rσ )/2), and we could take r+ = Rσ . For (13.2.5) to be of interest when θ > 0 is fixed, we would like the right-hand side to be N. This is the case when Rσ −r− ( N 1 −1 , in particular when Rσ − r− ( N 2 −1 for some 2 > 1 . Then r− /(N(1 − r− )) = O(N − 1 ) in the left-hand side. Thus for any such 2 , we have a uniform angular distribution of the eigenvalues in a N 2 −1 -neighborhood of the circle |z| = Rσ . Choosing θ = 2π, we see that most of the N eigenvalues belong to such a neighborhood. Larger Perturbations Now assume, N 0 − 2 ≤ δ 5

N− 2 . 3

(13.2.6)

Then, when Q ≤ C1 N (which holds with probability ≥ 1 − e−N ) we have σ (Aδ ) ⊂ δ 1/2 N 1/4 , N −1 δ 1/2 N 1/4 . D(0, 1 + C1 δN) and we observe that δN 2

13.3 Study in the Region |z| > R

249 1

1

Theorem 13.2.2 Let 0 ≤ r− ≤ 1 − 4(C1 δ) 2 N 4 , r+ ≥ 1 + C1 δN. Let 0 < 1 < 1. Then

with probability ≥ 1 − O(N)e−N 1 , we have       r− #(σ (Aδ ) ∩ ) − θ 1 − N  ≤  2π N(1 − r− )   1

0 /2 δN 2 N 1 −1 − NO(1) + + O(θ )N e (1 − r− )2 1 − r−   1 1 + O(1)N N 1 −1 ln N + δ 2 N 4 . The right-hand side in (13.2.7) is Since 1 − r− ≥

(13.2.7)

N when 1 − r− ( δ 1/2 N 1/4 and 1 − r− ( N 1 −1 .

0 1 1 1 1 1 5 −1 1 δ2N 4 = δ2N 4N ≥ N 2 −1 , O(1) O(1) O(1)

the last requirement on 1 − r− follows from the first, when 0 /2 > 1 , i.e., when 1 is chosen small enough. Again we have uniform angular distribution of the eigenvalues in r− 1 for r− in the range in the theorem. r− ≤ |z| ≤ r+ , since we also have N(1−r −) While in Theorem 13.2.1 we get uniform distribution in shells of thickness close to 1/N around the critical circle |z| = Rσ , we now need thicker shells that increase with δ and now contain the unit circle. In the remainder of this chapter we prove the two theorems above. In Sect. 13.5, we also give an upper bound on the number of eigenvalues in discs inside the critical one.

13.3

Study in the Region |z| > R

Let F and R be as at the end of Sect. 2.4, satisfying (2.4.16). Recall that det(z − A0 ) = zN . We get the following upper bound: | det(z − Aδ )| = | det(z − A0 )|| det(1 − (z − A0 )−1 δQ)| ≤ |z|N exp (z − A0 )−1 δQtr ≤ |z|N exp (F (|z|)δQtr )) . For the first inequality above we refer to [48] or Sect. 8.4. Using that by (2.4.13), (z − Aδ )−1  ≤ F (|z|)(1 − F (|z|)δQ)−1 ,

250

13 Perturbations of Jordan Blocks

when F (|z|)δQ < 1 (i.e., when |z| > R) we can permute the roles of A0 and Aδ , z − A0 = (z − Aδ )(1 + δ(z − Aδ )−1 Q), and get a lower bound: 

 F (|z|)δQtr |z| = | det(z − A0 )| ≤ | det(z − Aδ )| exp . (1 − F (|z|)δQ) N

In conclusion, under the assumption that |z| > R we have   F (|z|)δQtr ≤ | det(z − Aδ )| ≤ |z|N exp(F (|z|)δQtr ). |z| exp − 1 − F (|z|)δQ N

(13.3.1)

Later, we shall impose the condition (13.7.22), saying that QHS ≤ C1 N for a certain constant C1 > 0. Then by the Cauchy–Schwarz inequality for the singular values of Q we know that Qtr ≤ C1 N 3/2 and (13.3.1) tells us that     F (|z|)δC1 N 3/2 |z| exp − ≤ | det(z − Aδ )| ≤ |z|N exp F (|z|)δC1 N 3/2 . 1 − F (|z|)δQ (13.3.2) N

Here and in (13.3.1), the second inequality does not require the assumption that |z| > R.

13.4

Study in the Region |z| < 1

Following [148], we introduce an auxiliary Grushin problem. Define R+ : CN → C by R+ u = u1 , u = (u1 u2 . . . uN )t ∈ CN .

(13.4.1)

Let R− : C → CN be defined by R− u− = (0 0 . . . u− )t ∈ CN .

(13.4.2)

Here, we identify vectors in CN with column matrices. Then for |z| < 1, the operator  0 R+ A0 = A0 − z R−

 : CN+1 → C1+N

(13.4.3)

13.4 Study in the Region |z| < 1

251

is bijective. In fact, its matrix is lower triangular with the entry 1 everywhere on the main diagonal, the entry −z on the subdiagonal, and all other entries equal to 0. It has the inverse 

E0 :=

A−1 0

 0 E0 E+ =: , 0 E0 E−+ −

given by a lower triangular matrix with (E0 )j,k = zj −k for 1 ≤ k ≤ j ≤ N + 1. Then for 1 ≤ j ≤ N, 2 ≤ k ≤ N + 1, 0 = Ej,k

⎧ ⎨0,

if j < k, . ⎩zj −k , if k ≤ j

(13.4.4)

Moreover, ⎛ 0 E+

⎜ ⎜ =⎜ ⎜ ⎝

1 z .. .

⎞ ⎟   ⎟ ⎟ , E 0 = zN−1 zN−2 . . . 1 , − ⎟ ⎠

(13.4.5)

zN−1 0 E−+ = zN .

(13.4.6) 

A0 − z R− Notice that this is equivalent to the more traditional approach: the inverse of R+ 0   0 0 E E+ . is 0 0 E− E−+ As in Sect. 2.4, we see that 1

0 0  ≤ G(|z|) 2 , E−+  ≤ 1. E 0  ≤ G(|z|), E±



(13.4.7)

where  ·  denote the natural operator norms and  G(|z|) := min N,

1 1 − |z|

 " 1 + |z| + |z|2 + · · · + |z|N−1 .

(13.4.8)

The functions F (cf. (2.4.9)) and G are related by   1 1 F (R) = G , R ≥ 1, R R and we use this relation to extend the definition of G to ]0, +∞[.

(13.4.9)

252

13 Perturbations of Jordan Blocks

Next, consider the natural Grushin problem for Aδ . If δQG(|z|) < 1, we see that 

 0 R+ Aδ = Aδ − z R−

(13.4.10)

is bijective with inverse 

 δ Eδ E+ Eδ = , δ δ E−+ E− where E δ =E 0 − E 0 δQE 0 + E 0 (δQE 0 )2 − · · · = E 0 (1 + δQE 0 )−1 , δ 0 0 0 0 E+ =E+ − E 0 δQE+ + (E 0 δQ)2 E+ − · · · = (1 + E 0 δQ)−1 E+ , δ 0 0 0 0 =E− − E− δQE 0 + E− (δQE 0 )2 − · · · = E− (1 + δQE 0 )−1 , E−

(13.4.11)

δ 0 0 0 0 0 E−+ =E−+ − E− δQE+ + E− δQE 0 δQE+ − ··· 0 0 0 − E− δQ(1 + E 0 δQ)−1 E+ . = E−+

We get G(|z|) E  ≤ , 1 − δQG(|z|) δ

δ |E−+

0 − E−+ |

1

δ E± 

G(|z|) 2 , ≤ 1 − δQG(|z|)

(13.4.12)

δQG(|z|) . ≤ 1 − δQG(|z|)

Indicating derivatives with respect to δ with dots and omitting sometimes the super/subscript δ, we have 

 EQE EQE + ˙ =− E˙ = −E AE . E− QE+ E− QE

(13.4.13)

Integrating this from 0 to δ yields E δ − E 0  ≤

G(|z|)2 δQ , (1 − δQG(|z|))2

3

δ 0 E± − E± ≤

G(|z|) 2 δQ . (1 − δQG(|z|))2

(13.4.14)

13.5 Upper Bounds on the Number of Eigenvalues in the Interior

253

Notice that det A0 = 1. Combining (13.4.12) and (13.4.13), we get ˙ = |tr QE| ≤ Qtr E ≤ Qtr |∂δ ln det A| = |tr (AE)|

G(|z|) , 1 − δQG(|z|)

and integration from 0 to δ yields | ln | det Eδ || = | ln | det Aδ || ≤

G(|z|) δQtr . 1 − δQG(|z|)

(13.4.15)

We now sharpen the assumption that δQG(|z|) < 1 to δQG(|z|) < 1/2.

(13.4.16)

Then 1

δ E δ  ≤ 2G(|z|), E±  ≤ 2G(|z|) 2 , δ 0 |E−+ − E−+ | ≤ 2δQG(|z|).

(13.4.17)

Combining this with the identity E˙ −+ = −E− QE+ (cf. (13.4.13)) and (13.4.14), we get 0 0 E˙ −+ + E− QE+  ≤ 16G(|z|)2δQ2 ,

(13.4.18)

and after integration from 0 to δ, δ 0 0 0 E−+ = E−+ − δE− QE+ + O(1)G(|z|)2(δQ)2 .

(13.4.19)

Using (13.4.5) and (13.4.6) we get, with Q = (qj,k ), δ E−+

=z −δ N

N 

qj,k zN−j +k−1 + O(1)G(|z|)2 (δQ)2 ,

(13.4.20)

j,k=1

still under the assumption (13.4.16). More explicitly, the modulus of the remainder in (13.4.19) and (13.4.20) is bounded by 8G(|z|)2δ 2 Q2 .

13.5

Upper Bounds on the Number of Eigenvalues in the Interior

This section is somewhat independent of the main theme of the chapter, and we will use some estimates from Sects. 13.7 and 13.8.

254

13 Perturbations of Jordan Blocks

 Let Q = qj,k 1≤j,k≤N be a random matrix where the entries are independent random variables ∼ NC (0, 1) and recall Proposition 3.4.1, which implies that QHS ≤ C1 N, with probability ≥ 1 − e−N .

(13.5.1)

δC1 N ≤ 1/2.

(13.5.2)

2

We assume that

In this section we establish upper bounds on the number of eigenvalues in discs D(0, r) with r not too large. We work in a disc D(0, R0 ), where R0 ≤ 1 is the largest number such that δC1 NG(R0 ) ≤

1 . 2

(13.5.3)

Then with probability ≥ 1 − e−N , we have (13.4.16) for every z ∈ D(0, R0 ). We can then 2 apply (13.4.17) and see that with probability ≥ 1 − e−N , we have 2

δ |E−+ (z)| ≤ |z|N + 2δC1 NG(|z|), z ∈ D(0, R0 ).

(13.5.4)

With the same probability, we have (13.4.20) with the explicit bound 8G(|z|)2δ 2 (C1 N)2 on the modulus of the remainder term and we apply it for z = 0: δ (0) + δqN,1 | ≤ 8δ 2 (C1 N)2 . |E−+

(13.5.5)

Since qn,1 ∼ NC (0, 1), we know that |qn,1 | ≥ with probability ≥ e− for every fixed 2 2

≥ 0. Hence with probability ≥ e− − e−N , we have 2

δ (0)| ≥ δ − 8(C1 Nδ)2 = δ( − 8C12 δN 2 ). |E−+

(13.5.6)

For this to be of interest, we strengthen the assumption (13.5.2) (working in the limit N ( 1) to 16C12 δN 2 <

1,

(13.5.7)

and in particular, δ
0 we have M(r) ln

R 2 2 ≤ ln (8C1 NG(R)/ ) , with probability ≥ e− − e−N . r

(13.5.13)

Summing up, we have  Theorem 13.5.1 Let Aδ = A0 + δQ, where Q = qj,k 1≤j,k≤N and qj,k are independent random variables ∼ NC (0, 1). Let δ be in the range (13.5.12) and let R = Rδ be the unique solution in ]0, 1 − 1/N] of the equation R N = 2C1 δNG(R), so that (13.8.3) holds:    1 1 2 1 1 N N N N (2C1 δ) N . N (2C1 δ) ≤ Rσ ≤ 1 + O N Then for every (r, R, ) with 0 < r < R ≤ Rσ , > 0, we have (13.5.13), where M(r) is the number of eigenvalues of Aδ in D(0, r), counted with their multiplicities. We shall next weaken the assumption (13.5.12) to δ

1 . N

(13.5.14)

Then Lemma 13.7.2 can be applied with z = 0 and we see that if 0 < ≤ C for any fixed C > 0, we have δ |E−+ (0)| ≥

δ

2 and |Q| ≤ C1 N, with probability ≥ 1 − O( 2 ) − e−N . 2

(13.5.15)

Recall that R0 ≤ 1 is the largest number satisfying (13.5.3) and that (13.5.4) holds 2 with probability ≥ 1 − e−N . Jensen’s formula now gives (13.5.11) with probability 2 ≥ 1 − O( 2 ) − e−N for every fixed (r, R) with 0 < r < R < R0 . Recall that we have defined R = Rσ in [0, 1 − 1/N] as the unique solution in that interval of the

13.6 Gaussian Elimination and Determinants

257

equation R N = 2C1 δNG(R) when m(1 − 1/N) ≥ 2C1 δN (asymptotically equivalent to δ ≤ (2C1 eN 2 )−1 (1 + O(1/N)). When m(1 − 1/N) < 2C1 δN, we define Rσ = 1 − 1/N and notice that Rσ < R0 also in this case. Again, we have R N < 2C1 δNG(R) when R < Rσ . Theorem 13.5.2 Instead of (13.5.12), we assume (13.5.14) and N ( 1. When δ > (1 − 1/N)N /(2C1 N 2 ), we extend the definition of Rσ by putting Rσ = 1 − 1/N. Let C > 0 be fixed. Then for 0 < ≤ C, 0 < r < R ≤ Rσ , we have M(r) ln

13.6

R 2 ≤ ln (8C1 NG(R)/ ) , with probability ≥ 1 − O( 2 ) − e−N . r

(13.5.16)

Gaussian Elimination and Determinants

We review some standard material, see for instance [148] and Section 4 in [144]. Let Hj , Gj , j = 1, 2 be complex separable Hilbert spaces. Consider a bounded linear operator   A11 A12 A= : H1 × H2 → G1 × G2 . A21 A22

(13.6.1)

When A is bijective (with bounded inverse) we denote its inverse by  E11 E12 . =E = E21 E22 

−1

A

(13.6.2)

Proposition 13.6.1 (1) Assume that A11 is bijective. Then by Gaussian elimination we have the standard factorization into lower and upper triangular matrices:   A12 1 A−1 A11 0 11 . A= 0 A22 − A21 A−1 A21 1 11 A12 

(13.6.3)

The first factor is bijective since A11 is, so the bijectivity of A is equivalent to that of the second factor, which in turn is equivalent to that of A22 − A21 A−1 11 A12 . When A is bijective, we have the formula, −1

A

     0 A−1 1 a E11 E12 11 =: E, = =: −1 E21 E22 b 1 0 (A22 − A21 A−1 11 A12 )

(13.6.4)

258

13 Perturbations of Jordan Blocks −1 −1 −1 where a = −A−1 11 A12 (A22 − A21 A11 A12 ) , b = −A21 A11 and in particular, −1 E22 = (A22 − A21 A−1 11 A12 ) .

(13.6.5)

(2) Now assume that A is bijective. Then A11 is bijective precisely when E22 is, and when that bijectivity holds we have −1 = A22 − A21 A−1 E22 11 A12 , −1 A−1 11 = E11 − E12 E22 E21 .

(13.6.6)

The first statement is clear. The second statement is also quite simple to verify, by solving for x1 in the first equation in the system ⎧ ⎨A x + A x = y , 11 1 12 2 1 ⎩A21 x1 + A22 x2 = y2 , and substituting the result in the second equation. Let now H1 = G1 , H2 = G2 be of finite dimension and assume that A is bijective. From (13.6.4) and (13.6.5) we get det A−1 = (det E22 ) det A−1 11 , provided that A11 is bijective. This can be written (det A)(det E22 ) = det A11 .

(13.6.7)

This identity extends to the case when A11 is not necessarily bijective, since det E22 = 0 iff det A11 = 0.   Aδ − z R− with inverse We apply this to A = Aδ in (13.4.10), or rather to R+ 0   δ E δ E+ δ , −A = z − A , under the assumption (13.4.16). Noticing that E22 = E−+ 11 δ δ Eδ E− −+ in this case and recalling (13.4.15), one obtains   ln | det(z − Aδ )| − ln |E δ | ≤ 2G(|z|)δQtr . −+

(13.6.8)

13.7 Random Perturbations and Determinants

13.7

259

Random Perturbations and Determinants

Here to some extent we follow arguments of Vogel [155], see also [156]. We now let Q = (qj,k (ω))N j,k=1 , where qj,k are independent random variables with the law NC (0, 1). According to Proposition 3.4.1 we have 

P(Q2HS

C0 2 x N − ≥ x) ≤ exp 2 2



and hence if C1 > 0 is large enough, Q2HS ≤ C12 N 2 , with probability ≥ 1 − e−N . 2

(13.7.1)

In particular (13.7.1) holds for the ordinary operator norm of Q. We continue to assume that |z| < 1 until further notice. We choose δ ≥ 0 so that δC1 NG(|z|)
s with probability ≥ 1 − (1 − |z|)2 e2(s−ln δ) − e−N . 2

(13.8.20)

Notice that |z|N ≤ 2C1 G(|z|)δ, for |z| ≤ Rσ . Here we have to reconciliate the wishes to have a large lower bound and a probability very close to 1. Recalling the upper bound (13.8.4) on δ with the corresponding fixed parameter 0 > 0, we fix a new parameter 1 and choose

1 with 0 < 1 s = ln δ − N 1 .

(13.8.21)

We get from (13.8.20) that U (z) ≥ ln δ − N 1 with probability ≥ 1 − e−2N − e−N .

1

2

(13.8.22)

This estimate can be written U (z) ≥ N (h(|z|) − (|z|)) , where

(|z|) =

  1 1 2C1 N rN 2C1 δN 2 + ln + + N 1 −1 . 1−r N 1−r δ

Now, restrict r to the region r ≤ Rσ −

1 . N

(13.8.23)

272

13 Perturbations of Jordan Blocks

Then 2C1 N/(1 − r) dominates over r N /δ by a factor close to e or larger, so 

rN 2C1 N + ln 1−r δ

 " ln

N ≤ O(ln N), 1−r

and we get 1

2C1 δN 2 O(N 1 )

(r) ≤ + . 1−r N

(13.8.24)

Summing up, we have Proposition 13.8.4 For each z ∈ D(0, Rσ − 1/N), we have U (z) ≥ N(h(|z|) − (r)), with probability ≥ 1 − e−2N

1

− e−N , 2

(13.8.25)

where (r) > 0 is a function satisfying (13.8.24). We also want to study (h(|z|)) = (r −2 (r∂r )2 h)(|z|) and more precisely its integral over annuli centered at z = 0. We have  r2  (h(|z|))L(dz) = 2π ∂r r∂r h(r)dr = 2π [r∂r h(r)]rr21 , (13.8.26) r1 0. Also recall that since (13.9.2) we have simplified the notations by writing δ for 2C1 δ. We now reinstate the original δ. Let us review the estimates in Theorem 13.7.4 that will be used: 2 With probability ≥ 1 − e−N we have the interior upper bound (13.7.26):   2C1 δN 3/2 2C1 δN + =: Nf (r), ln | det(z − Aδ )| ≤ ln r N + 1−r 1−r when r < 1,

C1 δN 1−r

(13.9.31)

< 12 . Here and below, we frequently write r = |z|.

With probability ≥ 1 − e−N , we have the exterior upper bound (13.7.24): 2

ln | det(z − Aδ )| ≤ N ln r +

2C1 δN 3/2 =: Ng(r), r −1

(13.9.32)

when r > 1. Recall that f = f1 , g = g1 in the notation of the beginning of this section. Let h(r) = h1 (r) be the largest subharmonic function (of z) such that h(r) ≤ f (r), 0 < r < 1,

h(r) ≤ g(r), r > 1.

If we identify functions of r with functions of t via the substitution r = et , Proposition 13.9.2 gives,

h(r) =

⎧ ⎨f (r)

for r ≤ 1 − (1 + o(1))(4C1δ)1/2 N 1/4 ,

⎩g(r)

for r ≥ 1 + (1 + o(1))(4C1δ)1/2 N 1/4 ,

(13.9.33)

now with the original δ reinstated. Concerning (13.9.31), we notice that if r ≤ 1 − (1 + o(1))(4C1δ)1/2 N 1/4 , then by (13.9.5) 1 + o(1) C1 δN ≤ (C1 δ)1/2 N 3/4 < 1/2, 1−r 2 and hence (13.9.31) applies. Thus we can apply the maximum principle and conclude that 2 with probability ≥ 1 − e−N , we have ln | det(z − Aδ )| ≤ Nh(r), z ∈ C.

(13.9.34)

13.9 Eigenvalue Distribution for Larger Random Perturbations

287

With probability ≥ 1 − e−N the exterior lower bound in (13.7.25) 2

ln | det(z − Aδ )| ≥ Ng(r) − 6

C1 δN 3/2 , r −1

(13.9.35)

1 δN holds for r > 1 and Cr−1 < 12 . As before, the last condition on r is satisfied when 1/2 r ≥ 1 + (1 + o(1))(4C1δ) N 1/4 . We also recall the interior probabilistic lower bound (13.7.28), which says that if

|z| < 1 −

δ 1 , |z|N ≤ O(1) , N 1−r

t ≤ O(1)

δ δN , 1−r 1−r

1,

then ln |det(z − Aδ )| > ln t − O(1)

  t (1 − r) 2 δN 3/2 2 − e−N . , with probability ≥ 1 − O(1) 1−r δ

(13.9.36) −

We choose t = e−N 1 for some small fixed 1 . (This is almost the same choice as in the case of small random perturbations, the factor δ is now squeezed between two powers of N and is therefore superfluous.) Then we see that ln |det(z − Aδ )| > −N 1 − O(1)

(1 − r)2 −2N 1 δN 3/2 with probability ≥ 1 − O(1) e , 1−r δ2 (13.9.37)

for every fixed z with r < 1 − 1/N, r N ≤ O(δ)/(1 − r), provided that e−N 1 ≤ O(δ)/(1 − r), δN/(1 − r) 1. When r ≤ 1 − (1 + o(1))(4C1δ)1/2 N 1/4 , these conditions on r are fulfilled, so (13.9.37) applies. Recall that f = h precisely when t ≤ tmin (fs0 ). From (13.9.30) and (13.9.1), we get in that region, |f | ≤ O(δN 1/2 /(1 − r) + N 1 −1 ), so (13.9.37) implies that

ln | det(z − Aδ )| ≥ N(f (r) − (r)),

(13.9.38)

  δN 1/2

(r) ≤ O N 1 −1 + , r < 1, 1−r

(13.9.39)

where, as in (13.8.24),

288

13 Perturbations of Jordan Blocks

and we get Proposition 13.9.3 For every z ∈ D(0, 1 − (1 + o(1))(4C1δ)1/2 N 1/4 ), ln | det(z − Aδ )| ≥ N(h(r) − (r)),

(13.9.40)

with probability as in (13.9.37). Here (r) satisfies (13.9.39). We shall combine this with the upper bound (13.9.34) and the exterior lower bound (13.9.35), which gives ln | det(z − Aδ )| ≥ N(h(r) − (r)),

(13.9.41)

for r ≥ 1 + (1 + o(1))(4C1δ)1/2 N 1/4 , where C1 δN 1/2 , r > 1. (13.9.42) r −1  As in Sect. 13.8, we will study I (r1 , r2 ) = r1 . |z| −

The assumptions about P , and in particular the ellipticity assumption near infinity, imply that (P − z0 )−1 , Q(P − z0 )−1 and (P − z0 )−1 Q are compact when z0 ∈ σ (P ).1 For such a z0 (take for instance z0 away from the real axis) we write z − P = (z0 − P )(1 + (z0 − P )−1 (z − z0 − i Q)) = (1 + (z − z0 − i Q)(z0 − P )−1 )(z0 − P ). Here, (z0 − P )−1 (z − z0 − i Q), (z − z0 − i Q)(z0 − P )−1 are compact for z = z0 and since we can vary z0 , it follows from Fredholm theory that the spectrum of P is discrete. Proof of (14.1.5) This is the main part of the theorem and we shall follow the general strategy of [97] (see also [96, 134]) which consists in introducing a self-adjoint finite-rank perturbation of P which has no spectrum near ∂I . The use of the results in Chap. 12 will allow us to shorten the proof slightly. Without loss of generality we may assume that h ≤ , for if < h, we can write

Q = h(( / h)Q) and use h as a new perturbation parameter, since ( / h)Q ≤ 1. Let < δ 1. We define a δ-perturbation of P by moving the eigenvalues in ∂I + ] − δ, δ [ away from that set, by minimal displacements. Let λj , j ∈ N denote the eigenvalues of P repeated with their multiplicity and let ej , j ∈ N be a corresponding orthonormal basis of eigenvectors. Define,

 λj =

⎧  ⎪ ⎪ ⎨a point in ∂I + {−δ, δ} with dist (λj , ∂I + {−δ, δ}) = |λj − λj | ⎪ ⎪ ⎩

when λj ∈ ∂I + ] − δ, δ [,

λj when λj ∈ R \ (∂I + ] − δ, δ [ ).

We define our modified operator to be P δu =



 λj (u|ej )ej ,

(14.1.6)

where the spectral resolution of P is Pu =



λj (u|ej )ej .

1 (P − z )−1 Q is also compact: D(P ) → D(P ) since it is related to Q(P − z )−1 : L2 → L2 by 0 0 conjugation with P − z0 .

300

14 Weyl Asymptotics for the Damped Wave Equation

It follows that P δ − P  ≤ δ, P δ − P tr ≤ δ#(σ (P ) ∩ (∂I + ] − δ, δ[ )) ≤ O(1)δ 2 h−n ,

(14.1.7) (14.1.8)

σ (P δ ) ∩ (∂I + ] − δ, δ[ ) = ∅.

(14.1.9)

#(σ (P δ ) ∩ I ) = #(σ (P ) ∩ I ) + O(δh−n ).

(14.1.10)

By construction and (14.1.4),

Recalling that P = P + i Q, we put P δ = P δ + i Q.

(14.1.11)

Then P δ has a purely discrete spectrum contained in R + i[− , ] and more precisely, σ (P δ ) ⊂ (R \ (∂I + ] − δ, δ[ )) + D(0, ).

(14.1.12)

In particular (∂I + ] − δ + , δ − [ ) + iR is disjoint from the spectrum of P δ . By a simple deformation argument, we see that #(σ (P δ ) ∩ (I + iR)) = #(σ (P δ ) ∩ I ),

(14.1.13)

to be combined with (14.1.10). In the following, we choose δ = C0 ,

(14.1.14)

where C0 > 1 is a large constant to be fixed later. Let  = I + i[−1, 1],

(14.1.15)

and consider the following neighborhood of ∂: = W

 z∈∂

D(z, r(z)),

(14.1.16)

14.1 Eigenvalues of Perturbations of Self-adjoint Operators

301

where  r(z) = 4 + |z|/4.

(14.1.17)

If C0 in (14.1.14) is sufficiently large, then . D(z, 2 r(z)) ∩ ((R \ (∂I + ] − δ + , δ − [ )) + i[− , ]) = ∅, z ∈ W

(14.1.18)

 and that It follows that P δ has no spectrum in W . (P δ − z)−1  ≤ O(1/ r(z)), z ∈ W

(14.1.19)

We shall view the eigenvalues of P in  as the zeros of a relative determinant. Let 0 ≤ χ ∈ C0∞ (T ∗ X) be equal to 2 on a sufficiently large compact subset of T ∗ X. Then, if p = p+iχ, we see that ( p −z)−1 is a uniformly bounded function for z in a fixed neighborhood  . We can then quantize p of the closure of ∪W −p+P as an h-pseudodifferential operator  and get an operator P such that (for h > 0 small enough) • • •

P − P is uniformly bounded in L2 (X), P − P tr < +∞,  ∪  and the resolvent (P − z)−1 is uniformly bounded on that P has no spectrum in W  := P + i Q. set. The same holds for P

 are the zeros of the determinant of a trace class The eigenvalues of P in  ∪ W perturbation of the identity, namely,     D (z) := det (P − z)(P − z)−1 = det 1 − (P − P )(P − z)−1 .

(14.1.20)

 are the zeros of Similarly the eigenvalues of P δ in  ∪ W     D δ (z) = det (P δ − z)(P − z)−1 = det 1 − (P − P + P − P δ )(P − z)−1 (14.1.21) . and we have seen that there are no such values in W Remark 14.1.2 We can identify the algebraic multiplicities of the eigenvalues and the corresponding zeros of a determinant by expanding the discussion in Sect. 5.4 slightly: Let P (z) : F → G, z ∈ A be a holomorphic family of Fredholm operators of index zero, where A ⊂ C is open and connected. Assume that P (z) is bijective for at least one value of z. Then the set  ⊂ A of points where it is not bijective is discrete. Let z0 ∈ . We

302

14 Weyl Asymptotics for the Damped Wave Equation

can find 1 ≤ N0 ∈ N and bounded operators R+ (z) : F → CN0 , R− (z) : CN0 → G, depending holomorphically on z ∈ neigh (z0 , A), of maximal rank, such that 

 P (z) R− (z) P(z) = : F × CN0 → G × CN0 R+ (z) 0

(14.1.22)

is bijective for z ∈ neigh (z0 , A). Define the multiplicity 1 m(z0 , P ) = tr 2πi



P (z)−1 ∂z P (z)dz,

(14.1.23)

γ

where γ is the oriented boundary of the disc D(z0 , r) with r > 0 so small that D(z0 , r) ∩  = {z0 }. We shall see that m(z0 , P ) is well defined in general. When P (z) = z − P , it is the rank of the spectral projection, hence the usual multiplicity. Let   E(z) E+ (z) (14.1.24) E(z) = : G × CN0 → F × CN0 E− (z) E−+ (z) be the inverse of P(z). Then (cf. Sect. 5.4), we have −1 P (z)−1 = E(z) − E+ (z)E−+ E− (z),

(14.1.25)

and since E(z)∂z P (z) is holomorphic near D(z0 , r) when r > 0 is small enough, we get 1 2πi

 P (z)

−1

γ

1 ∂z P (z)dz = − 2πi

 γ

E+ (z)E−+ (z)−1 E− (z)∂z P (z)dz

which is of trace class. Hence, m(z0 , P ) is well defined and we can take the trace of the last integral, move the trace inside the integral, and get m(z0 , P ) = −

1 2πi

1 =− 2πi

 

γ

γ

  tr E+ (z)E−+ (z)−1 E− (z)∂z P (z) dz   −1 tr E−+ E− (∂z P )E+ dz,

(14.1.26)

using the cyclicity of the trace for the last equality. Using that ∂z E = −E(∂z P)E, we get − ∂z E−+ = E− (∂z P )E+ + E−+ (∂z R+ )E+ + E− (∂z R− )E−+ , −E− (∂z P )E+ = ∂z E−+ + E−+ (∂z R+ )E+ + E− (∂z R− )E−+ ,

(14.1.27)

14.1 Eigenvalues of Perturbations of Self-adjoint Operators

303

and substitution of the last identity in (14.1.26) gives m(z0 , P ) = I + II + III,

(14.1.28)

where I=

1 2πi

 γ

  −1 tr E−+ ∂z E−+ dz = m(z0 , det E−+ ),

the multiplicity of z0 as a zero of det E−+ ,     1 1 −1 II = tr E−+ E−+ (∂z R+ )E+ dz = tr((∂z R+ )E+ ) dz = 0, 2πi γ 2πi γ the integrand of the last integral being holomorphic,     1 1 −1 III = tr E−+ E− (∂z R− )E−+ dz = tr(E− (∂z R− )) dz = 0. 2πi γ 2πi γ Thus, m(z0 , P ) = m(z0 , det E−+ ).

(14.1.29)

Similarly, m(z0 , P ) = tr

1 2πi



(∂z P (z))P (z)−1 dz. γ

If P (z) = P1 (z)P2 (z), where P2 (z) : F → H, P1 (z) : H → G are Fredholm operators as above, one of which is bijective at z = z0 , then ⎧ ⎨m(z , P ), when P (z ) is bijective, 0 1 2 0 (14.1.30) m(z0 , P ) = ⎩m(z0 , P2 ), when P1 (z0 ) is bijective. The two cases are very similar and we only treat here the second one, when P1 (z0 ) is bijective. Let   2 (z) P2 (z) R− P2 (z) = 2 (z) 0 R+ be associated to P2 as in (14.1.22). Let 

P2−1

 2 E 2 E+ = E2 = . 2 E2 E− −+

304

14 Weyl Asymptotics for the Damped Wave Equation

Then, 

2 P P1 R− P= 2 R+ 0



  P1 0 = P2 0 1

is bijective. The inverse of P is 

2 E 2 E+ E= 2 E2 E− −+

    2 P1−1 0 E 2 P1−1 E+ , = 2 P −1 E 2 E− 0 1 −+ 1

2 . Therefore, we have (14.1.30) when and hence of the form (14.1.24) with E−+ = E−+ P1 (z0 ) is bijective. The other case can be treated similarly. Finally, let us recall that when F = G, P = 1 + K(z), where K is holomorphic and of trace class, then m(z0 , P ) as defined above, is equal to m(z0 , det P ); the multiplicity of z0 as a zero of det P (defined in Sect. 8.4).

We have   D (z)  − z)−1 (P − z)(P δ − z)−1 = det (P − z)( P

D δ (z)   = det (P − z)(P δ − z)−1   = det 1 − (P δ − P )(P δ − z)−1 .

(14.1.31)

  D (z)/D δ (z) ≤ exp (P δ − P )(P δ − z)−1 tr .



(14.1.32)

Thus,

Here, (P δ − P )(P δ − z)−1 tr ≤ (P δ − z)−1 P δ − P tr , : and by (14.1.19), (14.1.8), we get for z ∈ W (P δ − P )(P δ − z)−1 tr ≤ O(1)

δ2 .  r(z)hn

From (14.1.32), we conclude that ln |D (z)| − ln |D δ (z)| ≤ O(1)

δ2 . , z∈W  r(z)hn

(14.1.33)

14.1 Eigenvalues of Perturbations of Self-adjoint Operators

305

For every fixed θ > 0, we have (P − z)−1  ≤

Oθ (1)  , |z| ≥ (1 + θ ) . , z∈W  r(z)

(14.1.34)

Using this and a formula similar to (14.1.31) with P δ replaced by P , we get the upper bound ln |D δ (z)| − ln |D (z)| ≤

O(1)δ 2 ,  r(z)hn

in the same region as in (14.1.34), so in addition to (14.1.33) we have the lower bound ln |D (z)| − ln |D δ (z)| ≥ −

O(δ 2 )  , |z| ≥ (1 + θ ) . , z∈W  r(z)hn

(14.1.35)

 we have In W ln |D δ (z)| = !0 (z)/ hn ,

(14.1.36)

 . Extend !0 to a smooth where !0 depends on the various parameters and is harmonic in W  and notice that we have the exact formula function on  ∪ W  1 #(σ (P δ ) ∩ ) = !0 (z)L(dz). (14.1.37) 2πhn  Let r(z) =  r(z)/(1 + θ ) for some fixed θ with 0 < θ W =



1, and define (cf. (14.1.16))

D(z, r(z)).

