Partial Differential Equations II - Qualitative Studies of Linear Equations [3 ed.]
 9783031336997, 9783031337000

Table of contents :
Contents of Volumes I and III
Preface
Acknowledgments
Preface to the Second Edition
Preface to the Third Edition
Contents
7 Pseudodifferential Operators
1 The Fourier integral representation and symbol classes
2 Schwartz kernels of pseudodifferential operators
3 Adjoints and products
4 Elliptic operators and parametrices
5 L2-estimates
6 Gårding's inequality
7 Hyperbolic evolution equations
8 Egorov's theorem
9 Microlocal regularity
10 Operators on manifolds
11 The method of layer potentials
12 Parametrix for regular elliptic boundary problems
13 Parametrix for the heat equation
14 The Weyl calculus
15 Operators of harmonic oscillator type
16 Positive quantization of C(S*M)
References
8 Spectral Theory
1 The spectral theorem
2 Self-adjoint differential operators
3 Heat asymptotics and eigenvalue asymptotics
4 The Laplace operator on Sn
5 The Laplace operator on hyperbolic space
6 The harmonic oscillator
7 The quantum Coulomb problem
8 Potential well–quantum model of a deuteron
9 The Laplace operator on cones
10 Quantum adiabatic limit and parallel transport
11 A quantum ergodic theorem
A Von Neumann's mean ergodic theorem
B Wave equations and shifted wave equations
References
9 Scattering by Obstacles
1 The scattering problem
2 Eigenfunction expansions
3 The scattering operator
4 Connections with the wave equation
5 Wave operators
6 Translation representations and the Lax–Phillips semigroup Z(t)
7 Integral equations and scattering poles
8 Trace formulas; the scattering phase
9 Scattering by a sphere
10 Inverse problems I
11 Inverse problems II
12 Scattering by rough obstacles
A Lidskii's trace theorem
References
10 Dirac Operators and Index Theory
1 Operators of Dirac type
2 Clifford algebras
3 Spinors
4 Weitzenbock formulas
5 Index of Dirac operators
6 Proof of the local index formula
7 The Chern–Gauss–Bonnet theorem
8 Spinc manifolds
9 The Riemann–Roch theorem
10 Direct attack in 2-D
11 Index of operators of harmonic oscillator type
References
11 Brownian Motion and Potential Theory
1 Brownian motion and Wiener measure
2 The Feynman–Kac formula
3 The Dirichlet problem and diffusion on domains with boundary
4 Martingales, stopping times, and the strong Markov property
5 First exit time and the Poisson integral
6 Newtonian capacity
7 Stochastic integrals
8 Stochastic integrals, II
9 Stochastic differential equations
10 Application to equations of diffusion
11 Diffusion on Riemannian manifolds
A The Trotter product formula
References
12 The -Neumann Problem
A Elliptic complexes
1 The -complex
2 Morrey's inequality, the Levi form, and strong pseudoconvexity
3 The 12-estimate and some consequences
4 Higher-order subelliptic estimates
5 Regularity via elliptic regularization
6 The Hodge decomposition and the -equation
7 The Bergman projection and Toeplitz operators
8 The -Neumann problem on (0,q)-forms
9 Reduction to pseudodifferential equations on the boundary
10 The -equation on complex manifolds and almost complex manifolds
B Complements on the Levi form
C The Neumann operator for the Dirichlet problem
References
C Connections and Curvature
1 Covariant derivatives and curvature on general vector bundles
2 Second covariant derivatives and covariant-exterior derivatives
3 The curvature tensor of a Riemannian manifold
4 Geometry of submanifolds and subbundles
5 The Gauss–Bonnet theorem for surfaces
6 The principal bundle picture
7 The Chern–Weil construction
8 The Chern–Gauss–Bonnet theorem
9 Kahler manifolds and their curvature
References
Index

Citation preview

Applied Mathematical Sciences

Michael E. Taylor

Partial Differential Equations II Qualitative Studies of Linear Equations Third Edition