(14.1.38)

z∈∂

From (14.1.33), (14.1.17), we have , ln |D (z)| ≤ !(z)/ hn , z ∈ W

(14.1.39)

where !(z) = !0 (z) +

Cδ 2 , + δ 2 )1/2

((z)2

(14.1.40)

provided that C is large enough.  , [0, 1]), equal to 1 on W , such that We can find χ ∈ C0∞ (W α ∇z,z χ = O(1)r(z)−|α| , for all α ∈ N2 .

(14.1.41)

306

14 Weyl Asymptotics for the Damped Wave Equation

 ∪ , by putting We extend the definition of ! to W !(z) = !0 (z) +

Cδ 2 χ(z) . ((z)2 + δ 2 )1/2

(14.1.42)

Then, ! − !0 =

O(δ 2 ) , (|z| + δ)3

and it follows that 

 |! − !0 |L(dz) = O(δ 2 ) + O(1)

0

1

δ2 ds (s + δ)2

(14.1.43)

= O(δ). , In particular, since !0 is harmonic on W   W

|!|L(dz) = O(δ).

(14.1.44)

In view of the choice of r(z) in (14.1.17), if we take θ > 0 small enough in the definition of r(z), we can find distinct points zj0 ∈ ∂, j ∈ Z/NZ distributed in the positive sense so that j → arg (zj0 − (a + b)/2) is increasing, with the properties: 1. There are precisely four points zj0 that minimize the distance to R, namely a ± i5 /4, b ± i5 /4, 2. r(zj0 )/4 ≤ |zj0+1 − zj0 | ≤ r(zj0 )/2. It follows that N = O(1) ln(1/ ) and that ∂ ⊂



D(zj , r(zj )/2).

(14.1.45)

Cδ 2 , |zj | + δ

(14.1.46)

j

Let

j =

for C > 0 sufficiently large. Then by (14.1.35) and (14.1.42), ln |D (z)| ≥

1 (!(z) − j ), z ∈ D(zj , r(zj )/C), hn

(14.1.47)

14.2 The Damped Wave Equation

307

and we recall that ln |D (z)| ≤ !(z)/ hn on W , by (14.1.39). By construction, 

j = O(δ).

(14.1.48)

To sum up the discussion, we have, by (14.1.10) and (14.1.13), #(σ (P δ ) ∩ ) = #(σ (P δ ) ∩ I ) = #(σ (P ) ∩ I ) + O(δh−n ). Here #(σ (P δ ) ∩ ) is given by the integral formula (14.1.37), where by (14.1.43), 1 2πhn



1 !0 L(dz) = n 2πh 



!L(dz) + O(δh−n ). 

Thus, #(σ (P ) ∩ I ) =

1 2πhn



!L(dz) + O(δh−n ). 

On the other hand, Theorem 12.1.2, (14.1.39), (14.1.47), (14.1.44) and (14.1.48) show that      #(σ (P ) ∩ ) − 1  ≤ O(δh−n ), !L(dz)   n 2πh  so |#(σ (P ) ∩ ) − #(σ (P ) ∩ I )| ≤ O(δh−n ), which in view of (14.1.3) concludes the proof of (14.1.5).

 

Remark 14.1.3 Notice that Theorem 14.1.1 is an application of a more general abstract theorem, which we do not formulate in detail. In particular, there are extensions of the theorem to the case of boundary value problems. See [96, 97].

14.2

The Damped Wave Equation

Let X be compact Riemannian manifold of dimension n. The damped wave equation is of the form (∂t2 −  + 2a(x)∂t )v(t, x) = 0, (t, x) ∈ R × X.

(14.2.1)

Here  denotes the Laplace–Beltrami operator on X and we assume that a ∈ C ∞ (X; R). Because of the “damping term” 2a∂t v, this evolution is no longer energy conserving (for a

308

14 Weyl Asymptotics for the Damped Wave Equation

suitable energy) and we have to expect exponential growth or decay of the solutions when |t| → ∞. Clearly, this is related to the eigenfrequencies of the corresponding stationary problem. Put v(t, x) = eit τ u(x), τ ∈ C. Then v solves (14.2.1) iff (− − τ 2 + 2ia(x)τ )u(x) = 0.

(14.2.2)

When (u, τ ) is a non-trivial solution of (14.2.2), we call τ an eigenfrequency, or simply an eigenvalue, and call u “the” corresponding eigenfunction. It is easy to show that if τ is an eigenfrequency, then inf a ≤ τ ≤ sup a, when τ = 0, 2 min(inf a, 0) ≤ τ ≤ 2 max(sup a, 0), when τ = 0.

(14.2.3) (14.2.4)

In this chapter we establish the most basic result about the distribution of eigenfrequencies, namely that their real parts obey Weyl asymptotics [97]. Many other important results concern the distribution of imaginary parts and the growth-decay of solutions and they are closely related to the geometry and the interplay between the damping coefficient a and the classical trajectories (“rays”). We refer to [66, 92, 118], where more references can be found. Some very detailed results in two dimensions are applicable to the eigenfrequencies. (See [73] for a more recent work on that theme.) When a = 0, the eigenfrequencies are real and symmetrically distributed around 0. In fact, they are the square roots (with both signs) of the eigenvalues of −. In this case (neglecting the case of τ = 0 which has the multiplicity two, as we shall see), we define the multiplicity of an eigenfrequency τ to be equal to that of τ 2 as an eigenvalue of −. Applying the standard result on the Weyl asymptotics for the eigenvalues of − (see for instance [51] and further references given there) we have Proposition 14.2.1 When a = 0 the eigenfrequencies are real and symmetric around 0. The number of eigenfrequencies τ in [0, λ], counted with their multiplicities, is equal to 

λ 2π

n 

    vol p−1 ([0, 1]) + O λ−1 ,

(14.2.5)

when λ → +∞. Here p(x, ξ ) denotes the principal symbol of − (equal to the dual Riemannian metric on T ∗ X). Back to the general case, we notice that the set of eigenfrequencies is symmetric under reflection in the imaginary axis and can be identified with the set of eigenvalues of the

14.2 The Damped Wave Equation

309

unbounded operator 

 0 1 P= : H 1 (X) × H 0 (X) → H 1 (X) × H 0 (X), − 2ia(x)

(14.2.6)

with domain H 2 × H 1 . In fact, the relation between (14.2.2) and   u (P − τ ) 0 = 0, u1

(14.2.7)

is given by u0 = u, u1 = τ u. Let P0 be the operator in (14.2.6) with a = 0. Let 0 = λ0 < λ1 ≤ · · · → +∞ be the eigenvalues of − and let e0 , e1 , . . . be a corresponding orthonormal basis of eigenfunctions. For k = 0, we put fk±

  1 λk−1/2 ek = √ ∈ H 1 × H 0, ±ek 2

and for k = 0, f00

    e0 0 1 . , f0 = = 0 e0 −1/2

−1/2

Let (e0 )⊥ denote the subspace L2 -orthogonal to Ce0 . Then λ1 e1 , λ2 e2 , . . . is an orthonormal basis in H 1 ∩ (e0 )⊥ , equipped with the scalar product [u|v] = (−u|v). It follows that f1 , f2 , . . . is an orthonormal basis in H = (H 1 ∩ (e0 )⊥ ) × (H 0 ∩ (e0 )⊥ ), with the scalar product      u0 u0    u1  u1

 = H

− 0 0 1

      u0 u0    u1  u1

= [u0 | u0 ] + (u1 | u1 ). L2 ×L2

Also P0 fk± = ±λk fk± , so the restriction of P0 to H is self-adjoint with fk± as a corresponding orthonormal basis of eigenvectors. √ From this we see that P0 has a purely discrete spectrum, given by 0, ± λk , k ≥ 1; 0 is an eigenvalue of algebraic multiplicity two and the full spectral decomposition of H 1 ×H 0 is 1/2

H 1 × H 0 = Cf00 ⊕ Cf01 ⊕ H.

310

14 Weyl Asymptotics for the Damped Wave Equation

√ The algebraic multiplicity of ± λk is equal to the multiplicity of λk as an eigenvalue of −, when k ≥ 1. When τ 2 = λk for all k ≥ 0, the resolvent (P0 − τ )−1 is of the form (P0 − τ )−1 = (− − τ 2 )−1 (P0 + τ ).

(14.2.8)

Let   0 0 Q= , so that P = P0 + Q. 0 2ia Then we see that Q(P0 − τ )−1 , (P0 − τ )−1 Q are compact on H 1 × H 0 and H 2 × H 1 , respectively. The norms of these operators tend to zero when τ → ∞ in closed sectors that are disjoint from R away from 0. It follows that P has purely discrete spectrum and we define the multiplicity of an eigenfrequency to be equal to its algebraic multiplicity as an eigenvalue of P. Thus the eigenfrequencies form a discrete set, confined to the region defined by (14.2.3), (14.2.4) and the elements have a natural multiplicity as defined above. (In [134] two further equivalent definitions of the multiplicity are given.) The set of eigenfrequencies, counted with their multiplicities, is invariant under the map τ → −τ of reflexion in the imaginary axis. We can then state a theorem which is due to Markus and Matseev [97]: Theorem 14.2.2 Under the assumptions above, the number of eigenfrequencies of Eq. (14.2.1) with real part in [0, λ], counted with their multiplicity, is equal to 

λ 2π

n      vol p−1 ([0, 1]) + O λ−1 , λ → +∞.

(14.2.9)

Proof We reformulate the problem semi-classically. Let h " 1/λ and put τ = z/ h, so that |z| " 1, when |τ | " λ. Then (14.2.2) becomes (−h2  − z2 + 2iahz)u = 0,

(14.2.10)

and we define the semi-classical eigenfrequencies in the obvious way and define the multiplicity of the semi-classical eigenfrequency z to be that of the eigenfrequency τ = z/ h. We then have to show that the number of semi-classical eigenfrequencies (counted with their multiplicity) with real part in [0, b], where b " 1, is equal to     1 −1 ([0, b]) + O(h) , h → 0. vol p (2πh)n

(14.2.11)

14.2 The Damped Wave Equation

311

The set of semi-classical eigenfrequencies is symmetric with respect to the imaginary axis, and we have a uniform bound on the number of eigenfrequencies on the imaginary axis, so equivalently, we have to show that the number of semi-classical eigenfrequencies with real part in [a, b], where a := −b, is equal to     2 −1 vol p ([0, b]) + O(h) , h → 0. (2πh)n

(14.2.12)

As in the non-semi-classical case, the eigenfrequencies appear as eigenvalues of a matrix operator, namely 

 0 1 Ph = = P0 + ihQ : Hh1 × H 0 → Hh1 × H 0 , −h2  2iah

(14.2.13)

where 

 0 0 Q= , 0 2a

 0 1 , P0 = −h2  0 

(14.2.14)

and it is here convenient to endow H k = Hhk with the semi-classical norm uH k = hDk uL2 = (1 − h2 )k/2 uL2 . h

For z = 0, the resolvent of P0 is of the form (P0 − z)−1 = (−h2  − z2 )−1 (P0 + z),

(14.2.15)

and using the semi-classical Sobolev norms, we see that (P0 − z)−1 = O(1/|z|) : Hh1 × H 0 → Hh2 × Hh1 ,

(14.2.16)

uniformly when |z| " 1.  Again, P0 is self-adjoint on H = (Hh1 × H 0 ) ∩ (e0 )⊥ × (e0 )⊥ and has the spectral decomposition Cf00 ⊕Cf01 ⊕(Hh1 ×H 0)∩ (e0 )⊥ × (e0 )⊥ . The number of eigenvalues of 1 (and b " 1), and we P0 in b + [−δ, δ ] and in −b + [−δ, δ ] is O(δh−n ) when h < δ can construct a perturbation P0δ of P0 which satisfies (14.1.7)–(14.1.9) with I = [a, b ] = [−b, b ]. We are therefore basically in the same situation as in the proof of Theorem 14.1.1 (with there equal to h), and the remainder of the proof is then the same.   Notes We have established the Weyl law for the asymptotic distribution of the real parts of the eigenfrequencies for the damped wave equation, see Theorem 14.2.2, which is due to [97]. The problem of the distribution of the imaginary parts inside the spectral band is very deep. See [66, 73, 92, 118].

Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small Random Perturbations, Results and Outline

15

In this chapter we will state a result asserting that for elliptic semi-classical (pseudo-)differential operators the eigenvalues are distributed according to Weyl’s law “most of the time” in a probabilistic sense. The first three sections are devoted to the formulation of the results and in the last section we give an outline of the proof that will be carried out in Chaps. 16 and 17.

15.1

The Unperturbed Operator

In this section we describe the class of unperturbed operators. Let X be a smooth compact manifold of dimension n. It is also possible to treat the case when X = Rn (cf. [137]), but we will concentrate on the compact manifold case. Let P be an h-differential operator on X which in local coordinates takes the form P =



aα (x; h)(hD)α ,

(15.1.1)

|α|≤m ∂ where we use standard multiindex notation1 and let D = Dx = 1i ∂x . We assume that the ∞ 1. (We will also coefficients aα are uniformly bounded in C for h ∈ ]0, h0 ], 0 < h0

1 |α| = α , (hD)α = h|α| D α1 · · · D αn . x1 xn 1

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_15

313

314

15 Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small. . .

discuss the case when we only have some Sobolev space control of a0 (x).) Assume aα (x; h) = aα0 (x) + O(h) in C ∞ , aα (x; h) = aα (x) is independent of h for |α| = m.

(15.1.2)

Notice that this assumption is invariant under changes of local coordinates. The second part of (15.1.2) is for convenience only. Also assume that P is elliptic in the classical sense, uniformly with respect to h: |pm (x, ξ )| ≥

1 m |ξ | , C

(15.1.3)

for some positive constant C, where the classical principal symbol pm (x, ξ ) =



aα (x)ξ α

(15.1.4)

|α|=m

is invariantly defined as a function on T ∗ X. It follows that pm (T ∗ X) is a closed cone in C and we assume that pm (T ∗ X) = C.

(15.1.5)

If z0 ∈ C \ pm (T ∗ X), we see that λz0 ∈ (p) if λ ≥ 1 is sufficiently large and fixed, where (p) := p(T ∗ X) and p is the semiclassical principal symbol p(x, ξ ) =



aα0 (x)ξ α .

(15.1.6)

|α|≤m

Actually, (15.1.5) can be replaced by the weaker condition that (p) = C. Standard elliptic theory shows that P − z : H m (X) → L2 (X) is a holomorphic family of Fredholm operators, where H m (X) denotes the usual Soboloev space of order m. In particular, the index is independent of z ∈ C and the assumption (15.1.5) implies that P − z is bijective for at least one value of z. Hence the index is zero and P − z is bijective for z away from a discrete set. (When we only assume that (p) = C, we need to use the assumption that h > 0 is small enough.) It follows that if we consider P as an unbounded operator L2 (X) → L2 (X) with domain D(P ) = H m (X), then P has purely discrete spectrum. In the case of multiplicative random perturbations we will need the symmetry assumption P = P ∗ ,

(15.1.7)

15.2 The Random Perturbation

315

where P ∗ denotes the formal complex adjoint of P in L2 (X, dx), with dx denoting some fixed smooth positive density of integration and  is the antilinear operator of complex conjugation: u = u. Notice that the left-hand side in (15.1.7) is equal to the “real” transpose P t , defined by 

 (P u)vdx =

u(P t v)dx, u, v ∈ C ∞ (X).

and that the assumption implies that p(x, −ξ ) = p(x, ξ ).

15.2

(15.1.8)

The Random Perturbation

We will perturb our operator with a small constant factor times a potential which is a random linear combination of oscillatory functions. For convenience, the latter are chosen to be eigenfunctions of an elliptic differential operator.  be an elliptic differential operator on X of order 2, such that for any choice of Let R smooth local coordinates, = h2 R

 (hDxj )∗ rj,k (x)hDxk ,

(15.2.1)

where the star indicates that we take the adjoint with respect to some fixed positive smooth ≥ 0 density on X. We assume that the rj,k are smooth and independent of h and also that R 2 2  in the operator sense. Then h R is essentially self-adjoint with domain H (X) and it has an orthonormal basis of eigenfunctions, j ∈ L2 (X), j = 1, 2, . . . , with corresponding eigenvalues μ2j = (hμ0j )2 , where 0 ≤ μ0j # +∞. By Weyl’s law for the large eigenvalues of a positive elliptic operator of order 2, we know that #{j ; μ0j ≤ λ} = (1 + o(1))(2π)−n Cw λn ,

λ → +∞,

(15.2.2)

where Cw is the symplectic volume of {(x, ξ ) ∈ T ∗ X; r(x, ξ ) ≤ 1}. Here r(x, ξ ) =



rj,k (x)ξj ξk

(15.2.3)

j,k

 and also the semi-classical principal symbol of h2 R.  is the principal symbol of R Our random perturbation will be of the form δqω , where δ > 0 is a small parameter and qω (x) =

 0 n2 , so by Weyl’s law for the large eigenvalues of elliptic self-adjoint operators, the number of terms D in (15.2.4) is of the order of magnitude (L/ h)n . The small parameter δ will be of the form δ = τ0 hN2 −n , 0 < τ0 ≤ h2 , where ) is a sufficiently large constant. N2 ≥ N2 (n, s, , M, M The αj (ω) are random variables with a joint probability distribution P(dα) = C(h)e!(α;h)L(dα),

(15.2.6)

|∇α !| = O(h−N4 ),

(15.2.7)

where for some N4 > 0,

and L(dα) is the Lebesgue measure on CD . (C(h) is the normalizing constant, ensuring that the probability of BCD (0, R) is equal to 1.)

15.3

The Result

Let Pδ = P + δqω , δ = τ0 hN2 −n , where qω is as in (15.2.4)–(15.2.7). Let  N6 = max(N3 , N5 ). N3 = (M + 1)n, N5 = N4 + M, Let   C b a fixed open simply connected set with smooth boundary. In Sect. 16.4 below (see Proposition 16.4.2) we introduce a continuous subharmonic function φ on  satisfying (16.4.6): φ L(dz) = p∗ (dxdξ ), 2π

(15.3.1)

where the right-hand side denotes the direct image under p of the symplectic volume element. √ Let    be a Lipschitz domain as in Chap. 12 of constant scale r = h. Let  G(w, ) = ∂

h |dz| √ . 2 h + |w − z| h

(15.3.2)

15.3 The Result

317

Theorem 15.3.1 Let  δ > 0. Then with probability  ≥ 1 − O(1)h−N6 −n

1 ln τ0

     h −δ 1 1 2 ln G(w, )φ(w)L(dw) + |∂|h 2 e− O(1) , h 

the number of eigenvalues of Pδ in  (counted with their algebraic multiplicities) satisfies     1 −1 #(σ (Pδ ) ∩ ) − ≤ vol (p ())   (2πh)n      1 1 2 1 O(1) − δ h ln ln G(w, )φ(w)L(dw) + |∂|h 2 . hn τ0 h 

(15.3.3)

Remark 15.3.2 Actually, we shall prove the theorem for the slightly more general n operators, obtained by replacing P by P0 = P + δ0 (h 2 q10 + q20 ), for 0 ≤ δ0 ≤ h2 ,

q10 Hhs , q20 H s ≤ 1,

(15.3.4)

where  · Hhs is the natural semi-classical norm on H s that we shall review in Sect. 16.2. If we weaken (15.3.4) to 0 ≤ δ0 ≤ h2−ϑ ,

(15.3.5)

for some fixed ϑ ∈ ]0, 1/2 [, then we get the slightly weaker statement in the theorem 1 1 obtained by replacing |∂|h 2 with |∂|h 2 −ϑ both in the lower bound on the probability and in the estimate (15.3.3). As in [56], we also have a result valid simultaneously for a family C of domains  ⊂  satisfying the assumptions of Theorem 15.3.1 uniformly in the natural sense: With a probability as in Theorem 15.3.1 and after replacing h−N6 −n there by h−N6 −n−1/2 (or after increasing N6 by 1/2) the estimates (15.3.3) hold simultaneously for all  ∈ C.  has real coefficients, we may assume that the eigenfunctions j Remark 15.3.3 When R are real. Then, as will follow from the proofs below, we may restrict α in (15.2.4) to be in RD so that qω is real, still with |α| ≤ R, and change C(h) in (15.2.6) so that P becomes a probability measure on BRD (0, R). Then Theorem 15.3.1 remains valid. Remark 15.3.4 The assumption (15.1.7) cannot be completely eliminated. Indeed, let P = hDx + g(x) on T = R/(2πZ), where g is smooth and complex valued.  2π Then (cf. Hager [54] and Chap. 3) the spectrum of P is contained in the line z = 0 g(x)dx/(2π). This line will vary only very little under small multiplicative perturbations of P , so Theorem 15.3.1 cannot hold in this case.

318

15 Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small. . .

In order to have a better understanding of the statement in Theorem 15.3.1, we need to compare the Weyl term 1 W = 2π

 φ(z)L(dz)

(15.3.6)



and the main contribution to the remainder in (15.3.3), given by  R=

G(w, )φ(w)L(dw).

(15.3.7)



As we shall see in Sect. 17.5, if ∂ satisfies the regularity assumption (17.5.25):   √ √ R 1+κ |∂ ∩ D(w, R)| ≤ O(1) h √ , R ≥ h, h where |γ | denotes the length of the curve γ and 0 ≤ κ ≤ 1, then we have (17.5.26) as well as the improvement away from  in Remark 17.5.1, (17.5.27), (17.5.28): G(w, ; h) ⎧  κ−1 ⎪ ⎨O(1) 1 + d(w,∂) √ in general, h =  κ−2 ⎪ ⎩ diam √ () 1 + d(w,∂) √ , when d(w, ∂) ≥ diam (). h

(15.3.8)

h

Here d(w, ∂) denotes the distance from w to ∂. From this estimate we expect—unless  is very small or if a considerable fraction of p∗ (dxdξ ) is concentrated to a ∂—that R W and (15.3.3) indeed gives the asymptotic behaviour of the number of eigenvalues in  for moderate values of  δ and τ0 . In the remainder of this section we shall discuss three closely related examples when (17.5.25) holds with κ = 0, so that G(w, ; h) ⎧ −1 ⎪ ⎨ 1 + d(w,∂) √ in general, = O(1) h −2 ⎪ ⎩ diam √ () 1 + d(w,∂) √ , when d(w, ∂) ≥ diam (). h

(15.3.9)

h

Example 15.3.5 Let  be a fixed Lipschitz domain as in Chap. 12, independent of h with scale 1 (in the sense of that chapter), so (15.3.9) holds. Assume that 1

vol p−1 (∂ + D(0, t)) = O(t N0 ), t → 0,

(15.3.10)

15.3 The Result

319

for some N0 ≥ 1 and notice that this holds with N0 = 1 when p(ρ) ∈ ∂ 2⇒ dp(ρ), dp(ρ) are linearly independent. Then it is easy to check (as we shall do prior to Proposition 18.4.2) that 

  G(w, )φ(w)L(dw) = O h1/(2N0 ) , 

where for notational simplicity we have suppressed a factor ln(1/ h) on the right, when N0 = 1. Theorem 15.3.1 now tells us that with probability     h −δ 1 1 2 2N1 − O(1) h 0e , ≥ 1 − O(1)h−N6 −n ln ln τ0 h

(15.3.11)

    1 −1 #(σ (Pδ ) ∩ ) − vol (p ()) ≤  n (2πh)    O(1) −δ 1 1 2 2N1 h h 0, ln ln hn τ0 h

(15.3.12)

we have

which is a significant estimate if ln 1/τ0 = O(h−β ), where 0 < β < 1/(2N0 ), and we choose  δ small enough so that β +  δ < 1/(2N0 ).   p(T ∗ X) ∩  be open and with no critical value of p in its Example 15.3.6 Let  closure:  2⇒ dp(ρ), dp(ρ) are linearly independent. p(ρ) ∈  . Let 0 be a bounded closed Lipschitz domain as Then φL(dz) " L(dz) uniformly on  in Chap. 12, independent of h and with scale √ 1 (in the sense of that chapter). We fix some   1. Clearly ∂ satisfies (17.5.25) z0 ∈  and put  := z0 + α0 ⊂ , where h ≤ α with κ = 0, so we have (15.3.9) uniformly with respect to α. We see that W " α2 .

(15.3.13)

320

15 Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small. . .

In order to estimate R, assume for simplicity that z0 = 0. Then  α G(w, )φ(w)L(dw)  √ h + G(w, )φ(w)L(dw)  h      √ d(w, α∂0 ) −1  α h+ 1+ L(dw) √ ∩{d(w,α∂0 )≤α} h     d(w, α0 ) −2 α √ √ L(dw). + 1+ ∩{d(w,α∂0 )≥α} h h  

R=

(15.3.14)

Here “” means “≤ O(1) times”. The first integral in the last term is equal to   αd( w , ∂0 ) −1 √ L(d w ) 1+ h d( w,∂0 )≤1  √   α2 L(d w ) + α h √ 

α2

{d( w,∂0 )≤

√ {1≥d( w,∂0 )≥ αh }

h α }

  √ √ √ α α  α h + α h ln √ = α h 1 + ln √ . h h

1 L(d w ) d( w , ∂0 )

The second integral in the last member of (15.3.14) is equal to α √ α2 h



 1 w,∂0 )≥1} α ∩{d(

α3  √ h

1+

αd( w , ∂0 ) √ h



1 w,∂0 )≥1} α ∩{d(

−2

L(d w )

h L(d w ) α 2 d( w , ∂0 )2 √ 1  α h ln . α

Thus,     √ √ α 1 1 R ≤ O(1)α h 1 + ln √ + ln . = O(1)α h 1 + ln √ α h h In conclusion, we have α 1 R  hδ W when √ ≥ h−δ (1 + ln ), h h and the estimate (15.3.3) gives spectral asymptotics, provided that  δ < δ and τ0 is not too  δ < δ − δ. small, so that ln 1/τ0 = O(h−δ ) for some 

15.3 The Result

321

Example 15.3.7 Let z0 ∈ ∂ and assume for simplicity that p−1 (z0 ) consists of just one point ρ0 . Our final assumptions will imply that {p, {p, p}} = 0,

(15.3.15)

at ρ0 . dp and dp have to be collinear at the point ρ0 (since otherwise z0 would belong to the interior of ) and in particular {p, p}(ρ0 ) = 0. Without loss of generality, we may assume that dp(ρ0 ) = dp(ρ0 ) = 0.

(15.3.16)

We may also assume for notational reasons that z0 = 0. Since 0 belongs to ∂ , we see that one of the following holds: 1) p(ρ) ≥ 0 for all ρ ∈ (p)−1 (0) ∩ neigh (ρ0 ). 2) p(ρ) ≤ 0 for all ρ ∈ (p)−1 (0) ∩ neigh (ρ0 ). Suppose, in order to fix the ideas, that we are in the first case. We then make the following generic assumption (which implies (15.3.15) and is equivalent to that condition when n = 1). p|(p)−1 (0)∩neigh (ρ0 ) has a nondegenerate minimum at ρ0 .

(15.3.17)

Then for t ∈ neigh (0, R), p|(p)−1 (t )∩neigh (ρ0 ) has a nondegenerate minimum at a point ρ(t) depending smoothly on t with ρ(0) = ρ0 . We have p(ρ) − p(ρ(p(ρ))) " |ρ − ρ(p(ρ))|2

(15.3.18)

for ρ ∈ neigh (ρ0 , T ∗ X). If g(t) := p(ρ(t)), then near z = 0, is given by z ≥ g(z) (and from (15.3.16) we see that g  (0) = 0). Near ρ0 the hypersurfaces (p)−1 (t) carry a natural Liouville measure2 and the direct image of this measure under the map (p)−1 (z)  ρ −→ p(ρ) is of the form n− 32

f (z)(z − g(z))+

dz, where f > 0 is continuous,

2 When viewed as an 2n − 1 form λ on (p)−1 (t), it is given by λ ∧ dp = dx ∧ · · · ∧ dx ∧ n 1 dξ1 ∧ · · · ∧ dξn = the symplectic volume form.

322

15 Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small. . .

since the Liouville measure of the set p(ρ)−p(ρ(z)) ≤ r in (p)−1 (z) is " r n−1/2 . It follows that n− 32

p∗ (dxdξ ) = f (z)(z − g(z))+

(15.3.19)

L(dz)

near z = 0. Notice that when n = 1 the corresponding density tends to +∞ at the boundary of , while in the case n ≥ 2 it tends to 0. We will compare R() and W () for small domains centered at points of the boundary of . We may assume that g ≡ 0. Let  = (α) be the domain (α) = {z ∈ C; |z| < 2α, 0 ≤ z < α}, for

√

h≤α

1.

(15.3.20)

We have  W "

3

−α 0, there exists δ2 > 0 such that if δ ≥ e−h 2 is as in that theorem, then −δ with probability ≥ 1 − O(e−h 2 /r), we have     1 −1 #(σ (Pδ ) ∩ ) − vol (p ()) ≤  (2πh)n ⎛ ⎞   ⎟ κ−δ O(1) ⎜ 1 ⎜h 1 ⎟ + r + ln φ(w)L(dw) ⎜ ⎟, ⎠ hn ⎝ r r ∂+D(0,r) #$ % " =:F (r)

for any 0 < r

1.

(15.4.3)

326

15 Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small. . .

We neglect the factors h−δ1 and the ln factor in (15.4.2), (15.4.3). Then we are reduced to comparing 

1

A :=

G(w, )φL(dw) + h 2

(15.4.4)

 hκ + r + F (r) . r

(15.4.5)



and  B := inf 0 n/2. Then there exists a constant C = C(s) such that for all u, v ∈ Hhs (Rn ), we have u ∈ L∞ (Rn ), uv ∈ Hhs (Rn ) and uL∞ ≤ Ch−n/2 uHhs ,

(16.2.1)

uvHhs ≤ Ch−n/2 uHhs vHhs .

(16.2.2)

Proof The fact that u ∈ L∞ and the estimate (16.2.1) follow from Fourier’s inversion formula and the Cauchy–Schwarz inequality: 1 |u(x)| ≤ (2π)n



u(ξ )|)dξ ≤ hξ −s (hξ s |

It then suffices to use that h·−s  = C(s)h−n/2 .

1 h·−s uHhs . (2π)n/2

16.2 Sobolev Spaces and Multiplication

333

In order to prove (16.2.2), we pass to the Fourier transform side, and see that it suffices u|,  v = h·s | v |, to show that with  u = h·s | 

n

w(ξ )hξ s (h·−s  u ∗ h·−s  v )(ξ )dξ ≤ C(s)h− 2  u v w,

(16.2.3)

for all non-negative  u, v , w ∈ L2 , where ∗ denotes convolution. Here the left-hand side can be written as  hξ s w(ξ ) u(η) v (ζ )dξ dζ ≤ I + II, s s η+ζ =ξ hη hζ  where I and II denote the corresponding integrals over the sets {|η| ≥ |ξ |/2} and {|ζ | ≥ |ξ |/2}, respectively. Here I ≤ C(s)

   v (ζ ) dζ ( w(ξ ) u(ξ − ζ )dξ ) hζ s

≤ C(s)w u

 v  1. h·s L

 v − As in the proof of (16.2.1), we see that  h· v , so I is bounded by a s L1 ≤ C(s)h 2  n

n

constant times h− 2  u v w. The same estimate holds for II and (16.2.3) follows.

 

Let X be a compact n-dimensional manifold. We cover X by finitely many coordinate neighborhoods M1 , . . . , Mp and for each Mj , we let x1 , . . . , xn denote the corresponding p local coordinates on Mj . Let 0 ≤ χj ∈ C0∞ (Mj ) have the property that 1 χj > 0 on X. Define Hhs (X) to be the space of all u ∈ D (X) such that u2H s h

:=

p 

χj hDs χj u2 < ∞.

(16.2.4)

1

By straightforward arguments one can show that this definition does not depend on the choice of the coordinate neighborhoods or on χj , and that different choices of these objects give norms in (16.2.4) which are uniformly equivalent when h → 0. This also follows from the h-pseudodifferential calculus on manifolds with symbols in m  s/2 is , given in Sect. 16.1. We saw in Sect. 16.1 that (1 + h2 R) the Hörmander space S1,0 s an h-pseudodifferential operator with symbol in S1,0 and semiclassical principal symbol  given by (1 + r(x, ξ ))s/2 , where r(x, ξ ) = j,k rj,k (x)ξj ξk is the semiclassical principal  From Sect. 16.1, in particular Proposition 16.1.1, we get the following symbol of h2 R. result for every s ∈ R, which provides us with an equivalent definition of Hhs (X):

334

16 Proof I: Upper Bounds

 s/2 u ∈ L2 Proposition 16.2.2 Hhs (X) is the space of all u ∈ D (X) such that (1 + h2 R)  s/2 u, uniformly when h → 0. and the norm uHhs is equivalent to (1 + h2 R)  repeated according to their Let (μ0k )2 , k = 1, 2, . . . with μ0k > 0 be the eigenvalues of R multiplicity and let 1 , 2 , . . . be a corresponding orthonormal basis of eigenfunctions. It follows from the basic Weyl asymptotics for elliptic self-adjoint operators on compact manifolds (see for instance [51] and further references given there) that #{k; μ0k ≤ λ} " λn , uniformly for 0 < h

1, λ ≥ 1. If u ∈ D (X), we have u=



αk k , αk = (u| k ),

where the sequence of αk is of temperate growth and the series converges in the sense of  distributions. From Proposition 16.2.2 we see that u ∈ Hhs iff μk 2s is finite and u2H s " h



μk 2s |αk |2 ,

 uniformly with respect to h. Here μk := hμ0k , so that μ2k are the eigenvalues of h2 R. Remark 16.2.3 From the first definition of Hhs we see that Proposition 16.2.1 remains valid if we replace Rn by a compact n-dimensional manifold X. Of course, Hhs (X) coincides with the standard Sobolev space H s (X) = H1s (X) and the norms are equivalent for each fixed value of h. We have the following variant of Proposition 16.2.1: Proposition 16.2.4 Let s > n/2. Then there exists a constant C = Cs > 0 such that uvHhs ≤ CuH s vHhs , ∀u ∈ H s (Rn ), v ∈ Hhs (Rn ).

(16.2.5)

The result remains valid if we replace Rn by X. Proof The adaptation to the case of a compact manifold is immediate by working in local coordinates, so it is enough to prove (16.2.5) in the Rn -case. Let χ ∈ C0∞ (Rn ) be equal to 1 in a neighborhood of 0. Write u = u1 + u2 with u1 = χ(hD)u, u2 = (1 − χ(hD))u. Then, with hats indicating Fourier transforms, we have  1 hξ s hξ s u& v(ξ ) = v (η)dη. u(ξ − η)) hηs (χ(h(ξ − η)) 1 n (2π) hηs

16.3 Bounds on Small Singular Values and Determinants. . .

335

Here hξ /hη = O(1) on the support of (ξ, η) → χ(h(ξ − η)), so u1 vHhs ≤ O(1) uL1 vHhs ≤ O(1)uH s vHhs , where we also used that s > n/2 in the last estimate. When 1 − χ(hξ ) = 0, we have hξ s ≤ Chs ξ s , so u2 Hhs ≤ Chs uH s . From Proposition 16.2.1, we get  s− 2 uH s vH s ≤ Cu  H s vH s , u2 vHhs ≤ Ch− 2 u2 Hhs vHhs ≤ Ch h h n

n

when h ≤ 1.

16.3

 

Bounds on Small Singular Values and Determinants in the Unperturbed Case

Let X be a compact manifold and let P be a differential operator on X, satisfying the assumptions of the unperturbed operator in Theorem 15.3.1. Thus P is of the form (15.1.1), and satisfies (15.1.2), (15.1.3), (15.1.5) where the classical and semi-classical symbols pm and p are defined in (15.1.4) and (15.1.6) respectively. (We will not yet use the symmetry assumption (15.1.7).) Recall that the spectral parameter z varies in the open simply connected set   Cn with smooth boundary and that  contains a point z0 which is not in the image of p. We can (cf. the proof of Proposition 5.1.3) construct a symbol p  ∈ S m (T ∗ X) which is equal −1 (ρ) − z is non-vanishing to p outside any fixed given neighborhood of p (), such that p for every z ∈ . Indeed, there is a smooth map κ : C \ {z0 } → C \ {z0 } such that  = κ ◦ p. Let κ(C \ {z0 }) ∩  = ∅ and κ(z) = z for |z| large. It then suffices to put p p − p), where we use the h-quantization in Sect. 16.1. P = P + Oph ( Using the h-pseudodifferential calculus, we see that for h > 0 small enough,  − z : Hhm (X) → H 0 (X) P

(16.3.1)

is bijective for all z ∈  with a uniformly bounded inverse. Put Pz = (P − z)−1 (P − z) = 1 − (P − z)−1 (P − P ).

(16.3.2)

p − p) is a smoothing operator of trace class C1 (L2 , L2 ) with Notice that P − P = Oph ( the corresponding trace-class norm = O(h−n ) (see [41, 119]). In this section we shall estimate the number of small singular values of P − z and Pz and obtain closely related upper bounds on ln | det Pz |. We introduce the operator S = (P − z)∗ (P − z).

(16.3.3)

336

16 Proof I: Upper Bounds

(Later on we shall use the same symbol for a closely related bounded operator.) In order to obtain these estimates, we shall develop a slightly degenerate pseudodifferential calculus. Let h ≤ α 1. A basic weight function in our calculus will be   :=

α+s 1+s

1 2

, where s(ρ) = |p(ρ) − z|2 ,

√ and we see that α/2 ≤  ≤ 1. Consider first symbol properties of 1 +

s α

(16.3.4)

and its powers.

Proposition 16.3.1 For every choice of local coordinates x on X, let (x, ξ ) denote the α , β ∈ Nn , we have corresponding canonical coordinates on T ∗ X. Then for all  ∈ R,  uniformly in ξ and locally uniformly in x: α ∂x ∂ξ

β

  s  s  −| 1+ = O(1) 1 +  α |−|β| ξ −|β| . α α

(16.3.5)

Proof In the region |ξ | ( 1 we see that (1 + αs ) is an elliptic element of the Hörmander symbol class 2m α − S1,0 =: α − S(ξ 2m ),

and  " 1 there, so (16.3.5) holds. In the region |ξ | = O(1), we start with the case  = 1. 1 Since s ≥ 0, we have ∇s = O(s 2 ), so  1     1 s2 s  s s  −1  ≤ O(1) 1 + (α + s)− 2 = O(1) 1 +  . =O ∇ 1 + α α α α For k ≥ 2, we have       s  s  −2 s  −k 1  k = O(1) 1 +  ≤ O(1) 1 +  , ∇ 1 + =O α α α α and we get (16.3.5) when  = 1. α ∂ β (1 + s ) is a finite linear combination of terms If  ∈ R, then ∂x ξ α     s −k   s  s  β αk βk 1+ ∂xα1 ∂ξ 1 1 + · · · ∂x , ∂ξ 1 + α α α with  α = α1 + · · · +  αk , β = β1 + · · · + βk , and we get (16.3.5) in general.

 

16.3 Bounds on Small Singular Values and Determinants. . .

337

We next notice that when w belongs to a bounded subset of C,  s  s   s |w|  1+ ≤ w −  ≤ C 1 + . C α α α In fact, the second inequality is obvious, and so is the first one, when s α ≤ O(1), it follows from the fact that 1+

s = O(1), α

(16.3.6) s α

( 1. When

 s   w −  ≥ |w|. α

From (16.3.5), (16.3.6), we get    s  s  −|   β  α |−|β| ξ −|β| |w|−1 . ∂xα ∂ξ w −  ≤ O(1) w − α α

(16.3.7)

When passing to (w − αs ) and applying the proof of Proposition 16.3.1, we loose more powers of |w| which can still be counted precisely, but we refrain from doing so and simply state the following result: Proposition 16.3.2 For all  ∈ R,  α , β ∈ Nn , there exists J ∈ N, such that α ∂ξ ∂x

β

  s  s  −| w− = O(1) 1 +  α |−|β| ξ −|β| |w|−J , α α

(16.3.8)

uniformly in ξ and locally uniformly in (x, w). In this estimate we can replace (1 + s/α) with (1 + s/α)1 (w − s/α)2 for every (1 , 2 ) with 1 + 2 = . We next introduce some new symbol spaces. Definition 16.3.3 Let m (x, ξ ) be a weight function of the form m (x, ξ ) = ξ k  . We ∞ ∗ m) if for all say that the family a = aw ∈ C (T X), w ∈ D(0, C), belongs to S (  α , β ∈ Nn there exists J ∈ N such that α α |−|β| ∂ξ a = O(1) m(x, ξ )−| ξ −|β| |w|−J . ∂x β

(16.3.9)

Here, as in Proposition 16.3.2, it is understood that the estimate is expressed in canonical coordinates and is locally uniform in (x, w) and uniform in ξ . Notice that the set of estimates (16.3.9) is invariant under changes of local coordinates in X. Let U ⊂ X be a coordinate neighborhood that we shall view as a subset of Rn in the ) be a symbol as in Definition 16.3.3, so that (16.3.9) natural way. Let a ∈ S (T ∗ U, m holds uniformly in ξ and locally uniformly in (x, w). For fixed values of α, w the symbol

338

16 Proof I: Upper Bounds

k (T ∗ U ), so the classical h-quantization a belongs to S1,0

Au = Oph (a)u(x) =

1 (2πh)n



i

e h (x−y)·η a(x, η; h)u(y)dydη

(16.3.10)

is a well-defined operator C0∞ (U ) → C ∞ (U ), E  (U ) → D (U ). In order to develop our rudimentary calculus on X, we first establish a pseudolocal property for the distribution kernel KA (x, y): Proposition 16.3.4 For all  α , β ∈ Nn , N ∈ N, there exists M ∈ N such that α β ∂x ∂y KA (x, y) = O(hN |w|−M ),

(16.3.11)

locally uniformly on U × U \ diag(U × U ). Proof If γ ∈ Nn , then (x −y)γ KA (x, y) is the distribution kernel of Oph ((−hDξ )γ a) and √  1  (h/ξ )|γ | . By the observation after (16.3.4), h/ ≤ 2h/α 2 ≤ (−hDξ )γ a ∈ S m √ 1 2h 2 . Thus for any N ∈ N, there exists M ∈ N such that, (x − y)γ KA (x, y) = O(hN |w|−M ), if |γ | ≥ γ (N) is large enough. Equation (16.3.11) follows when  α = β = 0. Now, α ∂ β K can be viewed as the distribution kernel of a new pseudodifferential operator of ∂x y A the same kind, so we get (16.3.11) for all  α , β.   This means that if φ, ψ ∈ C0∞ (U ) have disjoint supports, then for every N ∈ N, there exists M ∈ N such that φAψ : H −N (Rn ) → H N (Rn ) with norm O(hN |w|−M ), and this leads to a simple way of introducing pseudodifferential operators on X: Let U1 , . . . , Us be coordinate neighborhoods that cover X. Let χj ∈ C0∞ (Uj ) form a partition of unity and j in the sense that χ j is equal to 1 near supp (χj ). Let let χ j ∈ C0∞ (Uj ) satisfy χj ≺ χ a = (a1 , . . . , as ), where aj ∈ S ( m). Then we quantize a by the formula A=

s 

χ j ◦ Oph (aj ) ◦ χj .

(16.3.12)

1

This is not an invariant quantization procedure, but it will suffice for our purposes. We next study the composition to the left with non-exotic pseudodifferential operators. Let U = Uj be one of the above coordinate neighborhoods, viewed as an open set in Rn , and take A = Oph (a), a ∈ S1,0 (m1 ), m1 = ξ r , B = Oph (b), b ∈ S (m2 ) with m2 = ξ k  as in Definition 16.3.3. We will assume that supp (b) ⊂ K × Rn , where K ⊂ U is compact. We are interested in C = A ◦ B.

16.3 Bounds on Small Singular Values and Determinants. . .

339

The symbol c of this composition is given by i

i

c(x, ξ ; h) = e− h x·ξ A(b(·, ξ )e h (·)·ξ )(x)  i 1 = a(x, η)b(y, ξ )e h (x−y)·(η−ξ )dydη n (2πh)

(16.3.13)

Let χ ∈ C0∞ (B(0, 12 )) be equal to 1 near 0, and let d(x, ξ ; h) be obtained from (16.3.13) after inserting a factor 1 − χ (η − ξ )/ξ ) in the integral. Then |η − ξ | ≥ C1 ξ  in the support of the integrand, so we can make repeated integrations by parts in the y-variables and get ∀N ∈ N,  α , β ∈ Nn , ∃M ∈ N, ∀K  U, ∃C > 0; α |∂x ∂ξ d(x, ξ ; h)| ≤ C β

hN ξ −N , (x, ξ ) ∈ K × Rn . |w|M

(16.3.14)

Up to such a term d, we get    i 1 η−ξ c(x, ξ ; h) ≡ a(x, η)b(y, ξ )χ e h (x−y)·(η−ξ )dydη (2πh)n ξ       −iξ ξ  n ξ = ) b(x + y, ξ )χ(η)e h y·η dydη. a x, ξ (η + 2πh ξ 

(16.3.15)

We shall apply the method of stationary phase (see for instance [41, 51]) and pause to recall some facts about the expansion of the integral 1 J (t, u) = (2πt)n



i

u(y, η)e− t y·η dydη, u ∈ S(R2n ).

The exponent is of the form     i y y Q · , 2t η η where 

 0 −1 Q= . −1 0 Then (cf. [51, Proposition 2.3]) we know that J (t, u) −→ u(0), t −→ 0.

340

16 Proof I: Upper Bounds

We also know (see [51, (2.4)]) that if F : L2 (R2n ) → L2 (R2n ) denotes the Fourier transform,  F

 i 1 ∗ ∗ t y·η (y ∗ , η ∗ ) = e −ity ·η , e n (2πt)

then we can apply Plancherel’s formula and write J (t, u) =

1 (2π)2n



(F u)(y ∗ , η∗ )eity

∗ ·η∗

dy ∗ dη∗

= exp(tDy · ∂η )(u)(0, 0). By Taylor’s formula with integral remainder, we have J (t, u) =

N−1  tk

k!

0

=

(∂η · Dy )k u(0, 0) + RN

 t |β| ∂ β D β u(0, 0) + RN , β! η y

|β| 0, put 1

 VN (α) =

1+

0

t α

−N



1/α

dV (t) =

(1 + τ )−N dV (ατ )

(16.3.37)

0

The contribution to the integral in (16.3.34) from the region 0 ≤ s ≤ 1 is equal to  O(1) 0

1

h α+t

N

 N h dV (t) = O(1) VN (α) α

(16.3.38)

Remark 16.3.6 The choice of 1 as the upper bound of integration was somewhat arbitrary. If we replace it by a fixed constant θ ∈ ]0, 1[ , then VN changes by a quantity which is O(1)α N (V (1) − V (θ )). The contribution to (16.3.34) from the region s(x, ξ ) > 1 is O(hN2 ) times some negative power of |w| for every N2 . In conclusion, we have  > 0, Proposition 16.3.7 For all N, N    N h  RN tr ≤ O(1)h−n VN (α) + hN |w|−M(N) . α

(16.3.39)

From (16.3.31), we get on the level of operators (w −

1 −1 1 S) = EN − (w − S)−1 RN . α α

Using the functional Cauchy–Riemann formula (sometimes named after Dynkin and/or Helffer and Sjöstrand, cf. [41]) we have 1 1 χ( S) = − α π



1 ∂ χ (w − S)−1 L(dw), ∂w α

(16.3.40)

16.3 Bounds on Small Singular Values and Determinants. . .

347

where χ ∈ C0∞ (R) and χ  ∈ C0∞ (C) is an almost holomorphic extension. We get 

1 1 χ( S) = − α π

1 ∂ χ EN L(dw) + ∂w π



  1 −1 ∂ χ w− S RN L(dw) =: I + II. ∂w α (16.3.41)

From (16.3.39), (16.3.41) and the fact that ∂ χ /∂w is in C0∞ and vanishes to infinite order on the real axis, we get −n

IItr = O(1)h

   h N  N . VN (α) + h α

It remains to study the term I and for that we return to the expression (16.3.27) for (the symbol of) EN (in local coordinates). Here z = αw and we see that EN is equal to 1 w−

  s α

β k,≥0



Cβ,γ ,δ h|β|

γ1 +···γk + δ1 +···+δ =(β,β)

∂ γ1 ( αs ) ∂ γk ( αs ) ∂ δ1 ( αh s−1 ) ∂ δ ( αh s−1 ) · · · · · · w − αs w − αs w − αs w − αs

The contribution from the general term to the symbol of I is Cβ,γ ,δ − π



∂ χ (w) 1 |β| s 1+k+ L(dw)h ∂w (w − α )

s s h h ∂ γ1 ( ) · · · ∂ γk ( )∂ δ1 ( s−1 ) · · · ∂ δ ( s−1 ) α α α α β,γ ,δ χ (k+) ( s )h|β| ∂ γ1 ( s ) · · · ∂ γk ( s )∂ δ1 ( h s−1 ) · · · ∂ δ ( h s−1 ). =C α α α α α

(16.3.42)

When β = 0,  = 0, we have Cβ,γ ,δ = 1 and we get χ( αs ). For every N ∈ N, the general term belongs to the symbol class  S

2 α

−N+k

h|β| |γ1 |+···+|γk |

   h . α

Here, we notice that |γ1 | + · · · + |γk | ≤ 2|β|, and since h ≤ α,  ≥ α 1/2 we conclude that the general term (16.3.42) belongs to  −N+k   |β|+ 2 h . S α α

348

16 Proof I: Upper Bounds

The trace-class norm of the contribution from (16.3.42) to I is bounded by O(1)h−n

  |β|+  1  h t −(N−k) 1+ dV (t) α α 0  |β|+ h ≤ O(1)h−n VN/2 (α), α

where we used that N can be chosen ≥ 2k.  > 0, the following holds. Summing up, we have proved that for all N, N Proposition 16.3.8 Let χ ∈ C0∞ (R). For 0 < h ≤ α ≤ 1/2, we have 1  χ( S)tr = O(1)h−n (VN (α) + hN ), α  1 1 h s(x, ξ )  )dxdξ + O(h−n ) (VN (α) + hN ). tr χ( S) = χ( n α (2πh) α α

(16.3.43)

(16.3.44)

Remark 16.3.9 Recall that Pz is defined in (16.3.2). Using simple h-pseudo-differential calculus (for instance as in the appendix of [139]), we see that if we redefine S as follows S = Pz∗ Pz ,

(16.3.45)

then in each local coordinate chart, S = Oph (S), where S ≡ s mod S1,0 (hξ −1 ) and s is now redefined as  s(x, ξ ) =

|p(x, ξ ) − z| | p(x, ξ ) − z|

2 (16.3.46)

.

The discussion goes through without any changes (now with m = 0) and we still have Proposition 16.3.8 with the new choice of S, s and the corresponding redefinition of V (t), now restricted to 0 ≤ t ≤ 1/2. Notice here that if Vnew denotes this new function, then for some C ≥ 1, we have t 1 V ( ) ≤ Vnew (t) ≤ CV (Ct), 0 ≤ t C C

1.

In the remainder of this section, we choose S, s as in (16.3.45), (16.3.46). In this case S is a trace-class perturbation of the identity, whose symbol is 1+O(h∞ /ξ ∞ ) and similarly for all its derivatives, in a region |ξ | ≥ Const.

16.3 Bounds on Small Singular Values and Determinants. . .

349

Let 0 ≤ χ ∈ C0∞ ([0, ∞[ ) with χ(0) > 0 and let α0 > 0 be small and fixed. From standard pseudodifferential calculus in the spirit of [104, Section 2], we get        1 1 1 ln s + α0 χ( s) dxdξ + O(h) . ln det S + α0 χ( S) = α0 (2πh)n α0 (16.3.47) Here is an outline of the Proof of (16.3.47) P = S + α0 χ(S/α0 ) is an h-pseudodifferential operator with symbol in S 0 . The principal symbol is s + α0 χ(s/α0 ) and the full symbol P of this operator for any system of local coordinates satisfies P − 1 ∈ hN S −N near ξ = ∞. Clearly P − 1 (on the operator level) is of trace class. Further, we have p ≥ 1/C. Let Pt = tP + (1 − t) for 0 ≤ t ≤ 1, so that P1 = P , P0 = 1. The principal symbol pt = tp + (1 − t) fulfills pt ≥ 1/C and Pt is uniformly invertible for 0 ≤ t ≤ 1 and h > 0 small enough. Now   ∂t ln det Pt = tr Pt−1 ∂t Pt , and by pseudodifferential operator calculus, we get 1 ∂t ln det Pt = (2πh)n 1 = (2πh)n

   −1 O(h) + pt ∂t pt dxdξ    O(h) + ∂t ln(pt )dxdξ .

Integration from t = 0 to t = 1 yields 1 ln det P = (2πh)n

   O(h) + ln(p)dxdξ ,  

which amounts to (16.3.47).

Extend χ to be an element of C0∞ (R; C) in such a way that t + χ(t) = 0 for all t ∈ R. Then (cf. [56]), we use that   E 1 E d ln E + tχ( ) = ψ( ), dt t t t

(16.3.48)

where ψ(E) =

χ(E) − Eχ  (E) , E + χ(E)

(16.3.49)

350

16 Proof I: Upper Bounds

so that ψ ∈ C0∞ (R). By standard functional calculus for self-adjoint operators and the classical identity d d ln(det At ) = tr A−1 At t dt dt for differentiable trace-class perturbations of the identity,1 we have S 1 S d ln det(S + tχ( )) = tr ψ( ). dt t t t

(16.3.50)

Using (16.3.44) (now with S as in that equation), we get for t ≥ α ≥ h > 0:      d 1 1 h 1 s  N ln det S + tχ( S) = ψ( )dxdξ + O(1) V (t) + O(h ) . N dt t (2πh)n t t t2 Integrating this from t = α0 to t = α and using (16.3.47), (16.3.48), we get   1 ln det S + αχ( S) = α     α0  1 h s  ) dxdξ + O(h + V (t)dt) . ln s + αχ( (2πh)n α t2 α

(16.3.51)

In fact, this follows from an estimation of  α

α0

h VN (t)dt = t2

 

α0 α

h t2



1 0

1 dV (s)dt (1 + st )N

1

=

(16.3.52)

J (s)dV (s), 0

where 

α0

J (s) = α

h tN h dt = t 2 (t + s)N s



α0 /s α/s

 tN  t. t −2 d ( t + 1)N

1 Equation (8.4.7) extends to the case when A and A are trace-class operators as in Sect. 8.4 and 0 1 the identity is valid for finite-rank perturbations of the identity. By Taylor expansion and partitions of unity we can approximate a C 1 family At of trace-class perturbations of the identity by a sequence (ν) (ν) of such perturbations At so that At → At uniformly in C1 and similarly for the derivatives (on (ν) (ν) any given compact interval), and so that N (At )⊥ ∪ R(At ) ⊂ H(ν) , where H(ν) is independent of t and of finite dimension. It then suffices to pass to the limit.

16.4 Subharmonicity and Symplectic Volume

351

Considering separately the three cases, α/s ( 1, α/s " 1, α/s J (s) "

1, we see that

h . max(α, s)

Thus the term (16.3.52) is 

1

" 0

h h dV (s) = V (1) + max(α, s) 1



1 α

h V (s)ds, s2

and (16.3.51) follows. Write     s(x, ξ ) ) dxdξ − ln s(x, ξ )dxdξ ln s(x, ξ ) + αχ( α  α 1  α  1 σ 1 s ψ( )dxdξ dt = ψ( )dV (σ )dt = t t t 0 0 0 t  α 1  α σ −N 1 dt ≤ O(1) 1+ VN (t)dt. dV (σ ) = O(1) t t 0 0 0 t Here we may notice that for t ≤ 1/2: 

1

VN (t) = 0

σ −N 1+ dV (σ ) ≥ t

 t σ −N 1+ dV (σ ) ≥ 2−N V (t). t 0

Combining the above computation with (16.3.51), we get Proposition 16.3.10 If 0 ≤ χ ∈ C0∞ ([0, ∞[), χ(0) > 0, we have uniformly for 0 < h ≤ α 1   1 ln det S + αχ( S) = α      α0  α dt 1 h dt + V (t) V (t) ln s(x, ξ )dxdξ + O(1) h + . N (2πh)n t t t α 0 (16.3.53)

16.4

Subharmonicity and Symplectic Volume

Recall that Pz is defined by (16.3.2) and has the semi-classical principal symbol pz (ρ) = ( p (ρ) − z)−1 (p − z), z ∈ .

352

16 Proof I: Upper Bounds

In Sect. 16.3 we encountered the function   1 φ(z) = ln |pz (x, ξ )|dxdξ = ln sz (x, ξ )dxdξ. 2

(16.4.1)

We also considered the function  V (t) = Vz (t) =

|p(x,ξ )−z|2≤t

(16.4.2)

dxdξ,

and the closely related one z (t) = V

 |pz (x,ξ )|2 ≤t

(16.4.3)

dxdξ.

These two functions will be used only for small values of t; they are equivalent, in the sense that 1 t z (t) ≤ CVz (Ct), 0 ≤ t ≤ 1/C, z ∈ , Vz ( ) ≤ V C C

(16.4.4)

for some C ( 1, as we saw after (16.3.46) where the notation was slightly different. For κ ∈ ]0, 1] consider the property that Vz (t) = O(t κ ),

0≤t

1.

(16.4.5)

Recall that m is the order of the elliptic differential operator P . Proposition 16.4.1 Equation (16.4.5) holds uniformly for z ∈  when κ =

1 2m .

Proof Because of the ellipticity, the set of points where |p(x, ξ ) − z| ≤ 1 is contained in a fixed compact set in T ∗ X, independent of z, that we can cover by finitely many sets of the form T ∗ Uj , j = 1, 2, . . . , N0 , where Uj are local coordinates charts that we can identify with open bounded subsets of Rn . On each such T ∗ Uj we know that ξn → p(x, ξ ) − z is a polynomial of degree m with leading term aj (x)ξnm , aj (x) = 0. By the standard factorization of this polynomial; p(x, ξ ) − z = aj (x)(ξn − λ1 (x, ξ  , z)) · · · (ξn − λm (x, ξ  , z)), ! we see that |p(x, ξ ) − z| ≥ t 1/2 when C  ξn ∈ j D(λj (x, ξ  , z), Ct 1/(2m) ) where C > 0 is independent of x, ξ  , z after shrinking Uj slightly. It follows that 1

L({ξn ∈ R; |p(x, ξ ) − z| ≤ t 1/2 }) = O(1)t 2m

16.4 Subharmonicity and Symplectic Volume

353

uniformly for x ∈ Uj , ξ  ∈ Rn−1 , where now L denotes the Lebesgue measure on the real line. The result then follows from Fubini’s theorem.   Proposition 16.4.2 The function φ(z) defined in (16.4.1) is continuous and subharmonic on . Moreover, φ(z) L(dz) = p∗ (dxdξ ), 2π

(16.4.6)

where the right-hand side is defined to be the direct image under the symbol map: T ∗ X  (x, ξ ) → p(x, ξ ) ∈ C of the symplectic volume element dxdξ (which we also denote dρ, ρ = (x, ξ )). Proof The property (16.4.5) for some κ > 0 (here with κ = 1/(2m)) implies that the integral (16.4.1) converges. Moreover, with 0 ≤ χ ∈ C0∞ (R), χ(0) > 0, 1 φ(z) = lim α→0 2



  sz (x, ξ ) ) dxdξ, ln sz (x, ξ ) + αχ( α

where the convergence is locally uniform, so that φ(z) is the locally uniform limit of continuous functions and hence continuous. p(ρ) − z)−1 (p(ρ) − z), where the first factor is holomorphic and Next, pz (ρ) = ( non-vanishing for z ∈ , so z ln |pz (ρ)| = z ln |p(ρ) − z| = 2πδ(z − p(ρ)). If ψ ∈ C0∞ (), we get 

 (z φ(z))ψ(z)L(dz) =  =

φ(z)z ψ(z)L(dz) 

 T ∗X C

ln |pz (ρ)|z ψ(z)L(dz)dρ = 

 = 2π

which establishes (16.4.6).

T ∗X

 T ∗X

z (ln |pz (ρ)|)ψ(z)L(dz)dρ C

δ(z − p(ρ))ψ(z)L(dz)dρ = 2π C

 T ∗X

ψ(p(ρ))dρ,  

354

16 Proof I: Upper Bounds

16.5

Extension of the Bounds to the Perturbed Case

For s > n/2, we consider the perturbed operator n

Pδ = P + δ(h 2 q1 + q2 ) = P + δ(Q1 + Q2 ) = P + δQ,

(16.5.1)

where qj ∈ H s (X), q1 Hhs ≤ 1, q2 H s ≤ 1, 0 ≤ δ

1.

(16.5.2)

According to Propositions 16.2.1 and 16.2.4, Q = O(1) : Hhs → Hhs , and hence by duality and interpolation, Q = O(1) : Hhσ → Hhσ , −s ≤ σ ≤ s.

(16.5.3)

Let Pδ := Pδ + P − P . Here (P − z)−1

⎧ ⎨H σ → H σ +m , h h = O(1) : ⎩H σ −m → H σ , h

h

for −s ≤ σ ≤ s and using (16.5.1), (16.5.3), the resolvent identity, and the fact that 0 ≤ δ 1, we get the same conclusion for (Pδ − z)−1 . The spectrum of Pδ in  is discrete and from Remark 14.1.2 it follows that it coincides with the set of zeros of det((Pδ − z)−1 (Pδ − z)) = det(1 − (Pδ − z)−1 (P − P )).

(16.5.4)

Pδ,z := (Pδ − z)−1 (Pδ − z) = 1 − (Pδ − z)−1 (P − P ) =: 1 − Kδ,z ,

(16.5.5)

Put

Sδ := (Pδ − z)∗ (Pδ − z), ∗ ∗ ∗ Sδ,z := Pδ,z Pδ,z = 1 − (Kδ,z + Kδ,z − Kδ,z Kδ,z ) =: 1 − Lδ,z .

(16.5.6) (16.5.7)

Then Kδ,z , Lδ,z = O(1) : Hh−s−m → Hhs+m .

(16.5.8)

Observe that Kδ,z tr ≤ (Pδ − z)−1 P − P tr ≤ O(h−n ), Lδ,z tr ≤ O(h−n ).

(16.5.9)

16.5 Extension of the Bounds to the Perturbed Case

355

We shall extend Proposition 16.3.8, Remark 16.3.9 and Proposition 16.3.10 to the perturbed case for δ ≥ 0 small enough.  ∈ C0∞ (C) be an almost As in Proposition 16.3.8, let χ ∈ C0∞ ([0, +∞[ ) and let χ holomorphic extension of χ. Then, working in the parameter range 0 < h ≤ α 1, we have  1 1 w −1 (16.5.10) )( )L(dw). χ(α Sδ ) = − (w − Sδ )−1 (∂z χ π α α This is not obviously of trace-class for all values of s, n, m, so we make a modification: We can find 0 ≤ ψ ∈ C0∞ (T ∗ X) such that |p(x, ξ ) − z|2 + ψ(x, ξ ) ≥

1 2m ξ  , on T ∗ X, C

for all z ∈ . Let ψ = ψ(x, hDx ) denote also a corresponding self-adjoint quantization. Then for h > 0 small enough, (P − z)∗ (P − z) + ψ ≥

1 hD2m . C

Now, Sδ = S0 + δQ∗ (P − z) + (P − z)∗ δQ + δ 2 Q∗ Q,

(16.5.11)

so for δ, h small enough we get Sδ + ψ ≥

1 hD2m . 2C

In particular, (Sδ + ψ − w)−1 exists and is uniformly bounded for w in a small neighborhood of 0 ∈ C. On the other hand, (w − Sδ )−1 = (w − (Sδ + ψ))−1 − (w − Sδ )−1 ψ(w − (Sδ + ψ))−1 .

(16.5.12)

When inserting this in (16.5.10) the first term gives the contribution zero, since it is holomorphic in a neighborhood of 0, and we get χ(α

−1

1 Sδ ) = π



1 w )( )L(dw), (w − Sδ )−1 ψ(w − (Sδ + ψ))−1 (∂z χ α α

which is of trace class, since ψ is.

(16.5.13)

356

16 Proof I: Upper Bounds

Differentiating this relation with respect to δ, we get ∂ χ(α −1 Sδ ) = ∂δ  1 1 w )( )L(dw) (w − Sδ )−1 S˙δ (w − Sδ )−1 ψ(w − (Sδ + ψ))−1 (∂z χ π α α  1 1 w + )( )L(dw), (w − Sδ )−1 ψ(w − (Sδ + ψ))−1 S˙δ (w − (Sδ + ψ))−1 (∂z χ π α α (16.5.14) where ∂ S˙δ = Sδ = Q∗ (P − z) + (P − z)∗ Q + 2δQ∗ Q = O(1) : Hhσ +m → Hhσ −m . ∂δ From (16.5.11) it follows that Sδ = S0 + R, where R = O(δ) : Hhσ +m → Hhσ −m , −s ≤ σ ≤ s. If w0 belongs to some compact set in C that is disjoint from R, it follows that for δ small enough, w0 − Sδ : Hhσ +m → Hhσ −m is bijective for −s ≤ σ ≤ s. In fact, w0 − Sδ = w0 − S0 − R = (w0 − S0 )(1 − (w0 − S0 )−1 R), and we see that (w0 − S0 )−1 = O(1) : Hhσ −m → Hhσ +m , so (w0 − S0 )−1 R = O(δ) : Hhσ +m → Hhσ +m , and hence (w0 − Sδ )−1 = (1 − (w0 − S0 )−1 R)−1 (w0 − S0 )−1 = O(1) : Hhσ −m → Hhσ +m , as claimed. Let w belong to a bounded set in C disjoint from R. Iterating the resolvent identity, we have (w − Sδ )−1 = (w0 − Sδ )−1 + (w0 − w)(w0 − Sδ )−2 + · · · + (w0 − w)2N−1 (w0 − Sδ )−2N + (w0 − w)2N (w − Sδ )−1 (w0 − Sδ )−2N . The last term can also be written in a sandwiched form as (w0 − w)2N (w0 − Sδ )−N (w − Sδ )−1 (w0 − Sδ )−N . For N large enough, we have (w0 − Sδ )−N = O(1) : Hh−s−m → Hhs+m . In particular, (w0 − Sδ )−N

⎧ ⎨H σ −m → H 0 , h = O(1) : ⎩H 0 → H σ +m . h

16.5 Extension of the Bounds to the Perturbed Case

357

Since (w − Sδ )−1 = O(|w|−1 ) : H 0 → H 0 , we conclude that (w − Sδ )−1 = O(|w|−1 ) : Hhσ −m → Hhσ +m .

(16.5.15)

Similarly, with ψ as in (16.5.12), positive on a sufficiently large set depending on the bounded set where we let w vary, (w − (Sδ + ψ))−1 = O(1) : Hhσ −m → Hhσ +m .

(16.5.16)

Using also that ψtr = O(h−n ), (∂z χ )(w/α) = ON ((|w|/α)N ) for all N ≥ 0 and choosing σ = 0, we see that the trace norm of the first integral in (16.5.14) is  O(1)

|w|≤O (α)

|w|

−2 −n

h

1 α



|w| α

N L(dw),

while the second integral satisfies the better bound with the exponent −2 replaced by −1. Choosing N = 2, we get 

∂ χ(α −1 Sδ )tr = O(1)α −1 h−n . ∂δ

(16.5.17)

Integrating this “from 0 to δ” and applying Proposition 16.3.8, we get Proposition 16.5.1 Let χ ∈ C0∞ (R). For 0 < h ≤ α χ(α

tr χ(α

−1

−1

−n

Sδ )tr ≤ O(1)h

1 Sδ ) = (2πh)n



1, we have

  δ  N VN (α) + h + , α

(16.5.18)

  δ s(x, ξ )  N −n )dxdξ + O(1)h , VN (α) + h + χ( α α (16.5.19)

where s(x, ξ ) = |p(x, ξ ) − z|2 . We next extend Remark 16.3.9 to the perturbed case. Recall the expressions for Kδ,z and Lδ,z in (16.5.5), (16.5.7) and also (16.5.9). We have K˙ δ,z = −(z − Pδ )−1 Q(z − Pδ )−1 (P − P ),

(16.5.20)

so K˙ δ,z  ≤ O(1), K˙ δ,z tr ≤ O(h−n ),

(16.5.21)

358

16 Proof I: Upper Bounds

since Q = O(1). Here the dot indicates differentiation with respect to δ. It follows that S˙δ,z tr ≤ O(h−n ).

(16.5.22)

With χ ∈ C0∞ (R), we get ∂ 1 χ(α −1 Sδ,z ) = − ∂δ π



1 w )( )L(dw), (w − Sδ,z )−1 S˙δ,z (w − Sδ,z )−1 (∂z χ α α

(16.5.23)

which again leads to 

∂ χ(α −1 Sδ,z )tr = O(1)α −1 h−n ∂δ

(16.5.24)

and combining this with Remark 16.3.9, we get Proposition 16.5.2 Proposition 16.5.1 remains valid if we replace (Sδ , s) with (Sδ,z , sz ), where sz (x, ξ ) =

|p(x, ξ ) − z|2 . | p(x, ξ ) − z|2

Finally, we shall extend Proposition 16.3.10 to the perturbed case. Let 0 ≤ χ ∈ C0∞ ([0, ∞[ ), χ(0) > 0. Put t f (t) = t + αχ( ) =: t + g(t), g = gα ∈ C0∞ . α For 0 < h ≤ α

1, we have f (Sδ,z ) = Sδ,z −

1 π



g (w)L(dw). (w − Sδ,z )−1 ∂

From this we see that 1 ∂ f (Sδ,z ) = S˙δ,z − ∂δ π



g (w)L(dw). (w − Sδ,z )−1 S˙δ,z (w − Sδ,z )−1 ∂

Now, ∂ ∂ ln det f (Sδ,z ) = tr f (Sδ,z )−1 f (Sδ,z ) = ∂δ ∂δ  1 tr (f (Sδ )−1 S˙δ ) − g (w)L(dw). tr (f (Sδ,z )−1 (w − Sδ,z )−1 S˙δ,z (w − Sδ,z )−1 )∂ π

16.5 Extension of the Bounds to the Perturbed Case

359

Here f (Sδ,z )−1 and (w − Sδ,z )−1 commute, so by the cyclicity of the trace and an integration by parts, we see that the last term is equal to tr (f (Sδ,z )−1

(−1) π



g (w)L(dw)S˙δ,z ) (w − Sδ,z )−2 ∂ w 

   −1 (−1) −1 ˙ = tr f (Sδ,z ) g (w)L(dw)Sδ,z (w − Sδ,z ) ∂ w ∂w π = tr (f (Sδ,z )−1 g  (Sδ,z )S˙δ,z ), leading to the general identity ∂ ln det f (Sδ,z ) = tr (f (Sδ,z )−1 f  (Sδ,z )S˙δ,z ). ∂δ With the above choice of f we have f |[0,+∞[ ≥ α/C, so f (Sδ,z )−1  ≤ C/α. Moreover, f  = O(1), so f  (Sδ,z ) = O(1). Using also (16.5.22), we conclude that ∂ ln det f (Sδ,z ) = O(α −1 h−n ). ∂δ Integrating from 0 to δ and using Proposition 16.3.10, we get Proposition 16.5.3 If 0 ≤ χ ∈ C0∞ ([0, +∞[ ), χ(0) > 0, we have, uniformly for 0 < h≤α 1,   1 ln sz (x, ξ )dxdξ ln det(Sδ,z + αχ(α Sδ,z )) = (2πh)n   α  α0 dt δ h dt V (t) + + h . VN (t) + + O(1) t t t α 0 α −1

We end the section with some remarks about H s -properties of eigenfunctions for low lying eigenvalues of Sδ and Sδ,z , here under the only assumption on δ that 0 ≤ δ 1. We start with Sδ . Let 0 ≤ t12 ≤ t22 ≤ · · · denote the increasing sequence of eigenvalues of Sδ , repeated according to their multiplicity and with tj ≥ 0. Let e1 , e2 , . . . be a corresponding orthonormal family of eigenfunctions. Proposition 16.5.1 shows that N0 := #{k; tk2 ≤ 1/2} ≤ O(h−n )

(16.5.25)

360

16 Proof I: Upper Bounds

Proposition 16.5.4 Under the assumptions above we have 

N0  j =1

→ − λj ej H s+m ≤ Os (1) λ 2 , h

(16.5.26)

t →  − where λ = λ1 λ2 . . . λN0 . Proof Using (16.5.15), (16.5.16) in (16.5.13) with α = 1, we see that for every χ ∈ C0∞ (R), χ(Sδ ) = O(1) : Hh−s−m → Hhs+m .