Applied Mathematical Sciences Founding Editors F. John J. P. LaSalle L. Sirovich

Volume 116

Series Editors Anthony Bloch, Department of Mathematics, University of Michigan, Ann Arbor, MI, USA C. L. Epstein, Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA Alain Goriely, Department of Mathematics, University of Oxford, Oxford, UK Leslie Greengard, New York University, New York, NY, USA Advisory Editors J. Bell, Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, CA, USA P. Constantin, Department of Mathematics, Princeton University, Princeton, NJ, USA R. Durrett, Department of Mathematics, Duke University, Durham, CA, USA R. Kohn, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA R. Pego, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA L. Ryzhik, Department of Mathematics, Stanford University, Stanford, CA, USA A. Singer, Department of Mathematics, Princeton University, Princeton, NJ, USA A. Stevens, Department of Applied Mathematics, University of Münster, Münster, Germany S. Wright, Computer Sciences Department, University of Wisconsin, Madison, WI, USA

The mathematization of all sciences, the fading of traditional scientific boundaries, the impact of computer technology, the growing importance of computer modeling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The purpose of this series is to provide such books, suitable for the user of mathematics, the mathematician interested in applications, and the student scientist. In particular, this series will provide an outlet for topics of immediate interest because of the novelty of its treatment of an application or of mathematics being applied or lying close to applications. These books should be accessible to readers versed in mathematics or science and engineering, and will feature a lively tutorial style, a focus on topics of current interest, and present clear exposition of broad appeal. A compliment to the Applied Mathematical Sciences series is the Texts in Applied Mathematics series, which publishes textbooks suitable for advanced undergraduate and beginning graduate courses.

Michael E. Taylor

Partial Differential Equations II Qualitative Studies of Linear Equations Third Edition

123

Michael E. Taylor Department of Mathematics University of North Carolina Chapel Hill, NC, USA

ISSN 0066-5452 ISSN 2196-968X (electronic) Applied Mathematical Sciences ISBN 978-3-031-33699-7 ISBN 978-3-031-33700-0 (eBook) https://doi.org/10.1007/978-3-031-33700-0 Mathematics Subject Classification: 35-01 1st & 2nd editions: © Springer Science+Business Media, LLC 1996, 2011 3rd edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my wife and daughter, Jane Hawkins and Diane Hart

Contents of Volumes I and III

Volume I: Basic Theory

1

Basic Theory of ODE and Vector Fields

2

The Laplace Equation and Wave Equation

3

Fourier Analysis, Distributions, and Constant-Coefficient Linear PDE

4

Sobolev Spaces

5

Linear Elliptic Equations

6

Linear Evolution Equations

A

Outline of Functional Analysis

B

Manifolds, Vector Bundles, and Lie Groups

Volume III: Nonlinear Equations

13 Function Space and Operator Theory for Nonlinear Analysis 14 Nonlinear Elliptic Equations 15 Nonlinear Parabolic Equations 16 Nonlinear Hyperbolic Equations 17 Euler and Navier–Stokes Equations for Incompressible Fluids 18 Einstein’s Equations