(16.5.27)

(We choose the set where ψ > 0 is sufficiently large depending on the support of χ and of its almost holomorphic extension, and use that ψ = O(1) : Hh−s+m → Hhs−m .) N N Let χ ≥ 0 be equal to 1 on [0, 1/2], so that χ(Sδ )( 1 0 λj ej ) = 1 0 λj ej . Using only   that χ(Sδ ) = O(1) : H 0 → Hhs+m , we get (16.5.26). Next consider Sδ,z in (16.5.7). From (16.5.20), the fact that P − P = O(1) : Hhs1 → s2 Hh , for all s1 , s2 and the observation after (16.5.3), we see that K˙ δ,z = O(1) : Hh−s−m → ∗ and Hhs+m and by duality and the explicit formula for Lδ,z , we get the same facts for K˙ δ,z −s−m s+m → Hh , and we conclude that L˙ δ,z . Thus, S˙δ,z = O(1) : Hh Sδ,z = S0,z + Rδ,z , Rδ,z = O(1) : Hh−s−m → Hhs+m . We then apply the sandwiched resolvent identity (w − Sδ,z )−1 =(w − S0,z )−1 + (w − S0,z )−1 Rδ,z (w − S0,z )−1 + (w − S0,z )−1 Rδ,z (w − Sδ,z )−1 Rδ,z (w − S0,z )−1 ,

(16.5.28)

to get for χ ∈ C0∞ ([0, 3/4[ ) that  1 χ(Sδ,z ) =χ(S0,z ) − χ (w) L(dw) (w − S0,z )−1 Rδ,z (w − S0,z )−1 ∂ π  1 χ (w)L(dw). − (w − S0,z )−1 Rδ,z (w − Sδ,z )−1 Rδ,z (w − S0,z )−1 ∂ π Here χ(S0,z ) is an h-pseudodifferential operator of order 0 in h and of order −∞ in ξ , so χ(S0,δ ) = O(1) : Hh−s−m → Hhs+m . Each of the two integrals define operators that are O(1) : Hh−s−m → Hhs+m . In fact, (w − S0,z )−1 = O(|w|−1 ) : Hhσ → Hhσ for every real σ .

16.5 Extension of the Bounds to the Perturbed Case

361

This is the analogue of (16.5.27), and completing the discussion as at the end of the proof of Proposition 16.5.4, we get Proposition 16.5.5 Proposition 16.5.4 remains valid if we let e1 , . . . , eN0 (still with N0 = O(h−n )) be an orthonormal family of eigenfunctions corresponding to the spectrum of Sδ,z in [0, 1/2]. Notes This chapter gives the first part of the proof of Theorem 15.3.1, following closely [137,139]. See Proposition 16.5.3 and the corresponding upper bound on ln det Sδ,z , which is bounded from above by ln det(Sδ,z + αχ(Sδ,z /α)). In Chap. 17 we give corresponding lower bounds, valid with probability close to 1—while the upper bounds are valid for all sufficiently small perturbations.

17

Proof II: Lower Bounds

In this chapter we give a lower bound on ln det Sδ,z which is valid with high probability, and then using also the upper bounds of Chap. 16, we conclude the proof of Theorem 15.3.1 with the help of Theorem 12.1.2.

17.1

Singular Values and Determinants of Certain Matrices Associated to δ Potentials

As in Chap. 16, let X be a compact smooth manifold of dimension n, equipped with a positive smooth density of integration dx. Let e1 (x), . . . , eN (x) be continuous functions on X. The discussion will remain valid if we replace X by a compact subset with nonempty interior and smooth boundary. We would like to find a continuous function q : X → C such that the matrix Mq = (Mq;j,k )1≤j,k≤N , given by  Mq;j,k =

q(x)ej (x)ek (x)dx, X

has nice lower bounds for its singular values. In the present section, we shall achieve such  a goal when q is replaced by a sum of Dirac masses of the form δa = N 1 δ(x − aj ) for a N suitable a = (a1 , a2 , . . . , aN ) ∈ X . Notice that Mδa = (Mj,k ), where Mj,k =

N 

ej (aν )ek (aν )

(17.1.1)

ν=1

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_17

363

364

17 Proof II: Lower Bounds

and that M = E ◦ Et,

(17.1.2)

where E = (ej (ak ))1≤j,k≤N . If we introduce the column vectors ⎛

⎞ e1 (x) ⎜ ⎟ ⎜ e2 (x) ⎟ ⎜ . ⎟ ⎜ ⎟ → − e (x) = ⎜ .. ⎟ ∈ CN , ⎜ . ⎟ ⎜ . ⎟ ⎝ . ⎠ eN (x) −e (a ), . . . , → −e (a )). → then E = (− e (a1 ), → 2 N Lemma 17.1.1 Let M ∈ [1, N] be an integer and let L ⊂ CN be a linear subspace of dimension M − 1, for some 1 ≤ M ≤ N. Then there exists x ∈ X such that → dist (− e (x), L)2 ≥

1 tr ((1 − πL )EX ). vol (X)

(17.1.3)

Here EX = ((ej |ek )L2 (X) )1≤j,k≤N is the Gramian of (e1 , . . . , eN ) and πL denotes the orthogonal projection from CN onto L. Proof Let ν1 , . . . , νN be an orthonormal basis in CN such that L is spanned by ν1 , . . . , νM−1 (and equal to 0 when M = 1). Let (·|·) denote the usual scalar product on CN and let (·|·)X be the scalar product on L2 (X). Write ⎞ ν1, ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ν = ⎜ . ⎟ . ⎜ . ⎟ ⎝ . ⎠ νN, ⎛

17.1 Singular Values and Determinants of Certain Matrices Associated. . .

365

We have → dist (− e (x), L)2 =

N  −e (x)|ν )|2 |(→  =M

=

N   | ej (x)ν j, |2 =M

=

j

N  

ν j, ej (x)ek (x)νk, .

=M j,k

It follows that 

→ dist (− e (x), L)2 dx =

X

N 

(EX ν |ν ) = tr ((1 − πL )EX ).

=M

It then suffices to estimate the integral from above by → vol (X) max dist (− e (x), L)2 . x∈X

  If we make the assumption that e1 , . . . , eN is an orthonormal family in L2 (X),

(17.1.4)

then EX = 1 and (17.1.3) simplifies to → e (x), L)2 ≥ max dist (− x∈X

N −M +1 . vol (X)

(17.1.5)

In the general case, let 0 ≤ ε1 ≤ ε2 ≤ · · · ≤ εN denote the eigenvalues of EX . Then we have inf

dim L=M−1

tr ((1 − πL )EX ) = ε1 + ε2 + · · · + εN−M+1 =: EM .

Indeed, the min-max principle shows that εk =

inf

dim L =k

sup (EX ν|ν),

ν∈L ν=1

(17.1.6)

366

17 Proof II: Lower Bounds

and so for a general subspace L of dimension M −1, the eigenvalues of (1−πL )EX (1−πL )  , with εj ≥ εj . are ε1 ≤ · · · ≤ εN−M+1 Now, we can use the lemma to choose successively a1 , . . . , aN ∈ X such that → − e (a1 )2 ≥

E1 , vol (X)

→ → dist (− e (a2 ), C− e (a1 ))2 ≥

E2 , vol (X)

... → → → dist (− e (aM ), C− e (a1 ) ⊕ · · · ⊕ C− e (aM−1 ))2 ≥

EM , vol (X)

... → Let ν1 , ν2 , . . . , νN be the Gram–Schmidt orthonormalization of the basis − e (a1 ), → − → − e (a2 ), . . . , e (aN ), so that −e (a ) ≡ c ν mod (ν , . . . , ν → M M M 1 M−1 ), where |cM | ≥



EM vol (X)

1 2

.

(17.1.7)

→ Recall that E is the N × N matrix with the columns − e (aj ). Expressing these vectors in the basis ν1 , . . . , νN will not change the absolute value of the determinant and E now becomes an upper triangular matrix with diagonal entries c1 , . . . , cN . Hence | det E| = |c1 · · · cN |,

(17.1.8)

and (17.1.7) implies that | det E| ≥

(E1 E2 · · · EN )1/2 . (vol (X))N/2

(17.1.9)

We now return to M in (17.1.1), (17.1.2). Since det M = (det E)2 , we have | det M| ≥

E1 E2 · · · EN . vol (X)N

(17.1.10)

Under the assumption (17.1.4), this simplifies to | det M| ≥

N! . vol (X)N

(17.1.11)

17.2 Singular Values of Matrices Associated to Suitable Admissible. . .

367

It will also be useful to estimate the singular values s1 (M) ≥ s2 (M) ≥ · · · ≥ sN (M) of the matrix M (by definition, the decreasing sequence of eigenvalues of the matrix 1 (M ∗ M) 2 ). Clearly, s1N ≥ s1k−1 skN−k+1 ≥

N 

sj = | det M|,

1 ≤ k ≤ N,

(17.1.12)

j =1

and we recall that s1 = M.

(17.1.13)

Combining (17.1.10) and (17.1.12), we get Proposition 17.1.2 Under the above assumptions, 1

s1 ≥ ⎛ sk ≥ s1 ⎝

(E1 · · · EN ) N , vol (X)

N  

j =1

17.2

⎞ 1  N−k+1 Ej ⎠ . s1 vol (X)

(17.1.14)

(17.1.15)

Singular Values of Matrices Associated to Suitable Admissible Potentials

We shall now carry over the results of the preceding section to potentials that are linear  Recall the definition of combinations of eigenfunctions of the auxiliary operator h2 R. 0 0 2  k = (μ ) k , μk = hμ . Also recall that D := #{k; μk ≤

k , μk , in Sect. 16.2: R

k k L} = O(Ln/2 / hn ) by Weyl asymptotics for elliptic self-adjoint operators on a compact manifold. Definition 17.2.1 An admissible potential is a potential of the form q(x) =



αk k (x), α ∈ CD .

(17.2.1)

0 n/2 + , > 0, as in (15.2.5).

368

17 Proof II: Lower Bounds

Proposition 17.2.2 Let a ∈ X. Then there exist αk ∈ C, 1 ≤ k < ∞, such that for L ≥ 1, there exists r ∈ H −s such that δa (x) =



αk k + r(x),

(17.2.2)

μk ≤L

where n

n

rH −s ≤ Cs, L−(s− 2 − ) h− 2 ,

(17.2.3)

h

(



⎛ 2

1 2

|αk | ) ≤ L

n 2 +

⎝



⎞1 2

−2( n2 + )

μk 

|αk |

2⎠

≤ CL 2 + h− 2 . n

n

(17.2.4)

μk ≤L

Proof Observe first that δa H −s = O(1)hξ −s L2 = Os (1)h− 2 . n

(17.2.5)

h

 then by Proposition 16.2.2 and the subsequent observation In general, if u ∈ H −s1 (X), (where s was arbitrary) we have u=

∞ 

 μk −2s1 |αk |2 " u2

αk k ,

−s1

Hh

1

.

Thus, if s > s1 > n/2: u=



αk k + r,

(17.2.6)

μk ≤L

where r2H −s " h

⎛ ⎝



μk ≤L



μk −2s |αk |2 ≤ CL−2(s−s1 ) u2

⎞1

⎛

2

|αk |

2⎠

−s1

Hh

μk >L

≤ L ⎝ s1



(17.2.7)

,

⎞1 2

−2s1

μk 

μk ≤L

|αk |

2⎠

≤ CLs1 u2

−s1

Hh

In particular, when u = δa get the proposition by taking s1 = + n/2.

.

(17.2.8)  

Let Pδ be as in (16.5.1), (16.5.2). Let 0 < h ≤ α ≤ 1/2, and let t12 ≤ t22 ≤ · · · ≤ tN2 α be the eigenvalues of Pδ∗ Pδ in [0, α[ with tj > 0. We call tj the singular values of Pδ ,

17.2 Singular Values of Matrices Associated to Suitable Admissible. . .

369

arranged in increasing order. Let e1 , . . . , eNα be a corresponding system of eigenfunctions. From (16.5.18), we know that −n

Nα ≤ O(1)h

  δ  N VN (α) + h + , α

but for the moment we will only use that Nα ≤ N1 ≤ O(h−n ) (when δ ≤ α). Let a = (a1 , . . . , aNα ) ∈ XNα and put qa (x) =

Nα 

(17.2.9)

δaj (x),

j =1

 Mqa ;j,k =

qa (x)ek (x)ej (x)dx, 1 ≤ j, k ≤ Nα .

(17.2.10)

We lighten the notation by writing N instead of Nα , when possible. Then using (16.2.2), (16.5.26), and the fact that qa H −s = O(1)Nh−n/2 , we get for all λ, μ ∈ CN , h

 Mqa λ|μ =

qa (x)



λk ek

 

 μj ej dx = O(1)Nh−n λμ

and hence s1 (Mqa ) = Mqa L(CN ,CN ) = O(1)Nh−n .

(17.2.11)

We now choose a so that (17.1.14) and (17.1.15) hold, where we recall that s1 ≥ s2 ≥ · · · ≥ sN are the singular values of Mqa and Ej is defined in (17.1.6), where 0 ≤ ε1 ≤ ε2 ≤ · · · ≤ εN are the eigenvalues of the Gramian EX = ((ej |ek )L2 (X) )1≤j,k≤N . Since {ej } is an orthonormal system, Ej = N − j + 1.

(17.2.12)

Then (17.1.14) gives the lower bound 1

ln N N (N!) N s1 ≥ = (1 + O( )) , vol (X) N e vol (X) where the last identity follows from Stirling’s formula. Rewriting (17.1.15) as

sk ≥

− k−1 s1 N−k+1



N  1

Ej vol (X)

1  N−k+1

,

(17.2.13)

370

17 Proof II: Lower Bounds

and using (17.2.11), we get 

1

sk ≥ C

k−1 N−k+1

(vol (X))

N N−k+1

hn N



k−1 N−k+1

1

(N!) N−k+1 .

(17.2.14)

Summing up, we get Proposition 17.2.3 There exist a1 , . . . , aN ∈ X, with N = Nα defined above, such that N qa (x)ek (x)ej (x)dx, then the singular values if qa = 1 δ(x − aj ) and Mqa ;j,k = s1 ≥ s2 ≥ . . . ≥ sN of Mqa , satisfy (17.2.11), (17.2.13) and (17.2.14). We shall next approximate qa by an admissible potential. Apply Proposition 17.2.2 to each δ-function in qa , to see that qa = q + r, q =



αk k ,

(17.2.15)

μk ≤L

where n

qH −s ≤ Ch− 2 N,

(17.2.16)

h

rH −s ≤ C L−(s− 2 − ) h− 2 N, n

n

h



|αk |2

1 2

≤ CL 2 + h− 2 N. n

n

(17.2.17)

(17.2.18)

In order to estimate Mr , we write  Mr β|γ  =

r(x)



βk ek

 

 γj ej dx,

so that 0 0 0 0 n 0 0 0 0 βk ek 0Hhs 0 γj ej 0Hhs . |Mr β|γ | ≤ CrH −s h− 2 0 h

Applying (16.5.26) to the last two factors, we get with a new constant C > 0: n

|Mr β|γ | ≤ CrH −s h− 2 βγ , h

so Mr  ≤ Ch− 2 rH −s . n

h

(17.2.19)

17.2 Singular Values of Matrices Associated to Suitable Admissible. . .

371

Using (17.2.17), we get for every > 0 n

Mr  ≤ C L−(s− 2 − ) h−n N.

(17.2.20)

For the admissible potential q in (17.2.15), we thus obtain from (17.2.14), (17.2.20) and the fact that sk (Mq ) ≥ sk (Mδa ) − Mr : 

1

sk (Mq ) ≥ C

k−1 N−k+1

(vol (X))

N N−k+1

hn N



k−1 N−k+1

1

n

(N!) N−k+1 − C L−(s− 2 − ) h−n N. (17.2.21)

Similarly, from (17.2.11), (17.2.20) we get for L ≥ 1: Mq  ≤ CNh−n .

(17.2.22)

Using Proposition 16.2.2 and the subsequent observations, we get for all s1 > n/2, ⎛ qHhs ≤ O(1) ⎝



⎞1 2

2s

μk  |αk |

2⎠

μk ≤L

⎛ ≤ O(1) ⎝



⎞1 2

−2s1

μk 

|αk |

2⎠

Ls+s1

μk ≤L

≤ O(1)h− 2 NLs+s1 , n

where we used (17.2.16) with s replaced by s1 in the last step. Thus for every > 0, qHhs ≤ O(1)NLs+ 2 + h− 2 , ∀ > 0. n

n

(17.2.23)

Summing up, we have obtained Proposition 17.2.4 We can find an admissible potential q as in (17.2.15), (17.2.18) such that the matrix Mq , defined by  Mq;j,k =

qek ej dx,

satisfies (17.2.21), (17.2.22). Moreover the Hhs -norm of q satisfies (17.2.23). Here N ≤ O(h−n ) is the number of singular values of Pδ in [0, α 1/2 [ and we assume that δ ≤ α.  with real coefficients, then we can choose q real-valued. Notice also that if we choose R

372

17.3

17 Proof II: Lower Bounds

Lower Bounds on the Small Singular Values for Suitable Perturbations

Let P , p be as in (15.1.1)–(15.1.8). Fix z ∈ . Our unperturbed operator will be as in (16.5.1)–(16.5.3), where we change the notations slightly: P0 = P + δ0 Q0 ,

0 ≤ δ0

1, (17.3.1)

Q0 = O(1) : Hhσ → Hhσ uniformly for − s ≤ σ ≤ s.

We also assume that Q0 is the operator of multiplication by a potential, so that P0 also satisfies the symmetry assumption (15.1.7). Proposition 16.5.1 gives: 1, the number N0 (α 1/2 ) of eigenvalues of (P0 −

Proposition 17.3.1 For 0 < h ≤ α z)∗ (P0 − z) in [0, α] satisfies N0 (α

1/2

−n

) = ON,N (1)h

  δ0  N . VN (α) + h + α

(17.3.2)

We can get a slightly different and more interesting bound when δ0 /α is not very small: Recall the max-min formula (cf. (17.A.6)) tj (P0 − z) =

sup

codim L=j −1

inf (P0 − z)u,

u∈L u=1

and the similar one for tj (P − z). Now, (P0 − z)u ≥ (P − z)u − δ0 Q0  when u = 1. Thus (which can also be viewed as a special case of the Ky Fan inequalities), tj (P0 − z) ≥ tj (P − z) − δ0 Q0 . In particular, if tj (P0 − z) ≤ α 1/2 , then tj (P − z) ≤ α 1/2 + δ0 Q0 , leading to N0 (α 1/2 ) ≤ N(α 1/2 + δ0 Q0 ), where the right-hand side denotes the number of singular values of P − z in [0, α 1/2 + δ0 Q0 ], or equivalently, the number of eigenvalues of (P − z)∗ (P − z) in [0, (α 1/2 + δ0 Q0 )2 ]. Combining this with Proposition 17.3.1 for δ0 there equal to 0 and assuming that Q ≤ 1 for simplicity, we get  > 0, Proposition 17.3.2 For all M, N 

N0 (α 1/2 ) ≤ OM,N (1)h−n (VM ((α 1/2 + δ0 )2 ) + hN ). This is of interest when δ0 ≤ α 1/2 .

17.3 Lower Bounds on the Small Singular Values for SuitablePerturbations

373

Assume from now on that 1

0 ≤ δ0 ≤ h 2 .

(17.3.3)

Choose τ0 ∈ ]0, h1/2], α = h, and let N = N0 (τ0 ), N ≤ O(1)(hM−n + h−n VM (4h)) =: M0

(17.3.4)

be the number of singular values of P0 − z in [0, τ0 [ , labeled in increasing order so that 0 ≤ t1 (P0 − z) ≤ · · · ≤ tN (P0 − z) < τ0 ≤ tN+1 (P0 − z), and let e1 , e2 , . . . , eN be an associated family of eigenfunctions of (P0 − z)∗ (P − z). Fix θ ∈ ]0, 1/4[ . Proposition 17.3.3 If q is an admissible potential as in (17.2.1) with L ≥ 1, then  − 2 Ls |α|. q∞ ≤ Ch− 2 qHhs ≤ Ch n

n

(17.3.5)

If N is sufficiently large, there exists an admissible potential q as in (17.2.1) such that a) n

n

|α| ≤ L 2 + h− 2 N, n

(17.3.6)

.

(17.3.7)

n

qHhs ≤ O(1)h− 2 NLs+ 2 + . 1 s− n −

2

L " M0

− s−2n n −

h

2

b) The conclusion of Proposition 17.2.4 applies to q, so (17.2.21) and (17.2.22) hold. 1. Then Put Pδ = P0 + δq, 0 ≤ δ c) tν (Pδ − z) ≥ tν (P0 − z) − Cδh−n L 2 + +s N, ν ≥ 1. n

d) There exists N2 > 0 depending only on n, s, , that we can choose arbitrarily large, such that if δ = Cτ0 hN2 −n , C ( 1, then either for q as above or for q = 0, we have, tν (Pδ ) ≥ τ0 hN2 ,

1 + [(1 − θ )N] ≤ ν ≤ N.

(17.3.8)

Here [x] = sup(] − ∞, x] ∩ Z) denotes the integer part of x ∈ R. When N belongs to a bounded interval, a)–d) are still valid, if we restrict ν in (17.3.8) to the value ν = N.

374

17 Proof II: Lower Bounds

Since N ≤ M0  h−n , we get from (17.3.6), (17.3.7) that L  h−Mmin ,

3n

Mmin := 3n

s−

n 2

−

,

n

|α| ≤ R := h− 2 h−( 2 + )Mmin , corresponding to the minimal orders of magnitude in (15.2.5). Proof We choose q as in Proposition 17.2.4 so that (17.3.5), (17.3.6) follow from (17.2.18) and the estimates leading to (17.2.23). Then we also have (17.2.21) and we choose L large enough to guarantee that the first term to the right is dominant for 1 ≤ k ≤ N/2: We have for such k: sk (Mq ) ≥

1 n 1 hn N (N!) N − C L−(s− 2 − ) n . CN h

By Stirling’s formula, (N!)1/N ≥ N/Const, so we get with a new constant C > 0: sk (Mq ) ≥

n N hn − C L−(s− 2 − ) n , C h

1≤k≤

N . 2

(17.3.9)

Thus, sk (Mq ) ≥

hn , 2C

(17.3.10)

provided that Ls− 2 − ( Nh−2n . In view of (17.3.4), this is achieved with a sufficiently large L satisfying (17.3.7). We have then verified all the statements up to and including b). Statement c) follows from the max-min description of tν (Pδ ) and the fact that n

q∞ ≤ Ch−n L 2 + +s N. n

(17.3.11)

Let e1 , . . . , eN be an orthonormal family of eigenfunctions corresponding to tν (P0 −z), so that (P0 − z)∗ (P0 − z)ej = (tj (P0 − z))2 ej .

(17.3.12)

Using the symmetry assumption (15.1.7) which extends to P0 , we see that a corresponding orthonormal family of eigenfunctions of (P0 − z)(P0 − z)∗ is given by fj = ej .

(17.3.13)

17.3 Lower Bounds on the Small Singular Values for SuitablePerturbations

375

If the non-vanishing tj are not all distinct it is not immediately clear that we can arrange so that fj = fj as in (17.A.10) (with P replaced by P0 ), but we know that f1 , . . . , fN and f1 , . . . , fN are orthonormal families that span the same space FN . Let EN be the span of e1 , . . . , eN . Then (P0 − z) : EN → FN and (P0 − z)∗ : FN → EN

(17.3.14)

have the same singular values 0 ≤ t1 ≤ t2 ≤ · · · ≤ tN . Define R+ : L2 → CN , R− : CN → L2 by R+ u(j ) = (u|ej ),

R− u− =

N 

u− (j )fj .

(17.3.15)

1

Then  P=

P0 − z R− R+ 0

 : Hhm × CN → L2 × CN

(17.3.16)

has a bounded inverse  E=

0 E 0 E+ 0 0 E− E−+

 .

0 = diag (t ), only Since we do not necessarily have (17.A.10), we cannot assert that E−+ j 0 0 are given by that E−+ is unitarily equivalent to diag (tj ), but the singular values of E−+ 0 ) = t (P − z), 1 ≤ j ≤ N, or equivalently by s (E 0 ) = t tj (E−+ j 0 j N+1−j (P0 − z), for −+ 1 ≤ j ≤ N. We here follow the convention of using the letter s for labelling the singular values in decreasing order and t for the increasing order. We will apply Sect. 17.A, and recall that N is assumed to be sufficiently large and that θ has been fixed in ]0, 1/4[ . Since z is fixed, it will also be notationally convenient to assume that z = 0. We consider two cases. 0 ) ≥ 8τ0 hN2 , for 1 ≤ j ≤ N − [(1 − θ )N]. Then from (17.A.37) we get Case 1. sj (E−+ the conclusion in d) with q = 0, Pδ = P0 . Case 2. 0 ) < 8τ0 hN2 for some j such that 1 ≤ j ≤ N − [(1 − θ )N]. sj (E−+

(17.3.17)

From (17.2.22) and the fact that N ≤ M0 in (17.3.4) we get s1 (Mq ) ≤ Mq  ≤ CM0 h−n .

(17.3.18)

376

17 Proof II: Lower Bounds

Put δQ, Pδ = P0 + δq = P0 + 

(17.3.19)

 δ, Q ≤ 1, δ = δCh−n L 2 + +s M0 , Q = δq/ n

where the last estimate comes from (17.3.11). Then, if  δ ≤ τ0 /2, we can replace P0 by Pδ in (17.3.16) and we still have a well-posed problem with inverse as in (17.A.25)– (17.A.31), satisfying (17.A.35)–(17.A.37) with Qω = Q as above and with δ replaced 0 QE 0 = M /(Ch−n L n2 + +s M ) and according to (17.3.10), we have with by  δ. Here E− q 0 + a new constant C 0 0 0 0 qE+ ) = sk ( δE− QE+ )≥ sk (δE−

δhn , C

1≤k≤

N . 2

(17.3.20)

Playing with the general estimate (17.A.15), we get sν (A + B) ≥ sν+k−1 (A) − sk (B), and for a sum of three operators sν (A + B + C) ≥ sν+k+−2 (A) − sk (B) − s (C). 

δ in (17.A.36) and get We apply this to E−+

 δ2  0 0 0 δ ) ≥ sν+k−1 ( δE− QE+ ) − sk (E−+ )−2 . sν (E−+ τ0

(17.3.21)

Here we use (17.3.17), (17.3.20) with j = k = N − [(1 − θ )N], to get for ν ≤ N − [(1 − θ )N] 

δ sν (E−+ )≥

Recall that θ < when

1 4

 δhn δ2 − 8τ0 hN2 − 2 . C τ0

(17.3.22)

and that  δ is given in (17.3.19). The first term to the right dominates

δhn ( τ0 hN2 , δhn (

 δ2 , τ0

(17.3.23)

i.e., when τ0 hN2 −n

δ

τ0

h3n Ln+2 +2s M02

,

(17.3.24)

17.3 Lower Bounds on the Small Singular Values for SuitablePerturbations

377

where the last term is " τ0

3n+2n n+2 +2s s− n −

h

2

2+ n+2 +2s n

M0

s− 2 −

by (17.3.7) and M0 ≤ O(h−n ) by (17.3.4), so the last term in (17.3.24) is ≥ τ0 hN3 /O(1) for some N3 = N3 (n, s, ) > 0 that can be explicitly computed. It suffices to choose N2 > N3 + n and we have a non-empty range of δ satisfying (17.3.24). In particular we can take δ = Cτ0 hN2 −n for some large enough C. Then we get from (17.3.22) 

δ ) ≥ 8τ0 hN2 , 1 ≤ ν ≤ N − [(1 − θ )N]. sν (E−+ 



δ )=s δ Equivalently, for tν (E−+ N+1−ν (E−+ ) we get 

δ tν (E−+ ) ≥ 8τ0 hN2 , 1 + [(1 − θ )N] ≤ ν ≤ N.

(17.3.25)

From (17.3.25) and (17.A.37), we get (17.3.8). When N = O(1), we still get (17.3.22) with ν = k = 1 and this gives the last statement.   The conclusion of c) in the proposition (which of course is valid also when we take q = 0 in d)) implies that with the choice of δ as in d), we have tν (Pδ ) ≥ tν (P0 ) − τ0 hN2 −K , for some K which depends only on n, s, . Choosing N2 sufficiently large, we get tν (Pδ ) ≥ tν (P0 ) − τ0 hN1 , where N1 > 0 can be taken arbitrarily large (if we choose N2 sufficiently large). The construction can now be iterated. Assume that N ( 1 and replace (P0 , N, τ0 ) (1) by (Pδ , N (1) , τ0 hN2 ) =: (P (1) , N (1) , τ0 ), where N (1) ≤ [(1 − θ )N] is the number of (1) singular values of P (1) in [0, τ0 ] and keep on, using the same values for the exponents N1 , N2 . Then we get a sequence (P (k) , N (k) , τ0(k) ), k = 0, 1, . . . , k(N), where the last value k(N) is determined by the fact that N (k(N)) is of the order of magnitude of a large constant. Moreover, (k)

tν (P (k) ) ≥ τ0 , N (k) < ν ≤ N (k−1) ,

(17.3.26)

(k)

(17.3.27)

tν (P (k+1) ) ≥ tν (P (k) ) − τ0 hN1 , ν > N (k) ,

378

17 Proof II: Lower Bounds (k+1)

τ0

(k)

= τ0 hN2 ,

(17.3.28)

N (k+1) ≤ [(1 − θ )N (k) ] is the number of (k+1)

singular values of P (k+1) in [0, τ0

],

(17.3.29)

P (0) = P0 , N (0) = N, τ0(0) = τ0 . Here, δ (k+1) = Cτ0(k) hN2 −n ,

P (k+1) = P (k) + δ (k+1) q (k+1),

q (k+1)Hhs ≤ Ch−K .

Notice that N (k) decays exponentially fast with k: N (k) ≤ (1 − θ )k N,

(17.3.30)

so we get the condition on k that (1 − θ )k N ≥ C ( 1 which gives, k≤

ln N C 1 ln 1−θ

(17.3.31)

.

We also have (k)

τ0

 k = τ0 hN2 .

(17.3.32)

For ν > N, we iterate (17.3.27) to get   tν (P (k) ) ≥ tν (P ) − τ0 hN1 1 + hN2 + h2N2 + · · ·

(17.3.33)

≥ tν (P ) − 2τ0 hN1 . For 1 that

ν ≤ N, let  = (N) be the unique value for which N () < ν ≤ N (−1) , so

tν (P () ) ≥ τ0(),

(17.3.34)

by (17.3.26). If k > , we get ()

tν (P (k) ) ≥ tν (P () ) − 2τ0 hN1 .

(17.3.35)

17.3 Lower Bounds on the Small Singular Values for SuitablePerturbations

379

ln N (k0 ) = C 1 ) for which N ln 1−θ decreasing N (k) by one unit at each

The iteration above works until we reach a value k = k0 = O(

O(1). After that, we continue the iteration further by step.   1 )/ ln( ) + O(1), be the last k-value we get in this way. Then, by Let k1 = O ln( N C 1−θ construction,   P (k1 +1) = P0 + δ (1) q (1) + . . . + δ (k1 +1) q (k1 +1)   P0 + δ q (1) + hN2 q (2) + . . . + hk1 N2 q (k1 +1) ,

where δ = Cτ0(0) hN2 −n and q (j ) satisfy (17.3.6), (17.3.7) uniformly. In particular, q (k+1) Hhs ≤ Ch−K and this also holds for q (1) + hN2 q (2) + · · · + hk1 N2 q (k1 +1) . Summing up the discussion so far, we have obtained Proposition 17.3.4 Let P0 , z, s > + n/2, > 0 be as in Proposition 17.3.3. Choose τ0 ∈ ]0, h1/2], so that the number N of singular values of P0 −z in [0, τ0 [ satisfies (17.3.4). Let N2 > 0 be large enough. Let 0 < θ < 14 and let N(θ ) ≤ O(1) be sufficiently large. We can find N = N (0) > N (1) > · · · > N (k1 ) = 1 and an admissible potential q = qh (x) as in (17.2.1), satisfying (17.3.5)–(17.3.7), such that if Pδ = P0 + δq, δ = Cτ0 hN2 −n , the following holds: As long as N (k) ≥ N(θ ), we have N (k+1) ≤ [(1 − θ )N (k) ], N (0) = N. Let k0 ≥ 0 be the last k value we get in this way. For k > k0 we have N (k+1) = N (k) − 1, until we reach the value k1 for which N (k1 ) = 1. Put τ0(k) = τ0 hkN2 , 1 ≤ k ≤ k1 + 1. We have the following estimates on the singular values of Pδ − z: N −K

2 • If ν > N (0) , then tν (Pδ − z) ≥ (1 − h C )tν (P0 − z). (k) (k−1) • If N < ν ≤ N , 1 ≤ k ≤ k1 , then tν (Pδ − z) ≥ (1 − O(hN2 −K ))τ0(k) . (k +1) • Finally, if ν = N (k1 ) = 1, then t1 (Pδ − z) ≥ (1 − O(hN2 −K ))τ0 1 .

Here K = K(n, s, ) is independent of the other parameters. We shall now obtain the corresponding estimates for the singular values of Pδ,z = (Pδ −  is defined as in the beginning of Sect. 16.3 and Pδ = Pδ + P − P . − z), where P Let√ e1 , . . . , eN be an orthonormal family corresponding to the singular values tj (Pδ ) in [0, h[ , put fj = ej , and let z)−1 (Pδ

(Pδ − z)u + R− u− = v,

R+ u = v+

380

17 Proof II: Lower Bounds

be the corresponding Grushin problem so that the solution operators fulfill 1 E ≤ √ , E±  ≤ 1, h

tj (E−+ ) = tj (Pδ − z) ≤

√

h, 1 ≤ j ≤ N.

(17.3.36)

− = P−1 R− . Then the In order to shorten the notation, we shall assume that z = 0. Put R δ problem − u− = v, Pδ,z u + R

R+ u = v+ ,

is well posed, with the solution  +E + v+ , u = Ev

− v + E −+ v+ , u− = E

where + = E+ ,  = EP δ , E E − = E− Pδ , E −+ = E−+ . E Adapting the estimate (17.A.18) to our situation, we get tk (Pδ,z ) ≥

tk (Pδ ) , 1 ≤ k ≤ N, δ   E Pδ tk (Pδ ) + E+ E− P

where we also recall that tk (Pδ ) ≤ Write

√ h.

E Pδ = EPδ + E(P − P ), E− Pδ = E− Pδ + E− (P − P ), and use that EPδ = 1 − E+ R+ = O(1) in L(L2 , L2 ), √ E− Pδ = −E−+ R+ = O( h) in L(L2 , 2 ), together with (17.3.36) and the fact that P − P  = O(1). It follows that 1 E Pδ  = O( √ ), E− Pδ  = O(1). h

(17.3.37)

17.3 Lower Bounds on the Small Singular Values for SuitablePerturbations

381

Using this in (17.3.37), we get tk (Pδ,z ) ≥

where used that tk (Pδ ) ≤

tk (Pδ ) tk√ C (Pδ ) + C h

≥

tk (Pδ ) , 2C

(17.3.38)

√ h when 1 ≤ k ≤ N. From Proposition 17.3.4 we get:

Proposition 17.3.5 Let P0 , z, s > + n/2, > 0 be as in Proposition 17.3.3. Choose τ0 ∈ ]0, h1/2 ], so that the number N of singular values of P0 −z in [0, τ0 [ satisfies (17.3.4). Let N2 > 0 be large enough. Let 0 < θ < 14 and let N(θ ) ≤ O(1) be sufficiently large. Let N = N (0) > N (1) > · · · > N (k1 ) = 1, q = qh (x), τ0(k) = τ0 hkN2 , 1 ≤ k ≤ k1 + 1 be as in Proposition 17.3.4. We have the following estimates on the singular values of Pδ,z (defined after Proposition 17.3.4) • If ν > N (0) , then tν (Pδ,z ) ≥ (1 −

hN2 −K C )tν (P0,z ).

(1−O (hN2 −K ))τ0 (2C) (k +1) (1−O (hN2 −K ))τ0 1 . (2C)

• If N (k) < ν ≤ N (k−1) , 1 ≤ k ≤ k1 , then tν (Pδ,z ) ≥ • Finally, if ν = N (k1 ) = 1, then t1 (Pδ − z) ≥

(k)

.

Here K = K(n, s, ) is independent of the other parameters. We recall that, by Proposition 16.5.3, ln det(Sδ,z + αχ(α −1 Sδ,z )) =   1 ln sz (x, ξ )dxdξ (2πh)n   α  α0 dt δ dt h V (t) + + h , + O(1) VN (t) + t t t α 0 α n

when Pδ = P + δ(h 2 q1 + q2 ) = P + δQ and q1 Hhs , q2 H1s ≤ 1, 0 < h ≤ α 1. ∞ −1  Here 0 ≤ χ ∈ C0 ([0, +∞[ ), χ(0) > 0 and Sδ,z = (Pδ − z) (Pδ − z). We apply this with P0 = P in (17.3.1) and Pδ as in Proposition 17.3.4 and with δ above δ Q, Q = O(1) : Hhσ → Hhσ for −s ≤ σ ≤ s. replaced by  δ in (17.3.19), so that Pδ = P + Thus we get ln det(Sδ,z + αχ(α −1 Sδ,z )) =   1 ln sz (x, ξ )dxdξ (2πh)n   α  α0 dt  δ + δ0 h dt V (t) + +h , VN (t) + + O(1) t t t α α 0

(17.3.39)

382

17 Proof II: Lower Bounds

where  δ = δCh−n L 2 + +s M0 with M0 given in (17.3.4), L is given in (17.3.7), and δ is given in d) in Proposition 17.3.3. Notice that n

 δ = O(hN2 −K τ0 ), α

(17.3.40)

where K is independent of N2 . Now choose α = h and set out to estimate ln det Sδ,z . First we have the upper bound ln det Sδ,z ≤ ln det(Sδ,z + αχ(α −1 Sδ,z )), which can be bounded from above by the right hand side of (17.3.39). As for the lower bound, we use that ln det Sδ,z = ln det(Sδ,z + αχ(α

−1

Sδ,z )) −

∞ 

 ln

tk2 + αχ(tk2 /α)



tk2

1

,

(17.3.41)

where 0 < t1 ≤ t2 ≤ · · · are the singular values of Pδ,z , treated in Proposition 17.3.5. The last sum in (17.3.41) has at most finitely many non-vanishing terms and is equal to ∞ −2 2 1 ln(1 + αχ(tk /α)tk ). Assuming that 0 ≤ χ ≤ 1, it can be bounded from above by 

ln(1 + αtk−2 ) ≤ (#{k; tk2 ≤ α sup supp χ}) ln(1 + αt1−2 ).

(17.3.42)

tk2 ∈αsupp χ

Combining (17.3.38) with Proposition 17.3.2 (cf. (17.3.4)), we have for every N > 0, #{k; tk2 ≤ α sup supp χ} ≤ ON (1)(hN−n + hn VN (Ch)). From Proposition 17.3.5 and the fact that k1 = O(ln(1/ h)) we see that t1 ≥ τ0 hN2 O(1) ln h ≥ τ0 hO(1) ln h , 1

1

so t1−2 ≤ τ0−2

 O(1) ln 1 h 1 , h

and hence ln(1 + αt1−2 )

  1 2 1 ≤ O(1) ln ln , h τ0

(17.3.43)

17.4 Estimating the Probability that det Pδ,z Is Small

383

which in conjunction with (17.3.43) and (17.3.42) gives  tk2 ≤α sup supp χ

   1 1 2 N−n −n ln(1+αtk−2 ) ≤ O(1) ln (h +h VN (Ch)). ln τ0 h

(17.3.44)

Finally, (17.3.41), (17.3.39) and (17.3.40) provide the lower bound for every N > 0, ln det Sδ,z ≥ (2πh)−n

.  ln sz (x, ξ )dxdξ

 /  α0 dt h dt δ0 + V (t) + O(τ0 hN2 −K + + h) t t t h 0 h   2   1 1 − O(1) ln hN−n + h−n VN (Ch) . ln τ0 h (17.3.45) 

+ O(1)

h

VN (t)

Here we have chosen α = h in (17.3.39), motivated by the fact that 

α

α −→

V (t) 0

dt + t



α0 α

dt h V (t) t t

is minimal at α = h. In the following, we assume that δ0 ≤ O(h2 ) and that N2 is large enough so that τ0 hN2 −K +

17.4

δ0 ≤ O(h). h

(17.3.46)

Estimating the Probability that det Pδ,z Is Small

In this section we keep the assumptions on (P , z) made in the beginning of Sect. 17.3 and choose P0 as in (17.3.1) with 0 ≤ δ0 ≤ h2 as in (15.3.4). As in the preceding section, we consider perturbations of P0 of the form Pδ = P0 + δq,

δ = Cτ0 hN2 −n , C ( 1,

(17.4.1)

 ) and q is an admissible potential of the form (17.2.1) where N2 ≥ N2 (n, s, , M, M q=



αk k ,

α ∈ CD ;

(17.4.2)

0 here 1 , 2 , . . . is an orthonormal basis of eigenfunctions of the auxiliary operator h2 R 0, which is an elliptic semi-classical differential operator of the form (15.2.1) with smooth

384

17 Proof II: Lower Bounds

 k = μ2 k , μk # +∞. We will assume that L ≥ 1 and recall that coefficients, and h2 R

k D := {k; μk ≤ L} = O((L/ h)n ) = O(h−N3 ), N3 = (M + 1)n. Here we impose the following bounds on the sum in (17.4.2) (cf. (15.2.5)): −M

Lmin ≤ L ≤ h

,

1 s− n −

2

Lmin " M0



|α| ≤ R, Rmin ≤ R ≤ h−M ,

− s−2n n −

h

2

n

+

,

M≥

n

2 Rmin := Lmin M0 h− 2 ,

3n s−

n 2

−

,

(17.4.3)

 ≥ 3n + ( n + )M. M 2 2 (17.4.4)

Recall from (17.3.4) that M0 = hM−n + h−n VM (4h) ≤ O(h−n ),

(17.4.5)

and that the number N of singular values of Pδ − z in [0, τ0 [ fulfills (17.3.4): N ≤ O(1)M0 ,

(17.4.6)

for any M > 0. The main conclusion of the preceding section is that there exists q as in (17.2.1) with L = Lmin , |α| ≤ Rmin , such that (17.3.45) holds (and q is the random potential in Pδ in the Propositions 17.3.4 and 17.3.5). We next review the upper bounds of Chap. 16 culminating in Proposition 16.5.3 that we apply with α = h, recalling that “δ” there refers to sup Pδ − P L(Hhσ ,Hhσ ) .

−s≤σ ≤s

With the notations of the present chapter, we get for general q as in (17.4.2)–(17.4.4): ln det(Sδ,z ) ≤  h .    α0 dt 1 dt h + V (t) (x, ξ )dxdξ + O(1) V (t) ln s z N (2πh)n t t t 0 h  / δ −n + O(1) h + h 2 qHhs . h

(17.4.7)

17.4 Estimating the Probability that det Pδ,z Is Small

385

Recall that δ is given in (17.4.1) and that 

qHhs ≤ O(1)RLs ≤ O(1)h−M−sM , so (17.4.7) gives ln det(Sδ,z ) ≤  h .    α0 dt h 1 dt V (t) VN (t) + ln sz (x, ξ )dxdξ + O(1) (2πh)n t t t h 0 1  n  + O(1) h + τ0 hN2 −n−1− 2 −M−sM ,

(17.4.8)

and this still holds if we allow α to vary in the larger ball 

|α|CD ≤ 2R = O(h−M ).

(17.4.9)

(Our probability measure will be supported in BCD (0, R), but we will need to work in a larger ball.) We consider the holomorphic function  F (α) = (det Pδ,z ) exp −

1 (2πh)n





ln |pz |dxdξ ,

(17.4.10)

where we recall that det Sδ,z = | det Pδ,z |2 , sz (x, ξ ) = |pz (x, ξ )|2 . Then by (17.4.8), we have ln |F (α)| ≤ 0 (h)h−n , |α| < 2R,

(17.4.11)

and for one particular value α = α 0 with |α 0 | ≤ 12 R, corresponding to the special potential in (17.3.45): ln |F (α 0 )| ≥ − 0 (h)h−n ,

(17.4.12)

where we put with C > 0 sufficiently large, 

h

0 (h)/C = 0



α0

dt h V (t) t t h  2 1 1 (hN−n + VN (Ch)) + τ0 hN2 −K , + h + ln ln τ0 h

VN (t)

dt + t

for an arbitrary fixed N > 0. Here K is independent of the choice of N2 ( 1.

(17.4.13)

386

17 Proof II: Lower Bounds

Let α 1 ∈ CD with |α 1 | = R and consider the holomorphic function of one complex variable f (w) = F (α 0 + wα 1 ).

(17.4.14)

We will mainly consider this function for w in the disc Dα 0 ,α 1 = D(w0 , r0 ) determined by the condition |α 0 + wα 1 | < R:   0 2  0 1 2  0 1 2  α    α α    +  α | α  =: r 2 , Dα 0 ,α 1 : w + | < 1 − 0     R R R R R 

(17.4.15)

√ √ with center w0 = −(α 0 /R|α 1 /R) and radius r0 ∈ [ 3/2, 5/2]. From (17.4.11), (17.4.12) we get ln |f (0)| ≥ − 0 (h)h−n , ln |f (w)| ≤ 0 (h)h−n .

(17.4.16)

By (17.4.11), we may assume that the last estimate holds in a larger disc, say 0 1 0 1 D(−( αR | αR ), 2r0 ). Let w1 , . . . , wM be the zeros of f in D(−( αR | αR ), 3r0 /2). Then it is standard to get the factorization f (w) = eg(w)

M 

(w − wj ),

j =1

w ∈ D(−(

α0 α1 | ), 4r0 /3), R R

(17.4.17)

together with the bounds |g(w)| ≤ O( 0 (h)h−n ),

M = O( 0 (h)h−n ).

(17.4.18)

See for instance Section 5 in [135] where further references are given. For 0 <

1, put ( ) = {r ∈ [0, 2r0 [; ∃w ∈ Dα 0 ,α 1 such that |w| = r and |f (w)| < }.

(17.4.19)

If r ∈ ( ) and w is a corresponding point in Dα 0 ,α 1 , we have with rj = |wj |, M  j =1

|r − rj | ≤

M 

|w − wj | ≤ exp(O( 0 (h)h−n )).

(17.4.20)

j =1 −n

Then at least one of the factors |r − rj | is bounded by ( eO( 0(h)h ) )1/M . In particular, −n the Lebesgue measure λ(( )) of the set ( ) is bounded by 2M( eO( 0(h)h ) )1/M .

17.4 Estimating the Probability that det Pδ,z Is Small

387

Noticing that the last bound increases with M when the last member of (17.4.20) is ≤ 1 and that {r ∈ [0, +∞[∩Dα 0 ,α 1 ; f (r) < } ⊂ ( ), we get Proposition 17.4.1 Let α 1 ∈ CD with |α 1 | = R and assume that > 0 is small enough so that the last member of (17.4.20) is ≤ 1. Then λ({r ∈ [0, +∞[ ; |α 0 + rα 1 | < R, |F (α 0 + rα 1 )| < }) ≤  

0 (h) hn ln . exp O(1) + hn O(1) 0 (h)

(17.4.21)

Here and in the following, the symbol O(1) in a denominator indicates a bounded positive quantity.   (h) Typically, we can choose = exp − h0n+α for some small α > 0 and then the upper bound in (17.4.21) becomes  

0 (h) 1 exp O(1) − . hn O(1)hα Now we equip BCD (0, R) with a probability measure of the form P(dα) = C(h)e!(α) L(dα),

(17.4.22)

where L(dα) is the Lebesgue measure, ! is a C 1 function which depends on h and satisfies |∇!| = O(h−N4 ), and C(h) is the appropriate normalization constant. Writing α = α 0 + Rrα 2 , 0 ≤ r < r0 (α 2 ), α 2 ∈ S 2D−1 ,

(17.4.23)

1 2

≤ r0 ≤ 32 , we get

φ(r) 2D−1  r drS(dα 2 ), P(dα) = C(h)e

(17.4.24)

 Here where φ(r) = φα 0 ,α 2 (r) = !(α 0 + rRα 2 ) so that φ  (r) = O(h−N5 ), N5 = N4 + M. S(dα 2 ) denotes the Lebesgue measure on S 2D−1 . For a fixed α 2 , we consider the normalized measure φ(r) 2D−1  r dr μ(dr) = C(h)e

(17.4.25)

on [0, r0 (α 2 )] and we want to show an estimate similar to (17.4.21) for μ instead of λ. Write eφ(r)r 2D−1 = exp(φ(r)+(2D −1) ln r) and consider the derivative of the exponent: φ  (r) +

2D − 1 . r

388

17 Proof II: Lower Bounds

This derivative is ≥ 0 for r ≤ 2 r0 , where  r0 = C −1 min(1, DhN5 ) for some large constant C, and we may assume that 2 r0 ≤ 1/2. Introduce the measure  μ ≥ μ by φ(rmax ) 2D−1   μ(dr) = C(h)e rmax dr,

rmax := max(r, r0 ).

(17.4.26)

Since  μ([0, r0 ]) ≤ μ([ r0 , 2 r0 ]), we get μ([0, r(α 2 )]) ≤ O(1). 

(17.4.27)

ψ(r)   μ(dr) = C(h)e dr,

(17.4.28)

We can write

where ψ  (r) = O(max(D, h−N5 )) = O(h−N6 ), N6 = max(N3 , N5 ),

(17.4.29)

where D = O(h−N3 ) by the estimate on D prior to (17.4.3). Decompose [0, r0 (α 2 )] into " h−N6 intervals of length " hN6 . If I is such an interval, we see that  μ(dr) λ(dr) λ(dr) ≤ ≤C Cλ(I )  μ(I ) λ(I )

on I.

(17.4.30)

From (17.4.21), (17.4.30) we get when the right-hand side of (17.4.20) is ≤ 1, μ({r ∈ I ; |F (α 0 + rRα 2 )| < })/  μ(I ) ≤ =

0 (h) O(1)h−N6 n h

  O(1) 0 (h) hn ln

exp λ(I ) hn O(1) 0 (h)

 hn ln . exp O(1) 0 (h) 

Use that  μ([0, r0 (α 2 )]) = O(1), multiply with  μ(I ) and sum the estimates over I , to get  μ({r ∈ [0, r(α 2 )]; |F (α 0 + rRα 1 )| < })   hn −N6 0 (h) ln . ≤ O(1)h exp hn O(1) 0 (h)

(17.4.31)

Since μ ≤  μ, we get the same estimate with  μ replaced by μ. Then from (17.4.24) we get

17.5 End of the Proof

389

Proposition 17.4.2 Let > 0 be small enough for the right-hand side of (17.4.20) to be ≤ 1. Then P(|F (α)| < ) ≤

0 (h) O(1)h−N6 n h

 hn ln . exp O(1) 0 (h) 

(17.4.32)

 has real coefficients, we may assume that the Remark 17.4.3 In the case when R eigenfunctions j are real, and from the observation after Proposition 17.2.4 we see that we can choose α0 above to be real. The foregoing discussion can then be restricted to the case of real α 2 and hence to real α. We can then introduce the probability measure P as in (17.4.22) on the real ball BRD (0, R). The subsequent discussion goes through without any changes, and we still have the conclusion of Proposition 17.4.2. Remark 17.4.4 Choosing N2 sufficiently large in the definition of δ, we see that 0 (h) in (17.4.13) satisfies 

h

0 (h) ≤ C 0

dt VM (t) + t



α0 h

dt 1 h V (t) + ln t t τ0

   1 2 VM (Ch) + h , ln h (17.4.33)

C = C(M), M > n. Remark 17.4.5 Let us write = exp(−

/ hn ). Proposition 17.4.2 shows that if 

≥ C 0 (h) > 0 for C > 0 sufficiently large, then  P(|F (α)| < exp(−

/ hn )) ≤ O(1)h−N6 −n 0 (h) exp −

17.5

 

. O(1) 0 (h)

(17.4.34)

End of the Proof

We first have a closer look at the two integrals appearing in the definition of 0 (h) in (17.4.13). By (16.3.37), 

h

VN (t) 0

dt = t



h 1

1+

0

0

σ2 t

−N dV (σ 2 )

dt . t

(17.5.1)

Recall from Proposition 16.4.2 (using also Proposition 16.4.1, see also [56, 104]) that the function   1 φ(z) := ln |pz (x, ξ )|dxdξ ln sz (x, ξ )dxdξ = 2

390

17 Proof II: Lower Bounds

is continuous and subharmonic with φ (w)L(dw) = p∗ (dxdξ ). 2π

(17.5.2)

Hence,  V (σ 2 ) =

|w−z|≤σ

φ (w)L(dw), 2π

so  VN (t) =

−N  |w − z|2 φ (w)L(dw). 1+ t 2π |w−z|≤1

(17.5.3)

This yields 

h

VN (t) 0

−N  dt |w − z|2 φ (w)L(dw) 1+ t 2π t 0 |w−z|≤1    |w − z|2 φ (w)L(dw), fN = h 2π |w−z|≤1

dt = t



h

(17.5.4)

where |w − z|2 fN ( )= h

h

 0

|w − z|2 1+ t

1/λ 

 fN (λ) =

1 1+ τ

0

1/λ 

 = 0

τ 1+τ

−N N

−N

dt = t



h |w−z|2

0

  1 −N dτ , 1+ τ τ

dτ τ

⎧ ⎨(1 + ln 1 ), dτ λ " ⎩λ−N , τ

if 0 < λ ≤ 1,

(17.5.5)

if λ ≥ 1.

Next, we have a look at 

1 h

dt h V (t) = t t



1

h dτ = V (τ 2 )2 √ 2 τ τ h  1  =



1



√ h |w−z|≤τ

2h dτ φ (w)L(dw) 2 2π τ τ

2h dτ φ (w)L(dw) τ 2 τ 2π     φ h = ,1 − h (w)L(dw). min 2 2π |w − z| |w−z|≤1

√ |w−z|≤1 max(|w−z|, h)

17.5 End of the Proof

391

Thus, 

1

h

dt h V (t) = t t



 g0

|w − z|2 h



φ (w)L(dw), 2π

⎧  ⎨1 − h, 1 g0 (λ) = min( , 1) − h = ⎩( 1 − h)+ , λ + λ

if λ ≤ 1,

(17.5.6)

(17.5.7)

if λ > 1.

Combining (17.5.4), (17.5.5), (17.5.3), (17.5.6) and (17.5.7) we get (cf (17.4.13), (17.4.33)):  α0 dt h dt V (t) + VN (Ch) JN (z; h) : = VN (t) + t t t 0 h    2 φ |z − w| (w)L(dw), kN = h 2π |z−w|≤1 

h

⎧ 1 ⎨(1 + ln λ1 ), 1 kN (λ)  k(λ) := + ln+ " 1+λ λ ⎩1, λ

if 0 < λ ≤ 1, if λ ≥ 1.

(17.5.8)

(17.5.9)

Applying (17.4.33),we get

0 (h)  ln

1 τ0

     1 2 |z − w|2 φ (w)L(dw) + O(h). ln k h h 2π |z−w|≤1

(17.5.10)

Let    be a √ domain with Lipschitz boundary as in the beginning of Chap. 12, with 0 ∈ ∂ be distributed on the boundary as in that constant scale r = h. Let z10 , z20 , . . . , zN  √ √ ! 0 , h), dist (z0 , z0 ) " D(z h, with the cyclic convention chapter, so that ∂ ⊂ N 1 j j j +1  + 1 = 1. Let J = JN (z; h) be defined in (17.5.8). We are interested in that N  N 

 J (zj0 ; h) 

G0 (w)

φ (w)L(dw), 2π





j =1

(17.5.11)

where G (w) = 0

 N  j =1

k

|zj0 − w|2 h

.

(17.5.12)

392

17 Proof II: Lower Bounds

√ We apply the averaging argument at the end of Sect. 12.3 with r equal to √ h and zj ∈ D(zj0 , h) for μ+ (dw) = (2π)−1 φ(w)L(dw) and conclude that there exist  any fixed > 0, such that 

 √ D( zj , h)

k

| zj − w|2 h



φ (w)L(dw) = O(1) 2π

 √ D( zj , h)

φ (w)L(dw). 2π (17.5.13)

We now replace the sum in (17.5.11) by  N 

 J ( zj ; h) =

G1 (w)

φ (w)L(dw), 2π





1

(17.5.14)

where G1 (w) =

 N 

k

1

| zj − w|2 h

,

(17.5.15)

so that by the preceding observation 

φ G (w) (w)L(dw) ≤ O(1) 2π 1

G (w) = 2

 N 



 G2 (w)

| zj − w|2  k h

φ (w)L(dw), 2π

 ,

(17.5.16)

  |w − z|2 h  k . = h h + |w − z|2

(17.5.17)

1

From the form of  k, we see that the order of magnitude of the right-hand side in (17.5.14) will not change if we replace  zj in (17.5.15) by zj0 . Thus,  N  j =1

 J ( zj ; h) ≤ O(1)

G(w)

φ (w)L(dw), 2π

(17.5.18)

17.5 End of the Proof

393

where G(w) =

 N 

  k

j =1

|zj0 − w|2 h

 (17.5.19)

.

From the geometric assumptions on , we see that the order of magnitude of G and hence the validity of (17.5.18) will not change if we replace G in (17.5.19) by   |z − w|2 |dz|  G(w) = G(w, ; h) = k √ . h h ∂ 

(17.5.20)

By a change of variables, w 1 G(w, ; h) = G( √ , √ ; 1). h h

(17.5.21)

Here   = h−1/2  is a Lipschitz domain as in Chap. 12 with constant scale r = 1, so the problem of studying G is reduced to that of studying G( w,  ; 1) =

 ∂ 

1 |dz|. 1 + |z − w |2

(17.5.22)

Consider the following regularity property for ∂   : For some fixed κ ∈ [0, 1[ , we have  ≤ O(1)R 1+κ , |∂   ∩ D( w , R)|

 ≥ 1, R

(17.5.23)

uniformly with respect to w . Then by writing G( w,  ; 1) =



+∞ d( w,∂  )

1  d|∂   ∩ D( w , R)|, 2 1+R

we see that G( w,  , 1) = O(1)(1 + d( w, ∂  ))κ−1 .

(17.5.24)

The most regular case is that with κ = 0. Returning to , we may assume that for some κ ∈ [0, 1[ :   √ R 1+κ , |∂ ∩ D(w, R)| ≤ O(1) h √ h

R≥

√ h.

(17.5.25)

394

17 Proof II: Lower Bounds

Then relations (17.5.21)–(17.5.24) lead to   d(x, ∂) κ−1 G(w, ; h) = O(1) 1 + √ . h

(17.5.26)

Remark 17.5.1 The estimate (17.5.26) can be improved when d(w, ) ≥ diam (). Let us first observe that (17.5.25) with w ∈ ∂ and R = diam () gives the estimate   √ diam () 1+κ |∂| ≤ O(1) h . √ h

(17.5.27)

When d(w, ∂) ≥ diam () we get from (17.5.20) |∂| 1 G(w, ; h) " √ . 2 d(w,∂) h 1+

(17.5.28)

h

From (17.5.27), we see that this estimate is sharper than (17.5.26) when d(w, ) ≥ diam (). Let us sum up the results obtained so far, √ in order to apply Theorem 12.1.2. We choose as in that theorem with r there equal to h. Apply Remark 17.4.5 with z there replaced by  zj and 0 (h) replaced by ( zj ; h), where

zj0

(z; h) = C ln

1 τ0

     |z − w|2 φ 1 2 (w)L(dw) + O(h), k ln h h 2π |z−w|≤1

(17.5.29)

cf (17.5.10) and where k can be replaced by  k, thanks to the choice of  zj .1 By Remark 17.4.5 we know that if 

j ≥ C ( zj ; h) ( 0, then 

j   1 − (φ( zj )−

j ) ≤ O(1)h−N6 −n ( zj ; h)e O(1) (zj ;h) . P | det Pδ,zj | < e (2π h)n

(17.5.30)

On the other hand, by Proposition 16.5.3, we know that for every admissible perturbation   1 (φ(z) +

(z; h)) . (17.5.31) | det Pδ,z | ≤ exp (2πh)n

1 We

could also replace 1D(0,1) (z − w) with a smooth cutoff χ(z − w), where 1, in order to make continuous in z. 1D(0,1)≤χ∈C ∞(D(0,1+θ)) for some 0 < θ 0

17.5 End of the Proof

395

In order to fix the ideas, we choose 

j = h−δ ( zj ; h) for some fixed small δ > 0. We can then apply Theorem 12.1.2 with the following substitutions: • “h” in the theorem should be replaced by (2πh)n , • “φ” in Theorem 12.1.2 should be replaced by the function φ(z) + (z; h),

j = h−δ ( zj ; h). • j in the theorem should be replaced with  We also recall the passage from Theorem 12.1.1 to Theorem 12.1.2 by an averaging procedure, and that we have applied the same procedure to see that  zj can be chosen so that      | zj − w|2 φ 1 1 2  (w)L(dw) + O(h).

( zj ; h) = O(1) ln ln k τ0 h h 2π | zj −w|≤1 (17.5.32) In view of Remark 12.3.2, we can choose the same  zj as in the proof of Theorem 12.1.2 (in our situation). From Theorems 12.1.2 and 17.5.30 we now conclude that with probability ≥ 1 − O(1)h−N6 −n



 h−δ −

( zj ; h) e O(1) ,

(17.5.33)

we have      (φ + (z; h)) 1  #(σ (Pδ ) ∩ ) − L(dz)   n (2πh)  2π ⎞ ⎛   N  C2 ⎝ ≤ n (φ + | |)L(dw) + h−δ

( zj ; h)⎠ , h γr 

(17.5.34)

j =1

where (as defined in Theorem 12.1.1) γr = 



D(z, r(z)), γ = ∂,

z∈γ

√  = O(h−1 ). zj satisfies N here with r = h. It is also clear that the number of points  Let us next review the remainder terms in (17.5.34): Combining (17.5.32) and (17.5.18), we see that 

( zj ; h) ≤ O(1) ln

1 τ0

   φ 1 2 h), (w)L(dw) + O(N G(w) ln h 2π 

 = O(|∂|h1/2 ). where G is given by (17.5.20). Here, Nh

(17.5.35)

396

17 Proof II: Lower Bounds

Noticing that the O(h) term in (17.5.32) can be assumed to be constant, we next look at   (z; h)L(dz) 

       φ(w) 1 2 |z − w|2 z k L(dw)L(dz) ln h h 2π z∈ w∈    1 φ(w) 1 2 L(dw), = O(1) ln H (w; h) ln τ0 h 2π 

= O(1) ln

1 τ0

where    |z − w|2 L(dz), z k h z∈

 H (w; h) =

and  is a fixed neighborhood of the closure of . By Green’s formula,  H (w; h) = ∂

  |z − w|2 ∂ k |dz|, ∂nz h

where nz is the exterior unit normal. An easy calculation shows that ∂ k ∂nz



|z − w|2 h





 1 h √ + 1{|z−w|≤ h} = O(1) , (h + |z − w|2 )3/2 |z − w|

so 

 H (w; h) = O(1) ∂

 h 1 √ + 1{|z−w|≤ h} |dz|. |z − w| (h + |z − w|2 )3/2

Comparing with (17.5.20) and (17.5.17),  G(w; h) = ∂

h |dz| √ , 2 h + |z − w| h

we see that  H ≤ O(1)G + ln+

h dist (w, ∂)2



17.5 End of the Proof

397

and conclude that   2       (z; h)L(dz)  ln 1 ln 1 G(w)φ(w)L(dw)   τ0 h       h + ln+ φ(w)L(dw) . dist (w, ∂)2 

(17.5.36)

The first term in the right-hand side is the same as in (17.5.35), while the second one will require an averaging argument. Finally, we look at  γr 

| |L(dz),

√ still with r = h and again with the O(h) term in (17.5.29) constant. This time, we use (17.5.29) with k in (17.5.9) regularized near λ = 1. Apply z to get z = O(1) ln

1 τ0

    1 2 h ln φ(w)L(dw) + φ(z) . 2 2 h  (h + |z − w| )

Consequently,  γr 

| |L(dz) = O(1) ln 

1 τ0

  1 2 ln h



K(w)φ(w)L(dw) + 

γr 

 φ(z)L(dz) ,

(17.5.37)

where  K(w) =

γr 

h L(dz) = O(1) (h + |z − w|2 )2

 ∂

h3/2 |dz| ≤ O(1)G(w). (h + |z − w|2 )2

Since G  1 on  γr , we have  γr 

| (z)|L(dz)  ln

1 τ0

   1 2 G(z)φ(z)L(dz). ln h 

(17.5.38)

398

17 Proof II: Lower Bounds

Combining (17.5.34)–(17.5.38), we get      φ 1 ≤ #(σ (Pδ ) ∩ ) − L(dz)   n (2πh)  2π      1 O(1) 1 2 1 −δ 2 h ln ln G(w, )φ(w)L(dw) + O(|∂|h ) hn τ0 h     h O(1) ln+ φ(w)L(dw), (17.5.39) + n h dist (w, ∂)2  valid with a probability √ √ as in (17.5.33). For t ∈ ] − h, h[, let ⎧ ⎨{z ∈ ; dist (z, ∂) > |t|}, if t ≤ 0, t = ⎩{z ∈ ; dist (z, ) < t}, if t > 0. Then ∂t is a uniformly Lipschitz √ curve of scale point z0 ∈ ∂, we have in D(z0 , 2 h)

√ h. Locally after rotation near any given

t = {z ∈ neigh (z0 , C); z < f (t, z)} where f (t, ·) is Lipschitz, uniformly with respect to t, and f (t, z) − f (s, z) " t − s. Let γt = ∂t . By construction, the curves γt are mutually disjoint and fill up  γr (r = Clearly,    

t

φ L(dz) − 2π

 

   φ φ L(dz) ≤ L(dz). 2π γr 2π 

√ h).

(17.5.40)

Applying the proof of (17.5.39) with  replaced by  γr , we can omit the troublesome last term there, coming from (17.5.36), and we conclude that with a probability as in (17.5.33), |#(σ (Pδ ) ∩  γr |       1 O(1) 1 1 2 −δ G(w, )φ(w)L(dw) + O(|∂|h 2 ) . h ln ln hn τ0 h 

(17.5.41)

17.5 End of the Proof

399

√ √ Consequently for each t ∈ ] − h, h [, after first replacing  by t in (17.5.39), we have with a probability as in (17.5.33):      φ 1 ≤ #(σ (Pδ ) ∩ ) − L(dz)   n (2πh)  2π   2   1 O(1) 1 1 h−δ ln ln G(w, )φ(w)L(dw) + O(|∂|h 2 ) hn τ0 h     h O(1) ln+ φ(w)L(dw). (17.5.42) + n h dist (w, γt )2  Now, 1 √ 2 h



√



h

√ ln+ − h

h dist (w, γt )2

 dt  1 γ2√h ,

so 1 √ 2 h



√ h





√ dt − h



so there exists t ∈ ] −



ln+



 φ(w)L(dw) 

γ2√h 

φ(w)L(dw),

√ √ h, h [ such that





ln+

h dist (w, γt )2

h dist (w, γt )2



 φ(w)L(dw) 

2√h γ

φ(w)L(dw).

As in (17.5.38), the last integral is  O(1)

G(w, )φ(w)L(dw), 

so choosing this value of t in (17.5.42), we can get rid of the last term there, and this completes the proof of Theorem 15.3.1.  We finally explain the extensions in Remark 15.3.2. The case when (15.3.4) holds is already included in the proof above. When we merely assume (15.3.5) for some ϑ ∈ ]0, 1/2 [ we get the corresponding weakening of the theorem after some modifications, basically just by multiplying some powers of h in some remainders with h−ϑ : • • • •

Before (17.3.46): Weaken the assumption to δ0 ≤ h2−ϑ . In (17.3.46), replace the last h with h1−ϑ . First sentence in Sect. 17.4: 0 ≤ δ0 ≤ h2−ϑ . Equation (17.4.7): Replace the first h with h1−ϑ in the last O(. . .).

400

• • • • • • • • • • •

17 Proof II: Lower Bounds

Equation (17.4.8): Idem. Equation (17.4.13): Replace the third term h to the right with h1−ϑ . Equation (17.4.33): Replace the last term h to the right with h1−ϑ . Equation (17.5.10): Replace the last term O(h) to the right with O(h1−ϑ ). Equation (17.5.29): Idem. Equation (17.5.32): Idem. h1−ϑ = O(|∂|h 12 −ϑ ). In (17.5.35) and on the following line: Read N On the following line: “Noticing that the O(h1−ϑ ) term . . . ” 1 1 Equation (17.5.39): Replace O(|∂|h 2 ) with O(|∂|h 2 −ϑ ) Equation (17.5.41): Idem. Equation (17.5.42): Idem.

17.A

Appendix: Grushin Problems and Singular Values

We mainly consider the case of an unbounded operator P = P0 + V , where P0 is an elliptic differential operator on X and V ∈ L∞ (X). The underlying Hilbert space is H = L2 (X) = H 0 and we will view P as an operator from H m to H 0 . The dual of H m is H −m and we shall keep in mind the variational point of view with the triple H m ⊂ H 0 ⊂ H −m . Consider  = P ∗ P : H m → H −m . For u ∈ H m , we have (u|u) = P u2 ≥ 0.

(17.A.1)

Proposition 17.A.1 If w ∈ ]0, +∞, [ , then  + w : H m → H −m is bijective with bounded inverse. Proof The injectivity is clear since (( + w)u|u) = wu2 + P u2 , u ∈ H m , and combining this with the standard elliptic apriori estimate, uH m ≤ C(P u + u), we get u2H m ≤ C(w)(( + w)u|u), implying uH m ≤ C(w)( + w)uH −m .

17.A Appendix: Grushin Problems and Singular Values

401

From this estimate and the fact that ∗ = , when we take the adjoint in the sense of bounded operators H m → H −m , it is standard to get the desired conclusion.   Notice that when P is injective, then by ellipticity and compactness, we have (u|u) ≥ for some C > 0 and we get the conclusion of the proposition also when w = 0. The operator ( + w)−1 : H −m → H m induces a compact self-adjoint operator H 0 → H 0 . The range consists of all u ∈ H m such that P u ∈ H m . The spectral theorem for compact self-adjoint operators tells us that there is an orthonormal basis of eigenfunctions e1 , e2 , . . . in H 0 such that 1 2 C u

( + w)−1 ej = μ2j (w)ej ,

(17.A.2)

where 0 < μj  0 when j → +∞. Clearly ej ∈ H m , P ej ∈ H m , so we can apply  + w to (17.A.2) and get ( + w)ej = μj (w)−2 ej ,

(17.A.3)

ej = (μj (w)−2 − w)ej .

(17.A.4)

which we write as

Here μj (w)−2 − w = (ej |ej ) ≥ 0, so we have found an orthonormal basis e1 , e2 , . . . ∈ H 0 with ej , P ej ∈ H m such that ej = tj2 ej ,

0 ≤ tj # +∞.

(17.A.5)

It is easy to check that tj2 are independent of w, that ej can be chosen independent of w, and we have μj (w)2 =

tj2

1 . +w

From Proposition 17.A.1 and its proof we know that  + w is self-adjoint as a bounded operator: H m → H −m . Consider  as a closed unbounded operator sa : L2 → L2 with domain D(sa ) = {u ∈ H m ; u ∈ L2 }. Then D(sa ) = ( + w)−1 L2 , which is dense in ( + w)−1 H −m = H m and hence dense in L2 . sa (or equivalently sa + w) is closed: If (sa + w)uj = vj , uj → u, vj → v in L2 , then vj → v in H −m , uj → u in H m , hence ( + w)u = v and since v ∈ L2 , we get u ∈ Dsa . Similar arguments show that sa is self-adjoint. We also know that sa has a purely discrete spectrum and that {ej }∞ j =1 is an orthonormal basis of eigenfunctions.

402

17 Proof II: Lower Bounds

We have the max-min principle tj2 =

sup

codim L=j −1

inf (u|u),

u∈L, u=1

(17.A.6)

where L varies in the set of closed subspaces of H 0 that are also contained in H m . Similarly, from (17.A.3), we have the mini-max principle μ2j =

inf

codim L=j −1

sup (( + w)−1 v|v),

(17.A.7)

v∈L v=1

where L varies in the set of closed subspaces of H 0 . When 0 ∈ σ (P ), so that P : H m → H 0 is bijective, we can extend (17.A.7) to the case w = 0 and then, as we have seen, μj (0)2 = tj−2 . Now assume P : H m → H 0 is a Fredholm operator of index 0.

(17.A.8)

The discussion above applies also to P P ∗ when P is viewed as an operator H 0 → H −m so that P ∗ : H m → H 0 . Put † = P P ∗ : H m −→ H −m . Then as in (17.A.5) we have an orthonormal basis f1 , f2 , . . . in H 0 with fj , P ∗ fj ∈ H m such that † fj =  tj2 fj ,

0 ≤ tj # +∞.

(17.A.9)

Proposition 17.A.2 We have  tj = tj and we can choose fj so that P ej = tj fj ,

P ∗ fj = tj ej .

(17.A.10)

Proof We have N () = N (P ), N († ) = N (P ∗ ) when P and P ∗ are viewed as operators H m → H 0 . Notice however that by elliptic regularity, the kernel of P ∗ : H 0 = H −m is the same as the one of P ∗ : H m → H 0 . Since P is Fredholm of index 0, the kernels of P and P ∗ have the same dimension, and consequently dim N () = dim N († ). Let t02 = tj20 be a non-vanishing eigenvalue of  of multiplicity k0 , so that tj0 −1 < tj0 = · · · = tj0 +k0 −1 < tj0 +k0

17.A Appendix: Grushin Problems and Singular Values

403

for some j0 , k0 ∈ N∗ and with the convention that the first inequality is absent when j0 = 1. If 0 = u ∈ N ( − t02 ), we know that u, P u ∈ H m , P u = 0 and we notice that in H −m .

† P u = P u = tj2 P u Thus v := P u ∈ H m is non-zero and satisfies † v = tj2 v,

so P gives an injective map from N ( − t02 ) into N († − t02 ). By the same argument P ∗ is injective from N († − t02 ) to N ( − t02 ) so the two spaces have the same dimension. It follows that  tj = tj for all j . Let ej , j0 ≤ j ≤ j0 + k0 − 1 be an orthonormal basis for N ( − t02 ) and put fj = t0−1 P ej ∈ N († − t02 ). Then (fj |fk ) = t0−2 (P ej |P ek ) = t0−2 (ej |ek ) = δj,k , so fj , j0 ≤ j ≤ j0 + k0 − 1 form an orthonormal basis for N († − t02 ). Also notice that P ∗ fj = t0−1 P ∗ P ej = t0 ej , and we get (17.A.10) in the non-trivial case when tj = 0.

 

Write tj (P ) = tj , so that tj (P ∗ ) = tj by the proposition. When P has a bounded inverse let s1 (P −1 ) ≥ s2 (P −1 ) ≥ · · · be the singular values of the inverse (as a compact operator in L2 ). We have sj (P −1 ) =

1 . tj (P )

(17.A.11)

Let 1 ≤ N < ∞ and let R+ : H m → CN , R− : CN → H 0 be bounded operators. Assume that   P R− : H m × CN → H 0 × CN (17.A.12) P= R+ 0 is bijective with a bounded inverse  E=

 E E+ E− E−+

.

(17.A.13)

404

17 Proof II: Lower Bounds

Recall that P has a bounded inverse precisely when E−+ does, and when this happens we have the relations, −1 P −1 = E − E+ E−+ E− ,

−1 E−+ = −R+ P −1 R− .

(17.A.14)

Cf. Sects. 3.2, 5.3, Chap. 6, Sect. 8.1 and Chap. 13. Recall ([48] and Proposition 8.2.2) that if A, B : H1 → H2 and C : H2 → H3 are bounded operators, where Hj are complex Hilbert spaces then we have the general estimates sn+k−1 (A + B) ≤ sn (A) + sk (B),

(17.A.15)

sn+k−1 (CA) ≤ sn (C)sk (A),

(17.A.16)

In particular, for k = 1, we get sn (CA) ≤ Csn (A), sn (CA) ≤ sn (C)A, sn (A + B) ≤ sn (A) + B. Applying this to the second part of (17.A.14), we get −1 sk (E−+ ) ≤ R− R+ sk (P −1 ),

1≤k≤N

whence tk (P ) ≤ R− R+ tk (E−+ ),

1 ≤ k ≤ N.

(17.A.17)

By a perturbation argument, we see that this holds also in the case when P , E−+ are non-invertible. Similarly, from the first part of (17.A.14) we get −1 ), sk (P −1 ) ≤ E + E+ E− sk (E−+

leading to tk (P ) ≥

tk (E−+ ) . Etk (E−+ ) + E+ E− 

(17.A.18)

Again this can be extended to the non-necessarily invertible case by means of small perturbations. Generalizing Sect. 3.2 (as in [56]), we get a natural construction of a Grushin problem associated to a given operator. Let P = P 0 : H m → H 0 be a Fredholm operator of index 0 as above. Choose N so that tN+1 (P 0 ) is strictly positive. In the following we sometimes write tj instead of tj (P 0 ) for short.

17.A Appendix: Grushin Problems and Singular Values

405 ∗

∗

Recall that tj2 are the first eigenvalues both for P 0 P 0 and P 0 P 0 . Let e1 , . . . , eN and ∗

∗

f1 , . . . , fN be corresponding orthonormal systems of eigenvectors of P 0 P 0 and P 0 P 0 , respectively. They can be chosen so that P 0 ej = tj fj ,

∗

P 0 fj = tj ej .

(17.A.19)

Define R+ : L2 → CN and R− : CN → L2 by R+ u(j ) = (u|ej ),

R− u− =

N 

u− (j )fj .

(17.A.20)

1

It is easy to see that the Grushin problem 2

P 0 u + R− u− = v, R+ u = v+ ,

(17.A.21)

has a unique solution (u, u− ) ∈ L2 × CN for every (v, v+ ) ∈ L2 × CN , given by 2

0 u = E 0 v + E+ v+ , 0 0 v , u− = E− v + E−+ +

(17.A.22)

where ⎧ ⎨E 0 v = N v (j )e , E 0 v(j ) = (v|f ), j j + + − 1 + 1 0 ⎩E 0 = −diag (tj ), E  ≤ tN+1 . −+

(17.A.23)

E 0 can be viewed as the inverse of P 0 as an operator from the orthogonal space (e1 , e2 , . . . , eN )⊥ to (f1 , f2 , . . . , fN )⊥ . We notice that in this case the norms of R+ and R− are equal to 1, so (17.A.17) tells us 0 ) for 1 ≤ k ≤ N, but of course the expression for E 0 in (17.A.23) that tk (P 0 ) ≤ tk (E−+ −+ implies equality. Let Q ∈ L(H 0 , H 0 ) and put P δ = P 0 − δQ (where we sometimes put a minus sign in front of the perturbation for notational convenience). We are particularly interested in the case when Q = Qω u = qu is the operator of multiplication by a function q. Here δ > 0 is a small parameter. Choose R± as in (17.A.20). Then if δ < tN+1 and Q ≤ 1, the perturbed Grushin problem 2

P δ u + R− u− = v, R+ u = v+ ,

(17.A.24)

406

17 Proof II: Lower Bounds

is well posed and has the solution 2

δv , u = E δ v + E+ + δ δ v , u− = E− + E−+ +

(17.A.25)

where  E = δ

δ E δ E+ δ δ E− E−+

 (17.A.26)

is obtained from E 0 by the formula  E =E δ

0

 1−δ

0 QE 0 QE+ 0 0

−1 .

(17.A.27)

Using the Neumann series, we get δ 0 0 0 E−+ = E−+ + E− δQ(1 − E 0 δQ)−1 E+ 0 0 0 0 0 0 0 = E−+ + δE− QE+ + δ 2 E− QE 0 QE+ + δ 3 E− Q(E 0 Q)2 E+ + ···

(17.A.28)

We also get E δ = E 0 (1 − δQE 0 )−1 = E 0 +

∞ 

δ k E 0 (QE 0 )k ,

(17.A.29)

k=1

δ 0 0 E+ = (1 − E 0 δQ)−1 E+ = E+ +

∞ 

0 δ k (E 0 Q)k E+ ,

(17.A.30)

0 δ k E− (QE 0 )k .

(17.A.31)

k=1

δ 0 0 E− = E− (1 − δQE 0 )−1 = E− +

∞  k=1

0 QE 0 : CN → CN has the δ The leading perturbation in E−+ is δM, where M = E− + matrix

M(ω)j,k = (Qek |fj ),

(17.A.32)

which in the multiplicative case reduces to  M(ω)j,k =

q(x)ek (x)fj (x)dx.

(17.A.33)

17.A Appendix: Grushin Problems and Singular Values

407

Put τ0 = tN+1 (P 0 ) and recall the assumption Q ≤ 1.

(17.A.34)

Then, if δ ≤ τ0 /2, the new Grushin problem is well posed with an inverse E δ given in (17.A.26)–(17.A.31). We get E δ  ≤

1 1−

δ τ0

E 0  ≤

2 , τ0

δ E± ≤

δ 0 0 0 − (E−+ + δE− QE+ ) ≤ E−+

1

≤ 2,

(17.A.35)

δ2 1 δ2 ≤ 2 . τ0 1 − τδ τ0

(17.A.36)

1−

δ τ0

0

δ ) ≤ 2τ ,2 we get Using this in (17.A.17), (17.A.18) together with the fact that tk (E−+ 0 δ ) tk (E−+ δ ≤ tk (P δ ) ≤ tk (E−+ ). 8

(17.A.37)

Notes This chapter concludes the proof of Theorem 15.3.1 and we have very much followed [137, 139]. Notice that the probabilistic lower bounds on the singular values of certain matrices in the Sects. 17.1 and 17.2 play a crucial role. It would be very interesting and probably very important to enrich this passage with different, perhaps more efficient methods. That would make it possible to generalize Theorem 15.3.1 and treat larger classes of operators and perturbations.

2 Indeed, δ ) ≤ t (E δ ) + E δ − E 0  ≤ t (E 0 ) + δ + 2 tk (E−+ k −+ k −+ −+ −+ 0 ) + 2δ ≤ t (E 0 ) + τ ≤ 2τ . ≤ tk (E−+ k −+ 0 0

δ2 τ0

Distribution of Large Eigenvalues for Elliptic Operators

18.1

18

Introduction

In this chapter we consider elliptic differential operators on a compact manifold and rather than taking the semi-classical limit (h →), we let h = 1 and study the distribution of large eigenvalues. Bordeaux Montrieux [16, 17] studied elliptic systems of differential operators on S 1 with random perturbations of the coefficients, and under some additional assumptions, he showed that the large eigenvalues obey the Weyl law almost surely. His analysis was based on a reduction to the semi-classical case, where he could use and extend the methods of Hager [55]. In [19] Bordeaux Montrieux and the author considered scalar elliptic operators on a general smooth compact manifold, using the semi-classical results of [139]. The present chapter follows closely, [19], but we replace [139] by the results of Sect. 15.3, which leads to some modifications. Let X be a smooth compact manifold of dimension n. Let P 0 be an elliptic differential operator on X of order m ≥ 2 with smooth coefficients and with classical principal symbol p(x, ξ ). In local coordinates we have, P0 =

 |α|≤m

aα0 (x)D α ,

p(x, ξ ) =



aα0 (x)ξ α .

(18.1.1)

|α|=m

The ellipticity of P 0 means that p(x, ξ ) = 0 for real ξ = 0. We assume that p(T ∗ X) = C.

(18.1.2)

Let dx be a strictly positive smooth density of integration dx on X and use it to define the L2 norm  ·  and the inner product (·|·). Let  : L2 (X) → L2 (X) be the antilinear © Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_18

409

410

18 Distribution of Large Eigenvalues for Elliptic Operators

operator of complex conjugation, given by u = u. We need the symmetry assumption (cf. (15.1.7)), (P 0 )∗ = P 0 ,

(18.1.3)

where (P 0 )∗ is the formal complex adjoint of P 0 . As in Sect. 15.1 we observe that the property (18.1.3) implies that p(x, −ξ ) = p(x, ξ ),

(18.1.4)

and conversely, if (18.1.4) holds, then the operator 12 (P 0 + (P 0 )∗ ) has the same principal symbol p and satisfies (18.1.3).  be an elliptic second-order differential operator on X with smooth coefficients, Let R which is self-adjoint and strictly positive. Let 0 , 1 , . . . be an orthonormal basis of  so that eigenfunctions of R  j = (μ0j )2 j , R

0 < μ00 < μ01 ≤ μ02 ≤ · · ·

(18.1.5)

Our randomly perturbed operator is Pω0 = P 0 + qω0 (x),

(18.1.6)

where ω is the random parameter and qω0 (x) =

∞ 

αj0 (ω) j .

(18.1.7)

j =0

Here we assume that αj0 (ω) are independent complex Gaussian random variables of variance σj2 and mean value 0: αj0 ∼ NC (0, σj2 ),

(18.1.8)

where β

1 −(μ0 ) M+1 (μ0j )−ρ e j ≤ σj ≤ O(1)(μ0j )−ρ , O(1) M=

3n − 12 , s − n2 −

0≤β
n,

(18.1.9)

(18.1.10)

18.1 Introduction

411

with s, ρ, fixed constants such that n n n 0 such that almost surely: there exists C(ω) < ∞ such that for all λ ∈ [1, ∞[ and all (g, θ1 , θ2 ) ∈ G, we have the estimate (19.3.12). The Condition (19.3.9) allows us to choose σj decaying faster than any negative power of μ0j . Then as in Chap. 18 it follows that qω (x) is almost surely a smooth function. Theorem 19.3.2 says roughly that for almost every PT symmetric elliptic operator of order ≥2 with smooth coefficients on a compact manifold which satisfies the conditions (19.3.2)–(19.3.4), the large eigenvalues distribute according to Weyl’s law in sectors with limiting directions that satisfy a weak non-degeneracy condition. Proof of Theorem 19.3.1 As already mentioned, the theorem is a variant of Theorem 18.1.1 (cf. Theorem 1.1 in [19]). The difference is just that we now use real random variables in the perturbation qω0 in order to guarantee the PT -symmetry and use Theorem 19.2.2 instead of Theorem 18.1.1.   The proof of Theorem 19.3.2 is a modification as in the proof of Theorem 18.1.2.

19.4

PT -Symmetric Potential Wells

In this section we discuss some recent results about real and non-real eigenvalues of PT symmetric deterministic perturbations of a P-symmetric self-adjoint Schrödinger operator in the semi-classical limit. Let X be either Rn or a compact smooth Riemannian manifold of dimension n. As the unperturbed operator, we take P0 = −h2  + V0 (x)

(19.4.1)

on X, where  is the Laplace-Beltrami operator and V0 ∈ C ∞ (X; R). Assume that V0 ◦ ι = V0 ,

(19.4.2)

ι2 = id = ι.

(19.4.3)

where ι : X → X is an isometry with

19.4 PT -Symmetric Potential Wells

435

We are interested in the spectrum of PT -symmetric perturbations of P0 near some fixed real energy E0 . To ensure that the spectrum is discrete near E0 when X = Rn , we assume in that case that α := lim inf V0 (x) > E0 .

(19.4.4)

Pδ = −h2  + Vδ (x),

(19.4.5)

Vδ (x) = V0 (x) + iδW (x),

(19.4.6)

x→∞

The perturbed operator is

where

δ ∈ R, |δ|

1, and W ∈ C ∞ (X; R) is bounded and odd with respect to ι, W ◦ ι = −W.

(19.4.7)

(We will also allow W to be unbounded in one of the results below.) We define P0 as a self-adjoint operator by taking the Friedrichs extension of (19.4.1) from C0∞ (X). Then σess (P0 ) ⊂ [α, +∞[ and σ (Pδ ) is contained in R + D(0, δW L∞ ) and is purely discrete in a fixed neighborhood of E0 when |δ|, h 1. In the self-adjoint case (δ = 0) very detailed informations about the eigenvalues can be obtained from “tunneling analysis”: Assume that V0−1 ( ] − ∞, 0]) =



Uj , #J < ∞,

(19.4.8)

j ∈J

where the potential wells Uj are closed (and hence compact by (19.4.4)) and pairwise disjoint: Uj ∩ Uk = ∅, j = k.

(19.4.9)

We refer to [59] where further references can be found. An important ingredient here is the Lithner–Agmon metric V0 (x)+ dx 2 [3, 95], where dx 2 denotes the Riemannian metric on X and as usual a+ = max(a, 0) for a ∈ R. The corresponding distance d(x, y) ≥ 0 is symmetric and satisfies the triangle inequality, but may be degenerate in the sense that d(x, y) = 0 ⇒ x = y.

19 Spectral Asymptotics for PT Symmetric Operators

436

We assume that Uj are “connected” in the sense that diamd (Uj ) = 0, j ∈ J.

(19.4.10)

19.4.1 Simple Well in One Dimension We describe a recent result by Boussekkine and Mecherout [23] which very roughly says that when X = R, ι(x) = −x, J = {0} and V0 and W are real analytic near U0 , then for |δ|, h small enough, the spectrum of Pδ in a fixed complex neighborhood of E0 is purely real. See also Rouby [120] for a more general result. We give a more detailed formulation. Let V0 ∈ C ∞ (R; R) be smooth and satisfy (19.4.2)–(19.4.4), with ι(x) = −x. Assume (H1) there exists m0 ≥ 0 such that for any α ∈ N, there exists Cα > 0 such that |∂xα V0 | ≤ Cα xm0 −α ,

on R.

When m0 > 0 we strengthen (19.4.4) by assuming that V0 (x) ≥

1 |x|m0 C0

for |x| ≥ C0 ,

form some positive constant C0 , (H2) V0 − E0 has exactly one potential well, more precisely, V0−1 ( ] − ∞, E0 ]) = [α00 , β00 ] =: U0 , V0−1 ( ] − ∞, E0 [ ) = ]α00 , β00 [ , where −∞ < α00 < β00 < ∞. Moreover, V0 (α00 ) < 0, V0 (β00 ) > 0. Notice that α00 = −β00 . (H3) V0 is analytic near U0 . As before, Vδ (x) = V0 (x) + δW (x), where we assume (H4) W ∈ C ∞ (R; R) is odd, W (−x) = −W (x), (H5) W satisfies the first part of (H1) with the same m0 , (H6) W is analytic near U0 .

19.4 PT -Symmetric Potential Wells

437

Let V0 , W also denote holomorphic extensions to a complex neighborhood of U0 . Then for E, δ ∈ C with |E − E0 | and |δ| small enough, we have unique solutions α0 (E, δ), β0 (E, δ) in complex neighborhoods of α00 , β00 , respectively, of the equations Vδ (α0 (E, δ)) = E, Vδ (β0 (E, δ)) = E, and α0 , β0 are holomorphic functions of (E, δ). For (E, δ) ∈ neigh ((E0 , 0), C2 ), we put  I0 (E, δ) = 2

β0 (E,δ)

1

(E − Vδ (x)) 2 dx,

(19.4.11)

α0 (E,δ)

where we integrate along the straight line segment from α0 (E, δ) to β0 (E, δ) and choose the branch of the square root which has argument close to 0. (When E is real and δ = 0, this is a real integral with a positive integrand.) It is easy to check that I0 (E, δ) is a holomorphic function of (E, δ). We define Pδ = −h2  + Vδ as the unbounded closed operator L2 (R) → L2 (R) with domain, D(Pδ ) = {u ∈ L2 ; u , u , xm0 u ∈ L2 }, and again the spectrum of Pδ is purely discrete in a fixed complex neighborhood of E0 , when |δ|, h are small enough. Theorem 19.4.1 (Boussekkine and Mecherout [23]) Let V0 , W ∈ C ∞ (R; R) be even and odd, respectively, and satisfy (19.4.2)–(19.4.4), (H1)–(H6). Define I0 (E, δ) as above, where Vδ (x) = V0 (x) + iδW (x) and (E, δ) ∈ neigh ((E0 , 0), C2 ). There exists a function I (E, δ; h) on neigh ((E0 , 0, 0), C × C × R), holomorphic in (E, δ) with I (E, δ; h) ∼ I0 (E, δ) + hI1 (E, δ) + · · ·

(19.4.12)

in the space of holomorphic functions defined near (E0 , 0), such that I (E, δ; h) is real 1, the eigenvalues E when E, δ are real and such that for δ ∈ ] − δ0 , δ0 [ with 0 < δ0 in neigh (E0 , C) are given by the Bohr–Sommerfeld condition I (E, δ; h) = 2π(k + 1/2)h, k ∈ Z.

(19.4.13)

These eigenvalues are real. Outline of the Proof This is an application of the complex WKB method, explained in Chap. 7. Let A > sup U0 , so that −A < inf U0 by parity and assume that A is small enough so that V0 , W are analytic near [−A, A]. We can then consider the Dirichlet realization

438

19 Spectral Asymptotics for PT Symmetric Operators

PδD of Pδ on L2 ([−A, A]). Then, as explained at the end of Sect. 7.2, the eigenvalues of PδD near E0 are given by a Bohr–Sommerfeld condition I D (E, δ; h) = 2π(k + 1/2)h, k ∈ Z, where I D , denoted by I(E; h) in (7.2.50), has the general properties of I in the theorem. Keeping track of the PT -symmetry, it is easy to show that I D (E, δ; h) is real when E, δ are real, and we get the conclusion of the theorem for PδD . To get the theorem, we need to complete the complex WKB method with a study of the exponential asymptotics of null solutions of Pδ − E that are of class L2 near +∞ or near −∞. This can be done by applying the fundamental idea of the complex WKB-method, as explained in Sect. 7.1 and it is here that we use (H1), (H5). This extension is indicated in [23] and carried out in detail in the work [101] and allows to make an asymptotic study of the Wronskian of two null solutions of Pδ − E that are L2 near −∞ and +∞ respectively.   The analyticity assumptions on V0 , W are essential, as can be seen by applying Theorem 19.2.22 to −h2  + V0 + iδ0(W + δqω ), where δ0 > 0 is small and fixed and δqω is a random perturbation. (When extending Theorem 19.2.2 to the case of Rn , a suitable cutoff function has to be inserted in the random perturbation and we refer to [137].) On the other hand, if V , W are merely smooth but satisfy the other assumptions in Theorem 19.4.1, then it seems clear that the conclusion there remains valid for |δ| ≤ O(1)h ln(1/ h).

19.4.2 Double Wells in Arbitrary Dimension We consider the general situation described in the beginning of this section, in particular in (19.4.1)–(19.4.10). Also assume that V0 has a double-well structure at energy E0 = 0, and that the two wells are exchanged by ι. More precisely, J = {−1, 1}, U±1 = ∅, and ι(U−1 ) = U1 .

(19.4.14)

To describe the spectrum of P0 , we introduce two self-adjoint reference operators. Let χ±1 ∈ C0∞ (X; [0, 1]) satisfy: χj = 1 near Uj ,

(19.4.15) ρ

supp χj ⊂ B(Uj , ρ) =: Uj

(19.4.16)

2 We here neglect the fact that Theorem 19.2.2 was established for operators on compact manifolds and not on Rn .

19.4 PT -Symmetric Potential Wells

439

where ρ > 0 is small. Here B(Uj , ρ) = {x ∈ X; d(Uj , x) < ρ}. Put 0,j = P0 + λχ−j , j = ±1, Pj = P

(19.4.17)

where λ > 0 is a constant, large enough so that {x ∈ X; V0 (x) + λχ−j (x) ≤ 0} = Uj . If we define Pu = u ◦ ι, u ∈ L2 (X),

(19.4.18)

then P is unitary on L2 (X) with P = 1 = P 2 and we have P ◦ P0 = P0 ◦ P,

(19.4.19)

−j ◦ P, j = ±1. P ◦ Pj = P

(19.4.20)

1 have the same spectrum. The last relation implies that P−1 and P Assume that  μ(h) = o(h)

(19.4.21)

is a simple eigenvalue of P1 (and hence of P−1 ), and that there exist C0 , N0 > 0 such that σ (P±1 ) ∩ ]  μ(h) − hN0 /C0 ,  μ(h) + hN0 /C0 [ = { μ(h)}. For h > 0 small enough, P0 has exactly two eigenvalues in the interval ] μ(h) − hN0 /(2C0 ),  μ(h) + hN0 /(2C0 ) [ , namely the eigenvalues μ(h) ± |t (h)| of the interaction matrix,   μ(h) t (h) , t (h) μ(h)

(19.4.22)

19 Spectral Asymptotics for PT Symmetric Operators

440

where μ(h) ∈ R and t (h) ∈ C, satisfy μ(h) =  μ(h) + Oρ (e( (ρ)−2S0)/ h ), (ρ) → 0, ρ → 0, ∀α > 0, t (h) = Oα (e(α−S0 )/ h ). Here, S0 = d(U1 , U−1 )

(19.4.23)

is the Lithner–Agmon distance between the two wells U±1 . Quite often we also have a lower bound on |t (h)|: ∀α > 0, |t (h)|−1 = Oα (e(α+S0 )/ h ). See for example [59, 130] or the review paper [133] and the references therein. Concerning the perturbation W , we assume (19.4.7). The result of Benbernou et al. [12] is the following: Theorem 19.4.2 ([12]) Under the above assumptions, the operator Pδ has exactly two eigenvalues (counted with their algebraic multiplicity) in D( μ, hN0 /C) for C ( 0 and N 0 for δ real such that |δ| h . These eigenvalues are equal to the eigenvalues of the matrix  a(δ) b(δ) , Xδ = b(δ) a(δ) 

and hence of the form λ± = a ±

(

|b|2 − (a)2 .

Here a(δ) = a(δ; h), b(δ) = b(δ; h) satisfy, a(0; h) = μ(h), b(0; h) = t (h),  ∂δ a = i

W (x)| e1 (x)|2 dx + O(δh−N0 ) + Oδ (e( (δ)−2S0)/ h ), ∂δ b = Oδ e( (δ)−S0)/ h ,

for all δ > 0, where (δ) → 0, δ → 0. Further,  e1 is the normalized eigenfunction with (P1 −  μ(h)) e1 = 0.

19.4 PT -Symmetric Potential Wells

441

If W > 0 on U1 , then  W (x)| e1 (x)|2 dx " 1,

(19.4.24)

and if we assume that (19.4.24) holds, then there exists δ+ ≥ 0 with the asymptotics, δ+ = (1 + Oδ (e( (δ)−S0)/ h )) 

|t (h)| , (δ) → 0, δ → 0, W (x)| e1 (x)|2 dx

such that • The two eigenvalues are real and distinct for |δ| < δ+ . • They are double and real when |δ| = δ+ . • They are non-real and complex conjugate, when δ+ < |δ|

hN0 .

The proof is based on the analysis of multi-well Hamiltonians in [59] (cf. [41]) that we extend slightly to non-self-adjoint operators, keeping track of the PT -symmetry. See [12]. Notes In this chapter we have reviewed some recent results on the eigenvalues of PT symmetric operators. The spectrum of such an operator is invariant under reflection in the real line and from the point of view of Physics it is important to know when it is real or not. Our general results on Weyl asymptotics for differential operators with random perturbations in the lowest order coefficient tend to show that most of the time the spectrum is non-real. In the case of a Schrödinger operator we review a result of [23] generalized in [120] which states that the eigenvalues are real in the case of an analytic single-well potential with not too large imaginary part. We also recall a result of [12] for double-well potentials showing that the eigenvalues bifurcate out into the complex plane already for very small perturbations of a real potential (regardless of the analyticity or lack of it).

20

Numerical Illustrations

by Agnès E. Sjöstrand1 and Johannes Sjöstrand In this chapter we give some numerical illustrations to the results about the asymptotic distribution of eigenvalues. Such calculations have already been carried out by many people, Trefethen [151], Trefethen and Embree [152], Davies [33, 35], Davies and Hager [38], Zworski [161,162] and many others. Numerical calculations with special attention to Weyl asymptotics have been carried out by Hager [54, 55], Bordeaux Montrieux [16], Vogel [155, 156]. Many of the illustrations below are therefore well-known and even though we wrote our own Matlab programs, we have clearly benefitted from the preceding works.

20.1

Jordan Block

The easiest case is that of a large Jordan block. Let J be the Jordan block of size N (denoted A0 in (2.4.1)). In Fig. 20.1 we let N = 20 and trace 31 level curves for the norm of the resolvent, corresponding to the levels ek , k = 0, 1, . . . , 30. We see that the norm of the resolvent grows very fast towards the center of the unit disc. Let Jδ = J + δQ, where Q is an N × N-matrix whose entries are real independent Gaussian random variables ∼ N (0, 1). In Fig. 20.2 we trace the spectrum of Jδ and the level curves of the norm of the resolvent. There is a dramatic improvement.

1 [email protected]

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9_20

443

444

20 Numerical Illustrations

It is probably a general result, that the presence of a random perturbation improves the resolvent norm in the spectral region. This appears quite clearly in the proofs of Weyl asymptotics, and an explicit result in this direction in the one-dimensional case can be found in [16]. For this reason we can be quite confident that numerical calculations of spectra of perturbed operators are accurate, while the corresponding calculations for the unperturbed operators can give strange results deep inside the pseudospectrum. Naturally, the reduction of a differential operator to a large matrix can be a source of errors as well. In Figs. 20.3, 20.4, and 20.5 we give a series of computations for larger Jordan blocks (N = 100). The calculation of corresponding pseudospectral level curves becomes time consuming and is presented only in the last figure. In accordance with the results of Chap. 13, most of the eigenvalues are very close to a circle with radius close to 1. The radius of this circle decreases slightly when we decrease δ. This can be viewed as a result on Weyl asymptotics, with J = 1]0,N]∩Z eiD , on 2 ([1, . . . , N])  CN , with h = 2π/N. The semiclassical symbol of J is now eiξ , whose range is equal to S 1 . In addition to the eigenvalues close to a circle there is a “uniformly bounded” number of eigenvalues inside. The expected density of those interior eigenvalues has been determined in [147]. The results and corresponding simulations can be generalized to bidiagonal operators of the form P = 1]0,N]∩Z (aeiD + be−iD ). The symbol is now aeiξ + be−iξ whose range is an ellipse. See Sjöstrand and Vogel [145, 146].

20.2

Hager’s Operator

We consider Hager’s operator hD + g(x) on S 1 (cf. Chap. 3). We will study two choices of g.

20.2.1 The Case When g(x) = i cos x Then, P = hDx + i cos x.

(20.2.1)

As we saw in the beginning of Chap. 3, the spectrum of P is an arithmetic progression of real eigenvalues (since the mean value of g is zero). As a numerical approximation,  we restrict the operator to the space HN = {u = N uk eikx } and then project to the −N  same space. In other words, we study the spectrum of N P : HN → HN . In Fig. 20.6, we have plotted the eigenvalues of P or rather N P in the case when N is small = 20. By choosing h with Nh = 2 we can arrange so that the real parts of the eigenvalues are situated in [−2, 2]. We notice that the real eigenvalues in the middle (roughly the ones

20.2 Hager’s Operator

445

with real part between −0.7 and 0.7) try to mimic the true eigenvalues of (20.2.1), while the ones on the diagonal segments are influenced by the fact that we do numerics with a projected “Toeplitz” operator. (The ones to the left are reminiscent of the eigenvalues of (hD)2 + i cos(x), which we study in the next section. An even better model, within reach of semi-classical analysis would be 1R+ (hD)(hD + i cos x).) The same Fig. 20.6 also shows pseudospectral curves, more precisely the level curves (z − P )−1  = ek , for 0 ≤ k ≤ K, with K = 20. Naturally each eigenvalue is encircled by 21 such curves, but in order to save computing time, we have chosen a rather low resolution, so these fine details are not visible. What is important though and clearly visible is the very strong pseudospectral effect in the interior of the rectangle [−2, 2] + i[−1, 1]. Near the middle the norm of the resolvent is of the order at least e20 = 4.8517 × 108 . Hence there is a corresponding spectral instability under perturbations. For even stronger effects we expect Matlab to have difficulties and to make numerical errors. When adding a random perturbation, the resolvent becomes reasonable in the new spectral region, as can be seen in Fig. 20.7. Here we consider Pδ = P + δQ,

(20.2.2)

where Qu(x) =

N 

(2π)−1 qj,k (ω)(u|eik(·))eij x

(20.2.3)

j,k=−N

and qj,k are independent complex random variables ∼ NC (0, 1). (Again the numerical computations concern N Pδ .) In Fig. 20.7 we still take N = 20, h = 0.1. The parameter δ is equal to 0.05. The norm of the particular random perturbation Q is 12.3104. This norm varies very little from one simulation to another. As we see, the eigenvalues now wander around in the range of the symbol and the resolvent norm no longer attains extremely large values (except right near the eigenvalues, though not visible because of the limited resolution). In Figs. 20.8 and 20.9 we present results of analogous simulations for N = 50 and h = 0.04 (still with hN = 2). The pseudospectral effect is now much stronger and Matlab gives several warning messages about the extreme size of the resolvent norm, when calculating the pseudospectral curves. There are hardly no real numerical eigenvalues in the unperturbed case. One possible explanation is mathematical: This may be the case already for the exact operator N P . We rather think that it is a pseudospectral effect. The spectral instability is so extreme that we are far from the true eigenvalues in the interior region. In the perturbed case, the pseudospectral effect is moderate and we are confident that we get the true eigenvalues of N Pδ , and perhaps even those of Pδ when restricting the real part of the spectral parameter to an interval ] − 2 + 1/O(1), 2 − 1/O(1)[ .

446

20 Numerical Illustrations

We also did some simulations with N = 500, h = 0.004. The pseudospectral effect now becomes very strong even close to the boundary of the range of the symbol, and we did not try to draw pseudospectral curves. In Fig. 20.10 we have plotted the numerical eigenvalues when δ = 0. Apparently, the numerical instability sets in already for dist ((z), {−1, 1}) ≈ 0.15, and only the eigenvalues stretching in from the corners are likely to be true eigenvalues of N P0 . (One might be able to justify them semi-classically by comparing them with the eigenvalues of 1R+ (hD)(hD + i cos(x)).) In Figs. 20.11, 20.12, 20.13, 20.14, 20.15, and 20.16 we have plotted the eigenvalues of the perturbed operator with δ ranging from 10−2 down to 10−12 . When δ = 10−2 the eigenvalues fill up the range [−2, 2] + i[−1, 1] quite well and some are even slightly outside. (For larger perturbations the random part becomes dominant and we get a cloud of eigenvalues around this rectangle. We are then in the theory of random matrices which has less to do with Weyl asymptotics for differential operators.) In accordance with the Weyl asymptotics, we see that the eigenvalues are distributed more densely near the upper and lower boundaries of the rectangle. For smaller values of δ they are no longer able to fill up the whole rectangle, and consequently the Weyl law will no longer hold right near the upper and lower boundaries. One can say that some eigenvalues try to obey the Weyl law up to the boundary, get stuck by the weakness of the perturbation and accumulate quite densely on a curve. Inside the region formed by the two curves we have Weyl asymptotics (apparently). The precise shape of the accumulation was studied by Vogel [155] in terms of the expected eigenvalue density. He also verified the Weyl law by letting Matlab count the eigenvalues in different rectangles. In Figs. 20.17, 20.18, 20.19, 20.20, 20.21, and 20.22 we plotted the perturbed eigenvalues together with the unperturbed ones (dd stands for δ).

20.2.2 The Case g(x) = eix The unperturbed operator is now P = hDx + eix .

(20.2.4)

We define the corresponding perturbed operators Pδ = P + δQ as before. In Fig. 20.23 we have plotted the eigenvalues of N Pδ when N = 500 and δ = 10−2 . In the middle the picture is similar to the case of the potential i cos x, but there are two obvious differences: The spectrum now fills up a stadium shaped domain with two discs at the ends, where the eigenvalue density is lower. This is not a numerical artefact, but rather a new case of the Weyl law, and there is a difference between N Pδ and Pδ in the two discs. The reason becomes clear if we plot the range of the symbol ξ +eix restricted to x ∈ S 1 , −2 ≤ ξ ≤ 2. In Fig. 20.24 we plotted the values of the symbol ξ + ieix , by letting first x take values in a grid in S 1 of step “Big” and letting ξ go through a denser grid in [−2, 2], then reversing the procedure, letting ξ go through the points −2 + k Big ∈ [−2, 2], and

20.3 (hD)2 + i cos x

447

for each such ξ , letting x go through a denser grid in S 1 . Each point in the middle region has two preimages, and in the discs near the ends there is only one preimage per point. We then plotted the eigenvalues (in blue) against the range for δ = 10−k , k = 2, 4, 6, 8, 10, 12, see Figs. 20.25, 20.26, 20.27, 20.28, 20.29, and 20.30.

20.3

(hD)2 + i cos x

The operator P = (hD)2 + i cos(x), x ∈ S 1 ,

(20.3.5)

appears naturally as a model when applying the secular method to certain non-self-adjoint perturbations of completely integrable quantum Hamiltonians in two dimensions (Hitrik– Sjöstrand [73]). The range of the symbol p = ξ 2 +i cos(x) is equal to [0; +∞[ +i[−1, 1]. The lower corner of this image corresponds to (x, ξ ) = (π, 0) and the harmonic approximation of P near that point is equal to −i plus the non-self-adjoint oscillator (hD)2 + ix 2 /2. Near the point (0, 0) the harmonic approximation is equal to i plus (hD)2 − ix 2 /2. The random perturbation is now Pδ = P + δQ, where Qu(x) = q(x)u(x), q(x) =

N 

 qk eikx ,

(20.3.6)

k=−N

and  qk are independent complex random variables ∼ NC (0, 1) The numerical calculations concern the matrix N Pδ : HN → HN , with N , HN as defined in the preceding section. In Fig. 20.31 we plot the spectrum in the unperturbed case, when N = 50, h = 0.04. Except for the branching to the right (which would require more analysis), the eigenvalues behave as one could expect. To the left we have approximations of the eigenvalues of the two harmonic oscillators, and it has been proved by Hitrik [67] that the exact eigenvalues of P do indeed have that behaviour here. Further inside this is only an expectation, which should be possible to verify, using the complex WKB-method. We have also traced 21 pseudospectral curves in a subregion, where the first one happened to fall outside the figure. Already for N = 50, there is a strong spectral instability. In Fig. 20.32, we consider the unperturbed case with N = 100, h = 0.02 and trace the same number of pseudospectral curves. It seems to us that the “eigenvalues” in the middle are non-real because of very strong numerical instability. In Fig. 20.33 we see that the pseudospectral behaviour improves and we can have confidence in the computed eigenvalues, at least as eigenvalues of N Pδ . In the remaining Figs. 20.34, 20.35, 20.36 20.37 20.38 20.39, and 20.40, we take N = 500, h = 0.004. In Fig. 20.34 we consider the unperturbed case. Then in

448

20 Numerical Illustrations

Figs. 20.35, 20.36 20.37 20.38 20.39, and 20.40 we consider the perturbed cases with δ decreasing from 0.01 to 10−12. When δ = 0.01 the perturbation is so strong that many eigenvalues fall outside the range of the symbol and we can hardly say that Weyl asymptotics holds in that case. When δ = 10−4 there is a tendency towards the Weyl law with a higher density of eigenvalues near z = 0 and near z = ±1 mimicking the growth of the density of the direct image of the symplectic volume dxdξ under the symbol map. In the other figures this is still true to some extent, but the weakness of the perturbation keeps the eigenvalues away from z = ±1.

20.4

Second-Order Differential Operators on S 1

In the case of Hager’s operator and of (hD)2 + i cos the critical values of the symbol belong to the boundary of the range. They are folds, except for the two corner points in the range of ξ 2 + i cos(x), where the differential of the symbol vanishes at the preimages. In this section and the next one we give examples where the symbol has critical points inside its range, so that the Weyl law will predict an accumulation of eigenvalues near such points in the interior of the spectral region. In this section we consider elliptic operators on S 1 of the form P = hD ◦ (1 + aeix + am e−ix + be2ix + bm e−2ix ) ◦ hD + ceix + cm e−ix

(20.4.7)

We then looked for suitable values of the coefficient, leading to nice critical values in the interior of the range of the symbol p = (1 + aeix + am e−ix + be2ix + bm e−2ix )y 2 + ceix + cm e−ix ,

(20.4.8)

where for programming reasons the dual variable is called “y” rather than “ξ ”. Possibly one could do without the terms be2ix and bm e−2ix , but having found some interesting choices with b and bm = 0, we decided to keep them. We computed the range as before, letting x belong to a grid in ]0, 2π] of step “Big”, then for each such x, plotting the image curve by letting y ∈ [0, yM ] go through a finer grid. Then we reversed the order of the operations, for each y in a grid of step Big in [0, yM ], and we draw the image curve by letting x go through a finer grid. We did choose yM suitably in order to capture the critical values. When the coefficients in (20.4.7), (20.4.8) are real, the operator P is PT -symmetric respect to with the parity x → −x. Correspondingly, the range of p is symmetric with respect to the real line.

20.4 Second-Order Differential Operators on S 1

449

As in Sect. 20.3, we consider random perturbations of the form Pδ = P + δQ, where N ikx and q are independent random variables Qu = q(x)u(x), q(x) = k −N qk (ω)e ∼ N (0, 1). In the PT -symmetric case we take qk real-valued and otherwise complexvalued, so that Q is PT -symmetric in the first case. Then as in the preceding sections we approximate Pδ with N Pδ : HN → HN , where N , HN are introduced in Sect. 20.2.

20.4.1 A PT -Symmetric Case A suitable choice of parameters turned out to be (a, am , b, bm , c, cm , Big, yM ) = (−0.4, 0.1, 0.1, 0.01, 1, 0.5, 0.08976, 2.3562). Big = π/35, yM = 3π/4. Figure 20.41 shows the range. We distinguish clearly some folds inside the range and two stronger singular values. Figures 20.42, 20.43, 20.44, 20.45, and 20.46 show the (numerical) spectrum of Pδ for N = 1500 with the range of p in the background, for δ =“dd” ranging from 2.5 × 10−3 down to 2.5 × 10−9 . Weyl asymptotics seems to hold in the sense that the density of eigenvalues is stronger where the range is denser. Apparently, there is more spectral stability near the folds and caustic points, and many eigenvalues can’t make it up to these points, when δ becomes smaller. The choice of yM was made to have some approximate fitting with the spectrum. Of course, near the right end of the picture, the plotted eigenvalues seem to represent only a fraction of the true eigenvalues, since the truncation N here cuts away some eigenvalues (that would be visible with a larger value of N).

20.4.2 A Non-PT -Symmetric Case Here we chose the parameters (a, am , b, bm , c, cm , Big, yM ) = (−0.5, 0.08i, 0, 0, 1, −0.5, 0.08976, 2.7826), then did the same calculations as in PT -symmetric case with N = 1500. See Figs. 20.47, 20.48, 20.49, 20.50, 20.51, and 20.52

450

20.5

20 Numerical Illustrations

Finite Difference Operators on S 1

Here, we replace hD, y with sin(hD), sin(y) in (20.4.7), (20.4.8): P = sin(hD) ◦ (1 + aeix + am e−ix + be2ix + bm e−2ix ) ◦ sin(hD) + ceix + cm e−ix , (20.5.9) p = (1 + aeix + am e−ix + be2ix + bm e−2ix )(sin y)2 + ceix + cm e−ix .

(20.5.10)

This operator acts on 2 (hZ). For a given N ∈ N \ {0}, we choose h = 2π/N, so that h > 0 is the smallest positive number with Nh = 2π. Then P acts on the N-dimensional 1 ), S 1 = hZ/2πZ. It is well known that functions u on S 1 have Fourier sum space 2 (SN N N expansions of the form u=



 uk eikx

(20.5.11)

k∈Z/NZ

and sin(hD)u =



sin(hk) uk eikx .

k∈Z/NZ

It is therefore simple to write down the matrix of P in terms of the Fourier coefficients in (20.5.11). Contrary to the case of differential operators, this is an exact reduction to a finite matrix, and no projection operator is necessary. It is also well known that the phase space associated to our operators above is Sx1 ×Sy1 and p in (20.5.10) is naturally a function on that space. The random perturbation is again multiplicative of the form q=



qk (ω)eikx ,

(20.5.12)

k∈Z/NZ

where qk are independent N (0, 1) random variables. As before, we can represent graphically the range by discretizing in x and y, and then study the eigenvalues of the N × N-matrix of P . Again we choose values of the coefficients that give rise to interesting folds and focal points in the range. Below we present the results in two cases, one with PT -symmetry and one without. In both cases we took the random perturbation real (contrary to the non-PT symmetric case for differential

20.6 PT -Symmetric Double Well

451

operators). Recall however that the main results on Weyl asymptotics are the same for real and complex random perturbations (cf. Remark 15.3.3).

20.5.1 A PT -Symmetric Case To find suitable parameters took much less effort, maybe because the compactness of the phase space. We made the following choice: (a, am , b, bm , c, cm ) = (−0.4, 0.1, 0.1, 0.02, 0.5, 0.25) The range is given in Fig. 20.53. In Figs. 20.54, 20.55, 20.56, and 20.57, we give the spectrum for δ(“dd”) ranging from 2.5 × 10−4 down to 2.5 × 10−9 with N = 3000.

20.5.2 A Non-PT -Symmetric Case We did the same computations with N = 3000 and (a, am , b, bm , c, cm ) = (−0.4 + 0.2i, 0.1, 0, 0, 0.5, 0.25). See Fig. 20.58 for the range and Figs. 20.59, 20.60, 20.61, 20.62, 20.63, and 20.64 for the spectrum.

20.6

PT -Symmetric Double Well

Let Z = x + h∂x be the annihilation operator and Z ∗ = x − h∂x the creation operator acting on functions of one real variable. If e0 , e1 , e2 , . . . is the standard orthonormal basis x = h1/2 x from of semi-classical Hermite functions in L2 (R) (obtained by√the scaling  the usual Hermite functions), then we know that Zek = 2khek−1 (= 0 by definition √ when k = 0), Z ∗ ek = 2(k + 1)ek+1 . Thus, the infinite matrices of Z and Z ∗ are easy to write down explicitly. We can then write down the infinite matrices for the multiplication operator x = (Z + Z ∗ )/2 and the semi-classical differentiation, h∂x = (Z − Z ∗ )/2 For numerical purposes, we restrict our attention to the N-dimensional space HN , spanned by e0 , e1 , . . . , eN−1 and correspondingly we replace x with xN := N ◦ x : HN → HN and h∂x with (h∂)N := N h∂ : HN → HN . In other words, we replace the infinite matrices of x and h∂x by the N × N-matrices obtained by restricting the row and column indices to {0, . . . , N − 1}. Thus to treat spectrally the non-self-adjoint harmonic operator, we can 2 and this gives the correct spectrum near 0. Here, we give a replace it by −(h∂)2N + ixN

452

20 Numerical Illustrations

numerical illustration of Theorems 19.4.1 and 19.4.2 by considering P = −(h∂x )2 + x 4 − x 2 + i x,

(20.6.13)

for > 0 small. When = 0 this is a self-adjoint Schrödinger operator with an even double-well potential. The height of the separating barrier is 0 and we know (in the limit h → 0) that the eigenvalues in any region ] − ∞, −1/O(1)] form pairs of values such that the distance between the two elements in the same pair is exponentially small, while the distance between a pair and the next higher one is of the order of h. In any region [1/O(1), O(1)[ the eigenvalues are simple with a distance " h from any such value to the next higher neighbor. We made a series of computations to find a value of for which there are two complex conjugate eigenvalues, very close to be a double real eigenvalue. See Fig. 20.65. We see a number of eigenvalues with small real part, that have split off the real line. The remaining eigenvalues are real, except the ones fairly far to the right. The non-reality to the right is certainly due to errors introduced by the spectral truncation. The real eigenvalues up to roughly 0.7 seem to be regularly spaced and to obey the Bohr– Sommerfeld rule. However, a careful look at the picture shows that the 12th real eigenvalue from the left with real part between 0.34 and 0.35 is actually constituted by two eigenvalues very close to each other. On the pdf version we only see that the blue dot is slightly widened horizontally. On the .fig-file this is seen more clearly after several enlargements. Again we think that this is due to the spectral truncation.  We therefore improved the program to get first the matrix of P in a range 1 ≤ j, k ≤ N  > N and truncating this matrix down to one of size N × N. Then, we get N Pδ with N with only one spectral truncation. The result obtained with the new program is given in Fig. 20.66. The anomaly near 0.34 has now disappeared, but there is a new close to double real eigenvalue ≈0.6. We then plotted both sets of eigenvalues in Fig. 20.67, marking the original ones with a dot and the “improved” ones with an “×”. The coincidence in the region z < 0.34 is remarkable and forgetting about the anomaly near 0.34 for the originally computed ones, we have a nice coincidence up to z ≈ 0.6. The second, more refined computation gives apparently a slightly better overall result, but the improvement mainly concerns relatively few eigenvalues situated to the right, where the spectral truncation plays a rule. Since the coincidence is so great up to about 0.6, we conclude (boldly) that Matlab computes the eigenvalues of the differential operator up to that level with quite a good accuracy. In particular, this is the case for the eigenvalues with real part  0.3, and in particular the splitting of eigenvalues out into the complex is nicely illustrated.

20.7 The Figures

20.7

453

The Figures

−1

k

norm((J−z) )=e , for N=20, k=0,...,30 1.5

1

0.5

0

−0.5

−1

−1.5

−1.5

−1

−0.5

0

0.5

1

Fig. 20.1 The level curves (J − z)−1  = ek , for k = 0, . . . , 30, N = 20

1.5

454

20 Numerical Illustrations −1

k

norm((Jdd−z) )=e , for N=20, dd=0.01, k=0,...,30 1.5

1

0.5

0

−0.5

−1

−1.5

−1.5

−0.5

−1

0

1

0.5

1.5

Fig. 20.2 The level curves (Jδ − z)−1  = ek , for k = 0, . . . , 30, δ = 0.01, N = 20

Spectrum of Jdelta, N=100, delta=0.001 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

−0.5

Fig. 20.3 Spectrum of Jδ , N = 100, δ = 0.001

0

0.5

1

20.7 The Figures

455 Spectrum of Jdelta, N=100, delta=0.0001

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

−0.8 −0.6 −0.4 −0.2

0

0.2

0.4

0.6

0.8

1

Fig. 20.4 Spectrum of Jδ , N = 100, δ = 10−4

Spectrum and pseudosp of Jdelta, N=100, delta=1e−05 1.5

1

0.5

0

−0.5

−1

−1.5

−1.5

−1

−0.5

0

0.5

1

1.5

Fig. 20.5 Spectrum of Jδ , N = 100, δ = 10−5 and pseudospectral level curves as in Fig. 20.2

456

20 Numerical Illustrations Hagicos, N=20, h=0.1, dd=0, norm(Q)=12.1712, K=20 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Fig. 20.6 Numerical eigenvalues and pseudospectral curves (P − z)−1  = ek , k = 0, 1, . . . , 20, for P in (20.2.1), h = 10−1 , N = 20

Hagicos, N=20, h=0.1, dd=0.05, norm(Q)=12.3104, K=20 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Fig. 20.7 Numerical eigenvalues and K + 1 = 21 pseudospectral curves for Pδ in (20.2.2), h = 10−1 , N = 20, δ = 0.05. Q = 12.3104

20.7 The Figures

457 Hagicos, N=50, h=0.04, dd=0, norm(Q)=19.5917, K=20

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −3

−2

−1

0

1

2

3

Fig. 20.8 Numerical eigenvalues and pseudospectral curves for P in (20.2.1), h = 0.04, N = 50, K = 20 Hagicos, N=50, h=0.04, dd=0.0001, norm(Q)=19.971, K=20 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Fig. 20.9 Numerical eigenvalues and pseudospectral curves for Pδ in (20.2.2), h = 0.04, N = 50, δ = 10−4 , K = 20. Q = 19.971

458

20 Numerical Illustrations Hagicos, N=500, h=0.004, dd=0, norm(Q)=62.8395 1.5

1

0.5

0

−0.5

−1

−1.5 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Fig. 20.10 Numerical eigenvalues for P in (20.2.1), h = 0.004, N = 500

Hagicos, N=500, h=0.004, dd=0.01, norm(Q)=62.8138 1.5

1

0.5

0

−0.5

−1

−1.5 2 −2 Fig. 20.11 Numerical eigenvalues for Pδ in (20.2.2), h = 0.004, N = 500, δ = 10 . Q = −2

62.8138

−1.5

−1

−0.5

0

0.5

1

1.5

20.7 The Figures

459 Hagicos, N=500, h=0.004, dd=0.0001, norm(Q)=62.8697

1.5

1

0.5

0

−0.5

−1

−1.5 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Fig. 20.12 Numerical eigenvalues for Pδ in (20.2.2), h = 0.004, N = 500, δ = 10−4 . Q = 62.8697

Hagicos, N=500, h=0.004, dd=1e−06, norm(Q)=62.8558 1.5

1

0.5

0

−0.5

−1

−1.5 2 −6 Fig. 20.13 Numerical eigenvalues for Pδ in (20.2.2), h = 0.004, N = 500, δ = 10 . Q = −2

62.8558

−1.5

−1

−0.5

0

0.5

1

1.5

460

20 Numerical Illustrations Hagicos, N=500, h=0.004, dd=1e−08, norm(Q)=62.871 1.5

1

0.5

0

−0.5

−1

−1.5 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Fig. 20.14 Numerical eigenvalues for Qδ in (20.2.2), h = 0.004, N = 500, δ = 10−8 . Q = 62.871

Hagicos, N=500, h=0.004, dd=1e−10, norm(Q)=62.5757 1.5

1

0.5

0

−0.5

−1

−1.5 2 −10 Fig. 20.15 Numerical eigenvalues for Qδ in (20.2.2), h = 0.004, N = 500, δ = 10 . Q = −2

62.5757

−1.5

−1

−0.5

0

0.5

1

1.5

20.7 The Figures

461 Hagicos, N=500, h=0.004, dd=1e−12, norm(Q)=63.011

1.5

1

0.5

0

−0.5

−1

−1.5 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Fig. 20.16 Numerical eigenvalues for Pδ in (20.2.2), h = 0.004, N = 500, δ = 10−12 . Q = 63.011 Hager operator, N=500, h=0.004, dd=0.01, kappa=0, norm Q=62.8323 1.5

1

0.5

0

−0.5

−1

−1.5 −1.5

−1

−0.5

0

0.5

1

1.5

Fig. 20.17 Numerical eigenvalues for Pδ in (20.2.2), h = 0.004, N = 500, δ = 10−2 (blue) and δ = 0 (red). Q = 62.8323

462

20 Numerical Illustrations Hager operator, N=500, h=0.004, dd=0.0001, kappa=0, norm Q=62.7098 1.5

1

0.5

0

−0.5

−1

−1.5 −1.5

−1

−0.5

0

0.5

1

1.5

Fig. 20.18 Numerical eigenvalues for Pδ in (20.2.2), h = 0.004, N = 500, δ = 10−4 (blue) and δ = 0 (red). Q = 62.7098

Hager operator, N=500, h=0.004, dd=1e−06, kappa=0, norm Q=63.0679 1.5

1

0.5

0

−0.5

−1

−1.5 −1.5

−1

−0.5

0

0.5

1

1.5

Fig. 20.19 Numerical eigenvalues for Pδ in (20.2.2), h = 0.004, N = 500, δ = 10−6 (blue) and δ = 0 (red). Q = 63.0679

20.7 The Figures

463 Hager operator, N=500, h=0.004, dd=1e−08, kappa=0, norm Q=63.1941

1.5

1

0.5

0

−0.5

−1

−1.5 −1.5

−1

−0.5

0

0.5

1

1.5

Fig. 20.20 Numerical eigenvalues for Pδ in (20.2.2), h = 0.004, N = 500, δ = 10−8 (blue) and δ = 0 (red). Q = 63.1941

Hager operator, N=500, h=0.004, dd=1e−10, kappa=0, norm Q=62.86 1.5

1

0.5

0

−0.5

−1

−1.5 −1.5

−1

−0.5

0

0.5

1

1.5

Fig. 20.21 Numerical eigenvalues for Pδ in (20.2.2), h = 0.004, N = 500, δ = 10−10 (blue) and δ = 0 (red). Q = 62.86

464

20 Numerical Illustrations Hager operator, N=500, h=0.004, dd=1e−12, kappa=0, norm Q=62.9342 1.5

1

0.5

0

−0.5

−1

−1.5 −1.5

−1

−0.5

0

0.5

1

1.5

Fig. 20.22 Numerical eigenvalues for Pδ in (20.2.2), h = 0.004, N = 500, δ = 10−12 (blue) and δ = 0 (red). Q = 62.9342

Hager exp(iX), N=500, h=0.004, dd=0.01, norm(Q)=62.9162 1.5 1 0.5 0 −0.5 −1 −1.5 −3

−2

−1

0

1

2

3

Fig. 20.23 Numerical eigenvalues for Pδ = P + δQ with P as in (20.2.4), h = 0.004, N = 500, δ = 10−2 . Q = 62.9162

20.7 The Figures

465

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −3

−2

−1

0

1

2

3

Fig. 20.24 Range of ξ + iex , restricted to Sx1 × [−2, 2]

Hager exp(iX), N=500, h=0.004, dd=0.01, norm(Q)=62.9504 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −3

−2

−1

0

1

2

3

Fig. 20.25 Numerical eigenvalues of Pδ = P + δQ with P as in (20.2.4) for N = 500, h = 0.004, δ = 10−2 . Q = 62.9504

466

20 Numerical Illustrations Hager exp(iX), N=500, h=0.004, dd=0.0001, norm(Q)=62.7418 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −3

−2

−1

0

1

2

3

Fig. 20.26 Numerical eigenvalues of Pδ = P + δQ with P as in (20.2.4) for N = 500, h = 0.004, δ = 10−4 . Q = 62.7418

Hager exp(iX), N=500, h=0.004, dd=1e−06, norm(Q)=62.7498 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −3

−2

−1

0

1

2

3

Fig. 20.27 Numerical eigenvalues of Pδ = P + δQ with P as in (20.2.4) for N = 500, h = 0.004, δ = 10−6 . Q = 62.7498

20.7 The Figures

467 Hager exp(iX), N=500, h=0.004, dd=1e−08, norm(Q)=62.6085

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −3

−2

−1

0

1

2

3

Fig. 20.28 Numerical eigenvalues of Pδ = P + δQ with P as in (20.2.4) for N = 500, h = 0.004, δ = 10−8 . Q = 62.6085

Hager exp(iX), N=500, h=0.004, dd=1e−10, norm(Q)=62.8754 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −3

−2

−1

0

1

2

3

Fig. 20.29 Numerical eigenvalues of Pδ = P + δQ with P as in (20.2.4) for N = 500, h = 0.004, δ = 10−10 . Q = 62.8754

468

20 Numerical Illustrations Hager exp(iX), N=500, h=0.004, dd=1e−12, norm(Q)=62.7895 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −3

−2

−1

0

1

2

3

Fig. 20.30 Numerical eigenvalues of Pδ = P + δQ with P as in (20.2.4) for N = 500, h = 0.004, δ = 10−12 . Q = 62.7895

2

(hD) +icos, unpert, h=0.04, N=50, dd=0, K=20

1

0.5

0

−0.5

−1

−1.5 0

0.5

1

1.5

2

2.5

3

3.5

Fig. 20.31 Spectrum and pseudospectral level curves for P in (20.3.5), N = 50, h = 0.04, K = 20

20.7 The Figures

469 2

(hD) +icos, h=0.02, N=100, dd=0, norm(Q)=28.9285 K=20 1.5

1

0.5

0

−0.5

−1

−1.5 0.5

0

1

1.5

2

2.5

3

3.5

Fig. 20.32 Spectrum and pseudospectral level curves for P in (20.3.5), N = 100, h = 0.02, K = 20

2

(hD) +icos, h=0.02, N=100, dd=0.0001, norm(Q)=28.9285, K=20 1.5

1

0.5

0

−0.5

−1

−1.5 0

0.5

1

1.5

2

2.5

3

3.5

Fig. 20.33 Spectrum and pseudospectral level curves for Pδ = P + δQ with P , Q as in (20.3.5), (20.3.6), N = 100, h = 0.02, δ = 10−4 , K = 20. Q = 28.9285

470

20 Numerical Illustrations 2

(hD) +icos, h=0.004, N=500, dd=0, norm(Q)=69.4346 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 20.34 Spectrum for P as in (20.3.5). N = 500, h = 0.004, Q = 69.4346

2

(hD) +icos, h=0.004, N=500, dd=0.01, norm(Q)=78.4188 1.5

1

0.5

0

−0.5

−1

−1.5 −1

0

1

2

3

4

5

Fig. 20.35 Spectrum for Pδ = P + δQ with P , Q as in (20.3.5), (20.3.6), N = 500, h = 0.004, δ = 0.01. Q = 78.4188

20.7 The Figures

471 2

(hD) +icos, h=0.004, N=500, dd=0.0001, norm(Q)=73.6021 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

1

0.5

1.5

2

2.5

3

3.5

4

Fig. 20.36 Spectrum for Pδ = P + δQ with P , Q as in (20.3.5), (20.3.6), N = 500, h = 0.004, δ = 10−4 . Q = 73.6021

2

(hD) +icos, h=0.004, N=500, dd=1e−06, norm(Q)=75.8064 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 20.37 Spectrum for Pδ = P + δQ with P , Q as in (20.3.5), (20.3.6), N = 500, h = 0.004, δ = 10−6 . Q = 75.8064

472

20 Numerical Illustrations 2

(hD) +icos, h=0.004, N=500, dd=1e−08, norm(Q)=75.9588 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

1

0.5

1.5

2

2.5

3

3.5

4

Fig. 20.38 Spectrum for Pδ = P + δQ with P , Q as in (20.3.5), (20.3.6), N = 500, h = 0.004, δ = 10−8 . Q = 75.9588

2

(hD) +icos, h=0.004, N=500, dd=1e−10, norm(Q)=70.036 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 20.39 Spectrum for Pδ = P + δQ with P , Q as in (20.3.5), (20.3.6), N = 500, h = 0.004, δ = 10−10 . Q = 70.036

20.7 The Figures

473 2

(hD) +icos, h=0.004, N=500, dd=1e−12, norm(Q)=74.5392 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

3.5

3

2.5

2

1.5

1

0.5

4

Fig. 20.40 Spectrum for Pδ = P + δQ with P , Q as in (20.3.5), (20.3.6), N = 500, h = 0.004, δ = 10−12 . Q = 74.5392

a, am, b, bm, c, cm, Big, yM= −0.4, 0.1, 0.1, 0.01, 1, −0.5, 0.08976, 2.3562 1.5

1

0.5

0

−0.5

−1

−1.5 −1

0

1

2

3

4

5

6

7

8

Fig. 20.41 Range of p in (20.4.8), (a, am , b, bm , c, cm , Big, yM ) = (−0.4, 0.1, 0.1, 0.01, 1, −0.5, 0.08976, 2.3562)

474

20 Numerical Illustrations PT, N=1500, h=0.00175, dd=0.0025, norm(Q)=136.1744 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −2

0

2

4

6

8

10

Fig. 20.42 Eigenvalues of Pδ = P + δQ as in (20.4.7), h = 0.00175, N = 1500, δ = 2.5 × 10−3 . Q = 136.1744

PT, N=1500, h=0.00175, dd=0.00025, norm(Q)=142.2009 2 1.5 1

0.5 0

−0.5

−1 −1.5 −2 −2

0

2

4

6

8

10

Fig. 20.43 Eigenvalues of Pδ = P + δQ as in (20.4.7), h = 0.00175, N = 1500, δ = 2.5 × 10−4 . Q = 142.2009

20.7 The Figures

475 PT, N=1500, h=0.00175, dd=2.5e−05, norm(Q)=142.2009

2 1.5 1

0.5 0

−0.5

−1 −1.5 −2 −2

0

2

4

6

8

10

Fig. 20.44 Eigenvalues of Pδ = P + δQ as in (20.4.7), h = 0.00175, N = 1500, δ = 2.5 × 10−5 . Q = 142.2009

PT, N=1500, h=0.00175, dd=2.5e−07, norm(Q)=132.1431 2 1.5 1

0.5 0

−0.5

−1 −1.5 −2 −2

0

2

4

6

8

10

Fig. 20.45 Eigenvalues of Pδ = P + δQ as in (20.4.7), h = 0.00175, N = 1500, δ = 2.5 × 10−7 . Q = 132.1431

476

20 Numerical Illustrations PT, N=1500, h=0.00175, dd=2.5e−09, norm(Q)=143.5027 2 1.5 1

0.5 0

−0.5 −1 −1.5

−2 −2

0

2

4

6

8

10

Fig. 20.46 Eigenvalues of Pδ = P + δQ as in (20.4.7), h = 0.00175, N = 1500, δ = 2.5 × 10−9 . Q = 143.5027

a, am, b, bm, c, cm, Big, yM= −0.5, 0+0.08i, 0, 0, 1, −0.5, 0.08976, 2.7826 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −2

0

2

4

6

8

10

12

Fig. 20.47 Range of p in (20.4.8), (a, am , b, bm , c, cm , Big, yM ) = (−0.5, 0.08i, 0, 0, 1, −0.5, 0.08976, 2.7826)

20.7 The Figures

477 NonPT, N=1500, h=0.00175, dd=0.0025, norm(Q)=163.2232

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −2

0

2

4

6

8

10

12

Fig. 20.48 Eigenvalues of Pδ = P + δQ as in (20.4.7), h = 0.00175, N = 1500, δ = 2.5 × 10−3 . Q = 163.2232

NonPT, N=1500, h=0.00175, dd=0.00025, norm(Q)=163.2232

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −2

0

2

4

6

8

10

12

Fig. 20.49 Eigenvalues of Pδ = P + δQ as in (20.4.7), h = 0.00175, N = 1500, δ = 2.5 × 10−4 . Q = 163.2232

478

20 Numerical Illustrations NonPT, N=1500, h=0.00175, dd=2.5e−05, norm(Q)=131.2679

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −2

0

2

4

6

8

10

12

Fig. 20.50 Eigenvalues of Pδ = P + δQ as in (20.4.7), h = 0.00175, N = 1500, δ = 2.5 × 10−5 . Q = 131.2679

NonPT, N=1500, h=0.00175, dd=2.5e−07, norm(Q)=146.792

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −2

0

2

4

6

8

10

12

Fig. 20.51 Eigenvalues of Pδ = P + δQ as in (20.4.7), h = 0.00175, N = 1500, δ = 2.5 × 10−7 . Q = 146.792

20.7 The Figures

479 NonPT, N=1500, h=0.00175, dd=2.5e−09, norm(Q)=132.8328

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −2

0

2

4

6

8

10

12

Fig. 20.52 Eigenvalues of Pδ = P + δQ as in (20.4.7), h = 0.00175, N = 1500, δ = 2.5 × 10−9 . Q = 132.8328

a, am, b, bm, c, cm= −0.4, 0.1, 0.1, 0.02, 0.5, 0.25 0.3

0.2

0.1

0

−0.1

−0.2

−0.3

−0.4 −1

−0.5

0

0.5

1

1.5

2

Fig. 20.53 Range of p in (20.5.10), (a, am , b, bm , c, cm ) = (−0.4, 0.1, 0.1, 0.02, 0.5, 0.25)

480

20 Numerical Illustrations PT, N=3000, h=0.0020944, dd=0.00025, norm(Q)=164.2631 0.3

0.2

0.1

0

−0.1

−0.2

−0.3

−0.4 −1

−0.5

0

0.5

1

1.5

2

Fig. 20.54 Eigenvalues of Pδ = P + δQ as in (20.5.9) ff., h = 0.0020944, N = 3000, δ = 2.5 × 10−4 . Q = 164.2631

PT, N=3000, h=0.0020944, dd=2.5e−05, norm(Q)=154.4232 0.3

0.2

0.1

0

−0.1

−0.2

−0.3

−0.4 −1

−0.5

0

0.5

1

1.5

2

Fig. 20.55 Eigenvalues of Pδ = P + δQ as in (20.5.9) ff., h = 0.0020944, N = 3000, δ = 2.5 × 10−5 . Q = 154.4232

20.7 The Figures

481 PT, N=3000, h=0.0020944, dd=2.5e−07, norm(Q)=182.4224

0.3

0.2

0.1

0

−0.1

−0.2

−0.3

−0.4 −1

−0.5

0

0.5

1

1.5

2

Fig. 20.56 Eigenvalues of Pδ = P + δQ as in (20.5.9) ff., h = 0.0020944, N = 3000, δ = 2.5 × 10−7 . Q = 182.4224

PT, N=3000, h=0.0020944, dd=2.5e−09, norm(Q)=140.5965 0.3

0.2

0.1

0

−0.1

−0.2

−0.3

−0.4 −1

−0.5

0

0.5

1

1.5

2

Fig. 20.57 Eigenvalues of Pδ = P + δQ as in (20.5.9) ff., h = 0.0020944, N = 3000, δ = 2.5 × 10−9 . Q = 140.5965

482

20 Numerical Illustrations a, am, b, bm, c, cm= −0.4+0.2i, 0.1, 0, 0, 0.5, 0.25 0.4 0.3 0.2

0.1 0

−0.1 −0.2 −0.3

−0.4 −1

−0.5

0

0.5

1

1.5

Fig. 20.58 Range of p in (20.5.10), (a, am , b, bm , c, cm ) = (−0.4 + 0.2i, 0.1, 0, 0, 0.5, 0.25)

nonPT, N=3000, h=0.0020944, dd=0.0025, norm(Q)=146.4608 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −1

−0.5

0

0.5

1

1.5

2

Fig. 20.59 Eigenvalues of Pδ = P + δQ as in (20.5.9) ff., h = 0.0020944, N = 3000, δ = 2.5 × 10−3 . Q = 146.4608

20.7 The Figures

483 NonPT, N=3000, h=0.0020944, dd=0.00025, norm(Q)=143.9731

0.4 0.3 0.2

0.1 0

−0.1

−0.2 −0.3 −0.4 −1

−0.5

0

0.5

1

1.5

Fig. 20.60 Eigenvalues of Pδ = P + δQ as in (20.5.9) ff., h = 0.0020944, N = 3000, Q = 143.9731. δ = 2.5 × 10−4

nonPT, N=3000, h=0.0020944, dd=2.5e−05, norm(Q)=152.9775 0.4 0.3

0.2 0.1

0

−0.1 −0.2 −0.3

−0.4 −1

−0.5

0

0.5

1

1.5

Fig. 20.61 Eigenvalues of Pδ = P + δQ as in (20.5.9) ff., h = 0.0020944, N = 3000, δ = 2.5 × 10−5 . Q = 152.9775

484

20 Numerical Illustrations nonPT, N=3000, h=0.0020944, dd=2.5e−06, norm(Q)=160.2713 0.4 0.3 0.2

0.1 0

−0.1

−0.2 −0.3 −0.4 −1

−0.5

0

0.5

1

1.5

Fig. 20.62 Eigenvalues of Pδ = P + δQ as in (20.5.9) ff., h = 0.0020944, N = 3000, δ = 2.5 × 10−6 . Q = 160.2713

nonPT, N=3000, h=0.0020944, dd=2.5e−07, norm(Q)=146.0593 0.4 0.3 0.2 0.1 0

−0.1 −0.2 −0.3 −0.4 −1

−0.5

0

0.5

1

1.5

Fig. 20.63 Eigenvalues of Pδ = P + δQ as in (20.5.9) ff., h = 0.0020944, N = 3000, δ = 2.5 × 10−7 . Q = 146.0593

20.7 The Figures

485 nonPT, N=3000, h=0.0020944, dd=2.5e−09, norm(Q)=147.0841

0.4 0.3 0.2

0.1

0 −0.1

−0.2 −0.3

−0.4 −1

−0.5

0

0.5

1

1.5

Fig. 20.64 Eigenvalues of Pδ = P + δQ as in (20.5.9) ff., h = 0.0020944, N = 3000, δ = 2.5 × 10−9 . Q = 147.0841

h=0.02, N=50, eps=0.016926 0.015

0.01

0.005

0

−0.005

−0.01

−0.015 −0.5

0

0.5

1

1.5

2

Fig. 20.65 Spectrum of P in (20.6.13) with truncations of x and of h∂. h = 0.02, N = 50,

= 0.016926

486

20 Numerical Illustrations With only one truncation h=0.02, N=50, eps=0.016926 0.025 0.02 0.015 0.01 0.005 0 −0.005 −0.01 −0.015 −0.02 −0.025 −0.5

0

0.5

1

1.5

2

2.5

Fig. 20.66 Spectrum of P in (20.6.13) with one global truncation. h = 0.02, N = 50, = 0.016926

h=0.02, N=50, eps=0.016926 0.025 0.02 0.015 0.01 0.005 0 −0.005 −0.01 −0.015 −0.02 −0.025 −0.5

0

0.5

1

1.5

2

2.5

Fig. 20.67 Superposition of the eigenvalues in Figs. 20.65 and 20.66. The former are marked with dots and the latter with cross symbols

Bibliography

1. S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math 15, 119–147 (1962) 2. S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Mathematical Studies, No. 2 (D. Van Nostrand Co., Inc., Princeton, 1965) 3. S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schrödinger Operators. Mathematical Notes, vol. 29 (Princeton University Press/University of Tokyo Press, Princeton/Tokyo, 1982) 4. L. Ahlfors, Complex analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd edn. International Series in Pure and Applied Mathematics (McGrawHill Book Co., New York, 1978) 5. A. Aleman, J. Viola, Singular-value decomposition of solution operators to model evolution equations. Int. Math. Res. Not. IMRN 2015(17), 8275–8288 (2014). https://arxiv.org/abs/1409. 1255 6. A. Aleman, J. Viola, On weak and strong solution operators for evolution equations coming from quadratic operators. J. Spectr. Theory 8(1), 33–121 (2018). https://arxiv.org/abs/1409. 1262 7. Y. Almog, The stability of the normal state of superconductors in the presence of electric currents. SIAM J. Math. Anal. 40(2), 824–850 (2008) 8. V.G. Avakumovi´c, Uber die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. Math. Z. 65, 324–344 (1956) 9. M.S. Baouendi, J. Sjöstrand, Régularité analytique pour des opérateurs elliptiques singuliers en un point. Ark. Mat. 14(1), 9–33 (1976) 10. M.S. Baouendi, J. Sjöstrand, Analytic regularity for the Dirichlet problem in domains with conic singularities. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4(3), 515–530 (1977) 11. B. Bellis, Subelliptic resolvent estimates for non-self-adjoint semiclassical Schrödinger operators. J. Spectral Theory 9(1), 171–194 (2019). https://arxiv.org/abs/1609.00436 12. A. Benbernou, N. Boussekkine, N. Mecherout, T. Ramond, J. Sjöstrand, Non-real eigenvalues for PT -symmetric double wells. Lett. Math. Phys. 106(12), 1817–1835 (2016). http://arxiv. org/abs/1506.01898 13. C.M. Bender, S. Boettcher, P.N. Meisinger, PT -symmetric quantum mechanics. J. Math. Phys. 40(5), 2201–2229 (1999)

© Springer Nature Switzerland AG 2019 J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators 14, https://doi.org/10.1007/978-3-030-10819-9

487

488

Bibliography

14. C.M. Bender, P.D. Mannheim, PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues. Phys. Lett. A 374(15–16), 1616–1620 (2010) 15. P. Bleher, R. Mallison Jr., Zeros of sections of exponential sums. Int. Math. Res. Not. 2006, 1–49 (2006), Art. ID 38937 16. W. Bordeaux Montrieux, Loi de Weyl presque sûre et résolvante pour des opérateurs différentiels non-autoadjoints, thèse, CMLS, Ecole Polytechnique, 2008. https://pastel.archivesouvertes.fr/pastel-00005367 17. W. Bordeaux Montrieux, Loi de Weyl presque sûre pour un système différentiel en dimension 1. Ann. Henri Poincaré 12(1), 173–204 (2011) 18. W. Bordeaux Montrieux, Estimation de résolvante et construction de quasimode près du bord du pseudospectre (2013). http://arxiv.org/abs/1301.3102 19. W. Bordeaux Montrieux, J. Sjöstrand, Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds. Ann. Fac. Sci. Toulouse 19(3–4), 567–587 (2010). http://arxiv.org/abs/0903.2937 20. C. Bordenave, M. Capitaine, Outlier eigenvalues for deformed i.i.d. random matrices. Commun. Pure Appl. Math. 69(11), 2131–2194 (2016) 21. D. Borisov, P. Exner, Exponential splitting of bound states in a waveguide with a pair of distant windows. J. Phys. A 37(10), 3411–3428 (2004) 22. L.S. Boulton, Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra. J. Operator Theory 47(2), 413–429 (2002) 23. N. Boussekkine, N. Mecherout, PT -symmetry and Schrödinger operators – the simple well case. Math. Nachr. 289(1), 13–27 (2016), French version at http://arxiv.org/pdf/1310.7335 24. L. Boutet de Monvel, Hypoelliptic operators with double characteristics and related pseudodifferential operators. Commun. Pure Appl. Math. 27, 585–639 (1974) 25. N. Burq, M. Zworski, Geometric control in the presence of a black box. J. Am. Math. Soc. 17(2), 443–471 (2004) 26. E. Caliceti, S. Graffi, J. Sjöstrand, Spectra of PT -symmetric operators and perturbation theory. J. Phys. A: Math. Gen. 38(1), 185–193 (2005) 27. E. Caliceti, S. Graffi, J. Sjöstrand, PT symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum. J. Phys. A: Math. Theor. 40(33), 10155–10170 (2007) 28. T. Carleman, Über die asymptotische Verteilung der Eigenwerte partielle Differentialgleichungen, Berichten der mathematisch-physisch Klasse der Sächsischen Akad. der Wissenschaften zu Leipzig, LXXXVIII Band, Sitsung v. 15. Juni 1936 29. T. Christiansen, Several complex variables and the distribution of resonances in potential scattering. Commun. Math. Phys. 259(3), 711–728 (2005) 30. T. Christiansen, Several complex variables and the order of growth of the resonance counting function in Euclidean scattering. Int. Math. Res. Not. 2006, 36 pp. (2006), Art. ID 43160 31. T. Christiansen, P.D. Hislop, The resonance counting function for Schrödinger operators with generic potentials. Math. Res. Lett. 12(5–6), 821–826 (2005) 32. T.J. Christiansen, M. Zworski, Probabilistic Weyl laws for quantized tori. Commun. Math. Phys. 299(2), 305–334 (2010). http://arxiv.org/abs/0909.2014 33. E.B. Davies, Semi-classical states for non-self-adjoint Schrödinger operators. Commun. Math. Phys. 200(1), 35–41 (1999) 34. E.B. Davies, Pseudospectra, the harmonic oscillator and complex resonances. Proc. Roy. Soc. Lond. A 455, 585–599 (1999) 35. E.B. Davies, Pseudopectra of differential operators. J. Operator Theory 43, 243–262 (2000)

Bibliography

489

36. E.B. Davies, Eigenvalues of an elliptic system. Math. Z. 243(4), 719–743 (2003) 37. E.B. Davies, Semigroup growth bounds. J. Operator Theory 53(2), 225–249 (2005) 38. E.B. Davies, M. Hager, Perturbations of Jordan matrices. J. Approx. Theory 156(1), 82–94 (2009) 39. E.B. Davies, A.B.J. Kuijlaars, Spectral asymptotics of the non-self-adjoint harmonic oscillator. J. Lond. Math. Soc. (2) 70, 420–426 (2004) 40. N. Dencker, J. Sjöstrand, M. Zworski, Pseudospectra of semiclassical (pseudo-)differential operators, Commun. Pure Appl. Math. 57(3), 384–415 (2004) 41. M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Note Series, vol. 268 (Cambridge University Press, Cambridge, 1999) 42. E.M. Dyn’kin, An operator calculus based upon the Cauchy-Green formula. Zapiski Nauchn. semin. LOMI 30, 33–40 (1972), J. Soviet Math. 4(4), 329–334 (1975) 43. K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194 (Springer, New York, 2000) 44. M.V. Fedoryuk, Asymptotic Analysis. Linear Ordinary Differential Equations (Springer, Berlin, 1993) 45. T. Fischer, Existence, Uniqueness, and Minimality of the Jordan Measure Decomposition. http://arxiv.org/abs/1206.5449 46. I. Gallagher, Th. Gallay, F. Nier, Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator. Int. Math. Res. Not. IMRN 2009(12), 2147–2199 (2009) 47. V.I. Girko, Theory of Random Determinants. Mathematics and Its Applications (Kluwer Academic Publishers, Dordrecht, 1990) 48. I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Non-selfadjoint Operators. Translations of Mathematical Monographs, vol. 18 (AMS, Providence, 1969) 49. D. Grieser, D. Jerison, Asymptotics of the first nodal line of a convex domain. Invent. Math. 125(2), 197–219 (1996) 50. A. Grigis, Estimations asymptotiques des intervalles d’instabilité pour l’équation de Hill. Ann. Sci. École Norm. Sup. (4) 20(4), 641–672 (1987) 51. A. Grigis, J. Sjöstrand, Microlocal Analysis for Differential Operators. London Mathematical Society Lecture Notes Series, vol. 196 (Cambridge University Press, Cambridge, 1994) 52. A. Guionnet, P.M. Wood, O. Zeitouni, Convergence of the spectral measure of non-normal matrices. Proc. AMS 142(2), 667–679 (2014) 53. M. Hager, Instabilité spectrale semiclassique d’opérateurs non-autoadjoints, Thesis, 2005. https://tel.archives-ouvertes.fr/tel-00010848/ 54. M. Hager, Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints. I. Un modèle. Ann. Fac. Sci. Toulouse Math. 15(2), 243–280 (2006) 55. M. Hager, Instabilité spectrale semiclassique d’opérateurs non-autoadjoints. II. Ann. Henri Poincaré 7(6), 1035–1064 (2006) 56. M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. Math. Ann. 342(1), 177–243 (2008). http://arxiv.org/abs/math/0601381 57. B. Helffer, Spectral Theory and Its Applications. Cambridge Studies in Advanced Mathematics, vol. 139 (Cambridge University Press, Cambridge, 2013) 58. B. Helffer, F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians. Lecture Notes in Mathematics, vol. 1862 (Springer, New York, 2004) 59. B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit. I. Commun. Partial Differ. Equ. 9(4), 337–408 (1984) 60. B. Helffer, J. Sjöstrand, Résonances en limite semi-classique. Mém. Soc. Math. France (N.S.), vol. 24–25 (Gauthier-Villars, Paris, 1986), pp. 1–228

490

Bibliography

61. B. Helffer, J. Sjöstrand, Equation de Schrödinger avec champ magnétique et équation de Harper. Springer Lecture Notes in Physics, vol. 345, 118–197 (1989) 62. B. Helffer, J. Sjöstrand, From resolvent bounds to semigroup bounds, published as part III of the published version of Ref. [138] (2010). http://arxiv.org/abs/1001.4171 63. R. Henry, Spectral projections of the complex cubic oscillator. Ann. Henri Poincaré 15(10), 2025–2043 (2014) 64. R. Henry, Instability for even non-selfadjoint anharmonic oscillators. J. Spectr. Theory 4(2), 349–364 (2014) 65. F. Hérau, J. Sjöstrand, C. Stolk, Semiclassical analysis for the Kramers-Fokker-Planck equation. Commun. PDE 30(5–6), 689–760 (2005) 66. M. Hitrik, Eigenfrequencies and expansions for damped wave equations. Methods Appl. Anal. 10(4), 543–564 (2003) 67. M. Hitrik, Boundary spectral behavior for semiclassical operators in dimension one. Int. Math. Res. Not. 2004(64), 3417–3438 (2004) 68. M. Hitrik, L. Pravda-Starov, Spectra and semigroup smoothing for non-elliptic quadratic operators. Math. Ann. 344(4), 801–846 (2009) 69. M. Hitrik, K. Pravda-Starov, Semiclassical hypoelliptic estimates for non-selfadjoint operators with double characteristics. Commun. Partial Differ. Equ. 35(6), 988–1028 (2010) 70. M. Hitrik, K. Pravda-Starov, Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics. Ann. Inst. Fourier (Grenoble) 63(3), 985–1032 (2013) 71. M. Hitrik, J. Sjöstrand, Rational invariant tori, phase space tunneling, and spectra for nonselfadjoint operators in dimension 2. Ann. Sci. ENS 4 41(4), 511–571 (2008) 72. M. Hitrik, J. Sjöstrand, Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions IIIa. One branching point. Canad. J. Math. 60(3), 572–657 (2008) 73. M. Hitrik, J. Sjöstrand, Rational invariant tori and band edge spectra for non-selfadjoint operators. J. Eur. Math. Soc. 20(2), 391–457 (2018). http://arxiv.org/abs/1502.06138 74. M. Hitrik, J. Sjöstrand, Two minicourses on analytic microlocal analysis, in Algebraic and Analytic Microlocal Analysis, AAMA, Evanston, Illinois, USA, 2012 and 2013. Springer Proceedings in Mathematics & Statistics, vol. 269 (2018), pp. 483–540. http://arxiv.org/abs/ 1508.00649 75. M. Hitrik, J. Sjöstrand, J. Viola, Resolvent estimates for elliptic quadratic differential operators. Anal. PDE 6(1), 181–196 (2013) 76. M. Hitrik, K. Pravda-Starov, J. Viola, Short-time asymptotics of the regularizing effect for semigroups generated by quadratic operators. Bull. Sci. Math. 141(7), 615–675 (2017). https:// arxiv.org/abs/1510.01992 77. M. Hitrik, K. Pravda-Starov, J. Viola, From semigroups to subelliptic estimates for quadratic operators. Trans. Am. Math. Soc. 370(10), 7391–7415 (2018). https://arxiv.org/abs/1510.02072 78. L. Hörmander, Differential equations without solutions. Math. Ann. 140, 169–173 (1960) 79. L. Hörmander, Differential operators of principal type. Math. Ann. 140, 124–146 (1960) 80. L. Hörmander, The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1968) 81. L. Hörmander, Fourier integral operators, lectures at the Nordic summer school of mathematics, unpublished notes, 1969. Available at http://portal.research.lu.se/portal/files/50761067/ Hormander_Tjorn69.pdf 82. L. Hörmander, On the Existence and the Regularity of Solutions of Linear Pseudo-Differential Equations. Série des Conférences de l’Union Mathématique Internationale, No. 1. Monographie No. 18 de l’Enseignement Mathématique. Secrétariat de l’Enseignement Mathématique, 69 pp. (Université de Genève, Geneva, 1971)

Bibliography

491

83. L. Hörmander, The Analysis of Linear Partial Differential Operators. I–IV, Grundlehren der Mathematischen Wissenschaften, vols. 256, 257, 274, 275 (Springer, Berlin, 1983/1985) 84. L. Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd edn. NorthHolland Mathematical Library, vol. 7 (North-Holland Publishing Co., Amsterdam, 1990) 85. R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985) 86. V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics. Springer Monographs in Mathematics (Springer, Berlin, 1998) 87. O. Kallenberg, Foundations of Modern Probability. Probability and Its Applications (New York) (Springer, New York, 1997) 88. T. Kato, Perturbation Theory for Linear Operators. Die Grundlehren der mathematischen Wissenschaften, Band, vol. 132 (Springer, New York, 1966) 89. V.V. Kuˇcerenko, Asymptotic solutions of equations with complex characteristics (Russian). Mat. Sb. (N.S.) 95(137), 163–213, 327 (1974) 90. H.H. Kuo, Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics, vol. 463 (Springer, Berlin, 1975) 91. B. Lascar, J. Sjöstrand, Equation de Schrödinger et propagation des singularités pour des opérateurs pseudodifférentiels à caractéristiques réelles de multiplicité variable I. Astérisque 95, 467–523 (1982) 92. G. Lebeau, Equation des ondes amorties. Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993), pp. 73–109. Mathematical Physics Studies, vol. 19 (Kluwer Academic Publishers, Dordrecht, 1996) 93. B.Ya. Levin, Distribution of Zeros of Entire Functions, English translation (American Mathematical Society, Providence, 1980) 94. B.M. Levitan, Some questions of spectral theory of selfadjoint differential operators. Usp. Mat. Nauk (N.S.) 11(72), no. 6, 117–144 (1956) 95. L. Lithner, A theorem of the Phragmén-Lindelöf type for second-order elliptic operators. Ark. Mat. 5, 281–285 (1964) 96. A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Translated from the Russian by H.H. McFaden. Translation ed. by B. Silver. With an appendix by M.V. Keldysh. Translations of Mathematical Monographs, vol. 71 (American Mathematical Society, Providence, 1988) 97. A.S. Markus, V.I. Matseev, Asymptotic behavior of the spectrum of close-to-normal operators. Funktsional. Anal. i Prilozhen. 13(3), 93–94 (1979), Functional Anal. Appl. 13(3), 233–234 (1979) (1980) 98. J. Martinet, Sur les propriétés spectrales d’opérateurs nonautoadjoints provenant de la mécanique des fluides, Thèse de doctorat, Université de Paris Sud, 2009 99. V.P. Maslov, Operational Methods. Translated from the Russian by V. Golo, N. Kulman, G. Voropaeva (Mir Publishers, Moscow, 1976) 100. O. Matte, Correlation asymptotics for non-translation invariant lattice spin systems. Math. Nachr. 281(5), 721–759 (2008) 101. N. Mecherout, N. Boussekkine, T. Ramond, J. Sjöstrand, PT -symmetry and Schrödinger operators. The double well case. Math. Nachr. 289(7), 854–887 (2016). http://arxiv.org/abs/ 1502.06102 102. A. Melin, J. Sjöstrand, Fourier Integral Operators with Complex-Valued Phase Functions. Fourier Integral Operators and Partial Differential Equations (Colloq. Internat., Univ. Nice, Nice, 1974), pp. 120–223. Lecture Notes in Mathematics, vol. 459 (Springer, Berlin, 1975) 103. A. Melin, J. Sjöstrand, Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem. Commun. Partial Differ. Equ. 1(4), 313–400 (1976)

492

Bibliography

104. A. Melin, J. Sjöstrand, Determinants of pseudodifferential operators and complex deformations of phase space. Methods Appl. Anal. 9(2), 177–238 (2002). https://arxiv.org/abs/math/0111292 105. A. Melin, J. Sjöstrand, Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2. Astérique 284, 181–244 (2003). https://arxiv.org/abs/math/0111293 106. A. Menikoff, J. Sjöstrand, On the eigenvalues of a class of hypo-elliptic operators. Math. Ann. 235, 55–85 (1978) 107. D.V. Nekhaev, A.I. Shafarevich, Semiclassical limit of the spectrum of the Schrödinger operator with complex periodic potential. Mat. Sb. 208(10), 126–148 (2017) 108. L. Nirenberg, A proof of the Malgrange preparation theorem. In Proceedings of Liverpool Singularities-Symposium, I (1969/70), pp. 97–105. Lecture Notes in Mathematics, vol. 192 (Springer, Berlin, 1971) 109. M. Ottobre, G. Pavliotis, K. Pravda-Starov, Exponential return to equilibrium for hypoelliptic quadratic systems. J. Funct. Anal. 262, 4000–4039 (2012) 110. A. Pfluger, Die Wertverteilung und das Verhalten von Betrag und Argument einer speciellen Klasse analytischen Funktionen I, II. Comment. Math. Helv. 11, 180–214 (1938), 12, 25–65 (1939) 111. K. Pravda-Starov, Étude du pseudo-spectre d’opérateurs non auto-adjoints, thesis, 2006. http:// tel.archives-ouvertes.fr/docs/00/10/98/95/PDF/manuscrit.pdf 112. K. Pravda-Starov, A complete study of the pseudo-spectrum for the rotated harmonic oscillator. J. Lond. Math. Soc. (2) 73(3), 745–761 (2006) 113. K. Pravda-Starov, On the pseudospectrum of elliptic quadratic differential operators. Duke Math. J. 145(2), 249–279 (2008) 114. K. Pravda-Starov, Subelliptic estimates for quadratic differential operators. Am. J. Math. 133, 39–89 (2011) 115. P. Redparth, Spectral properties of non-self-adjoint operators in the semiclassical regime. J. Differ. Equ. 177(2), 307–330 (2001) 116. M. Reed, B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edn. (Academic, New York, 1980), Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness (Academic, New York, 1975), Methods of Modern Mathematical Physics. IV. Analysis of Operators (Academic Press, New York, 1978) 117. F. Riesz, B.-Sz. Nagy, Leçons d’Analyse Fonctionnelle, Quatrième édition. Académie des Sciences de Hongrie, Gauthier-Villars (Editeur-Imprimeur-Libraire/Akadémiai Kiadó, Paris/Budapest, 1965) 118. G. Rivière, Delocalization of slowly damped eigenmodes on Anosov manifolds. Commun. Math. Phys. 316(2), 555–593 (2012) 119. D. Robert, Autour de l’Approximation Semi-classique. Progress in Mathematics, vol. 68 (Birkhäuser, Boston, 1987) 120. O. Rouby, Bohr-Sommerfeld quantization conditions for non-selfadjoint perturbations of selfadjoint operators in dimension one. Int. Math. Res. Not. IMRN 2018(7), 2156–2207 (2018). http://arxiv.org/abs/1511.06237 121. O. Rouby, S. V˜u Ngo.c, J. Sjöstrand, Analytic Bergman operators in the semiclassical limit (2018). https://arxiv.org/abs/1808.00199 122. E. Schenk, Systèmes quantiques ouverts et méthodes semi-classiques, thèse novembre, 2009. https://www.ipht.fr/Docspht/articles/t09/266/public/thesis_schenck.pdf 123. R.T. Seeley, Complex powers of an elliptic operator, in Proceedings of Symposium on Singular Integrals, vol. 10, pp. 288–307 (American Mathematical Society, Providence, 1967)

Bibliography

493

124. R. Seeley, A simple example of spectral pathology for differential operators. Commun. Partial Differ. Equ. 11(6), 595–598 (1986) 125. B. Shiffman, S. Zelditch, S. Zrebiec, Overcrowding and hole probabilities for random zeros on complex manifolds. Ind. Univ. Math. J. 57(5), 1977–1997 (2008) 126. K.C. Shin, On the reality of the eigenvalues for a class of PT -symmetric oscillators. Commun. Math. Phys. 229(3), 543–564 (2002) 127. A.A. Shkalikov, Spectral portraits of the Orr-Sommerfeld operator at large Reynolds numbers (Russian). Sovrem. Mat. Fundam. Napravl. 3, 89–112 (2003), Translation in J. Math. Sci. (N. Y.) 124(6), 5417–5441 (2004) 128. A.A. Shkalikov, Eigenvalue asymptotics of perturbed self-adjoint operators. Methods Funct. Anal. Topol. 18(1), 79–89 (2012) 129. B. Simon, Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. H. Poincaré Sect. A (N.S.) 38(3), 295–308 (1983), Erratum in ibid 40(2), 224 (1984) 130. B. Simon, Semiclassical analysis of low lying eigenvalues. II. Tunneling. Ann. Math. (2) 120(1), 89–118 (1984) 131. J. Sjöstrand, Parametrices for pseudodifferential operators with multiple characteristics. Ark. Mat. 12(1), 85–130 (1974) 132. J. Sjöstrand, Singularités analytiques microlocales. Astérisque 95 (Société Mathématique de France, Paris, 1982), pp. 1–166 133. J. Sjöstrand, Puits multiples, (d’après des travaux avec B. Helffer), Sém. Goulaouic-MeyerSchwartz, 1983–1984, Exposé No. 7, École Polytech., Palaiseau, 1984. http://www.numdam. org/item?id=SEDP_1983-1984____A7_0 134. J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations. Publ. RIMS Kyoto Univ. 36(5), 573–611 (2000) 135. J. Sjöstrand, Resonances for bottles and trace formulae. Math. Nachr. 221, 95–149 (2001) 136. J. Sjöstrand, Lectures on Resonances (2007). http://sjostrand.perso.math.cnrs.fr/Coursgbg.pdf 137. J. Sjöstrand, Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations. Ann. Fac. Sci. Toulouse 18(4), 739–795 (2009). http://arxiv.org/abs/ 0802.3584 138. J. Sjöstrand, Spectral properties of non-self-adjoint operators, Notes d’un minicours à Evian les Bains, 8–12 juin, 2009, Actes des Journées d’é.d.p. d’Évian 2009, published: http://jedp. cedram.org:80/jedp-bin/feuilleter?id=JEDP_2009___ preprint: http://arxiv.org/abs/1002.4844 139. J. Sjöstrand, Eigenvalue distribution for non-self-adjoint operators on compact manifolds with small multiplicative random perturbations. Ann. Fac. Sci. Toulouse 19(2), 277–301 (2010). http://arxiv.org/abs/0809.4182 140. J. Sjöstrand, Resolvent Estimates for Non-self-adjoint Operators via Semi-groups. International Mathematical Series, vol. 13, pp. 359–384 Around the research of Vladimir Maz’ya, III (Springer/Tamara Rozhkovskaya Publisher, Novosibirsk, 2010). http://arxiv.org/abs/0906.0094 141. J. Sjöstrand, Counting zeros of holomorphic functions of exponential growth. J. Pseudodiffer. Operators Appl. 1(1), 75–100 (2010). http://arxiv.org/abs/0910.0346 142. J. Sjöstrand, Eigenvalue distributions and Weyl laws for semiclassical non-self-adjoint operators in 2 dimensions. Geometric Aspects of Analysis and Mechanics, vols. 345–352. Progress in Mathematics, vol. 292 (Birkhäuser/Springer, New York, 2011) 143. J. Sjöstrand, PT Symmetry and Weyl Asymptotics. The Mathematical Legacy of Leon Ehrenpreis, Springer Proceedings in Mathematics, vol. 16, pp. 299–308 (2012). http://arxiv.org/abs/ 1105.4746

494

Bibliography

144. J. Sjöstrand, Weyl Law for Semi-classical Resonances with Randomly Perturbed Potentials. Mém. de la SMF 136, 144 (2014). http://arxiv.org/abs/1111.3549 145. J. Sjöstrand, M. Vogel, Interior eigenvalue density of large bi-diagonal matrices subject to random perturbations, in Proceedings of the Conference “Microlocal Analysis and Singular Perturbation Theory”, RIMS, Kyoto University, October 5–9, 2015, RIMS Conference Proceedings series “RIMS Kokyuroku Bessatsu” (2015). http://arxiv.org/abs/1604.05558 146. J. Sjöstrand, M. Vogel, Large bi-diagonal matrices and random perturbations. J. Spectr. Theory 6(4), 977–1020 (2016). http://arxiv.org/abs/1512.06076 147. J. Sjöstrand, M. Vogel, Interior eigenvalue density of Jordan matrices with random perturbations. In Analysis Meets Geometry: A Tribute to Mikael Passare. Trends in Mathematics, pp. 439–466 (Springer, Cham, 2017). http://arxiv.org/abs/1412.2230 148. J. Sjöstrand, M. Zworski, Elementary linear algebra for advanced spectral problems. Ann. Inst. Fourier 57(7), 2095–2141 (2007) 149. M. Sodin, Zeros of Gaussian analytic functions. Math. Res. Lett. 7, 371–381 (2000) 150. M. Sodin, B. Tsirelson, Random complex zeroes. III. Decay of the hole probability. Isr. J. Math. 147, 371–379 (2005) 151. L.N. Trefethen, Pseudospectra of linear operators. SIAM Rev. 39(3), 383–406 (1997) 152. L.N. Trefethen, M. Embree, Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators (Princeton University Press, Princeton, 2005) 153. J. Viola, Spectral projections and resolvent bounds for partially elliptic quadratic differential operators. J. Pseudo-Differ. Oper. Appl. 4, 145–221 (2013) 154. J. Viola, The norm of the non-self-adjoint harmonic oscillator semigroup. Integr. Equ. Oper. Theory 85(4), 513–538 (2016) 155. M. Vogel, The precise shape of the eigenvalue intensity for a class of non-selfadjoint operators under random perturbations. Ann. Henri Poincaré 18(2), 435–517 (2017). http://arxiv.org/abs/ 1401.8134 156. M. Vogel, Two point eigenvalue correlation for a class of non-selfadjoint operators under random perturbations. Commun. Math. Phys. 350(1), 31–78 (2017). http://arxiv.org/abs/1412. 0414 157. A. Voros, Spectre de l’Équation de Schrödinger et Méthode BKW. Publications Mathématiques d’Orsay 81, vol. 9, 75 pp. (Université de Paris-Sud, Département de Mathématique, Orsay, 1982) 158. G. Weiss, The resolvent growth assumption for semigroups on Hilbert spaces. J. Math. Anal. Appl. 145, 154–171 (1990) 159. H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71(4), 441–479 (1912) 160. S. Zrebiec, The zeros of flat Gaussian random holomorphic functions on Cn , and hole probability. Mich. Math. J. 55(2), 269–284 (2007) 161. M. Zworski, A remark on a paper of E. B. Davies: “Semi-classical states for non-self-adjoint Schrödinger operators”. Proc. Am. Math. Soc. 129(10), 2955–2957 (2001) 162. M. Zworski, Numerical linear algebra and solvability of partial differential equations. Commun. Math. Phys. 229(2), 293–307 (2002)

Bibliography

495

Symbols Item

Page/formula

" D(z0 , r0 ) D, Dx Hh1 H s (Rn )  S(m)  · HS · NC (0, 1) NC (0, σ 2 ) σ

O(..) 1/O(..) O(h∞ ) N neigh (A, B) ∂ ∂zj

31 11 23 35

∂z , ∂ z {a, b} Pt # ∼ ∼ T

37 31, 61 315 30 30 32 67

207 68 42 67 30 41 13 21 21 34 23 31 133 133

496

Bibliography

Terms Item

Page/formula

Bohr–Sommerfeld condition Canonical transformation Classical principal symbol Cyclicity of the trace Determinant Eikonal equation

-pseudospectrum FBI Fourier transform Green kernel Hamilton field Index Ky Fan Lagrangian manifold Metaplectic Microlocally Multiindex Order function Poisson bracket Poisson kernel Pseudodifferential operator Semi-classical principal symbol Subharmonic Symplectic coordinates Symplectic form Transpose Weyl inequalities Weyl quantization

154 102 314 173, (8.4.4) 174 58, (4.1.17) 13 196 332 228 185 157 166 99, 186, 187 124, 134 195 313 67 31 228 68 54, 314 14 96 96, 186 315 167 68