vii

Preface

Partial differential equations are a many-faceted subject. Created to describe the mechanical behavior of objects such as vibrating strings and blowing winds, it has developed into a body of material that interacts with many branches of mathematics, such as differential geometry, complex analysis, and harmonic analysis, as well as a ubiquitous factor in the description and elucidation of problems in mathematical physics. This work is intended to provide a course of study of some of the major aspects of PDE. It is addressed to readers with a background in the basic introductory graduate mathematics courses in American universities: elementary real and complex analysis, differential geometry, and measure theory. Chapter 1 provides background material on the theory of ordinary differential equations (ODE). This includes both very basic material – on topics such as the existence and uniqueness of solutions to ODE and explicit solutions to equations with constant coefficients and relations to linear algebra – and more sophisticated results – on flows generated by vector fields, connections with differential geometry, the calculus of differential forms, stationary action principles in mechanics, and their relation to Hamiltonian systems. We discuss equations of relativistic motion as well as equations of classical Newtonian mechanics. There are also applications to topological results, such as degree theory, the Brouwer fixed-point theorem, and the Jordan-Brouwer separation theorem. In this chapter, we also treat scalar first-order PDE, via the Hamilton–Jacobi theory. Chapters 2–6 constitute a survey of basic linear PDE. Chapter 2 begins with the derivation of some equations of continuum mechanics in a fashion similar to the derivation of ODE in mechanics in Chap. 1, via variational principles. We obtain equations for vibrating strings and membranes; these equations are not necessarily linear, and hence they will also provide sources of problems later, when nonlinear PDE is taken up. Further material in Chap. 2 centers around the Laplace operator, which on Euclidean space Rn is (1)



@2 @2 þ  þ ; 2 @x2n @x1

and the linear wave equation,

ix

x

(2)

Preface

@2u  Du ¼ 0: @t2

We also consider the Laplace operator on a general Riemannian manifold and the wave equation on a general Lorentz manifold. We discuss the basic consequences of Green’s formula, including energy conservation and finite propagation speed for solutions to linear wave equations. We also discuss Maxwell’s equations for electromagnetic fields and their relation with special relativity. Before we can establish general results on the solvability of these equations, it is necessary to develop some analytical techniques. This is done in the next couple of chapters. Chapter 3 is devoted to Fourier analysis and the theory of distributions. These topics are crucial for the study of linear PDE. We give a number of basic applications to the study of linear PDE with constant coefficients. Among these applications are results on harmonic and holomorphic functions in the plane, including a short treatment of elementary complex function theory. We derive explicit formulas for solutions to Laplace and wave equations on Euclidean space, and also the heat equation, (3)

@u  Du ¼ 0: @t

We also produce solutions on certain subsets, such as rectangular regions, using the method of images. We include material on the discrete Fourier transform, germane to the discrete approximation of PDE, and on the fast evaluation of this transform, the FFT. Chapter 3 is the first chapter to make extensive use of functional analysis. Basic results on this topic are compiled in Appendix A, Outline of Functional Analysis. Sobolev spaces have proven to be a very effective tool in the existence theory of PDE, and in the study of regularity of solutions. In Chap. 4 we introduce Sobolev spaces and study some of their basic properties. We restrict attention to L2-Sobolev spaces, such as Hk(Rn ), which consists of L2 functions whose derivatives of order  k (defined in a distributional sense, in Chap. 3) belong to L2 ðRn Þ, when k is a positive integer. We also replace k by a general real number s. The Lp -Sobolev spaces, which are very useful for nonlinear PDE, are treated later, in Chap. 13 Chapter 5 is devoted to the study of the existence and regularity of solutions to linear elliptic PDE, on bounded regions. We begin with the Dirichlet problem for the Laplace operator, (4)

Du ¼ f on X;

u ¼ g on @X;

and then treat the Neumann problem and various other boundary problems, including some that apply to electromagnetic fields. We also study general boundary problems for linear elliptic operators, giving a condition that guarantees

Preface

xi

regularity and solvability (perhaps given a finite number of linear conditions on the data). Also in Chap. 5 are some applications to other areas, such as a proof of the Riemann mapping theorem, first for smooth simply connected domains in the complex plane C, then, after a treatment of the Dirichlet problem for the Laplace operator on domains with rough boundary, for general simply connected domains in C. We also develop the Hodge theory and apply it to de Rham cohomology, extending the study of topological applications of differential forms begun in Chap. 1. In Chap. 6 we study linear evolution equations, in which there is a “time” variable t, and initial data are given at t = 0. We discuss the heat and wave equations. We also treat Maxwell’s equations, for an electromagnetic field, and more general hyperbolic systems. We prove the Cauchy–Kowalewsky theorem, in the linear case, establishing local solvability of the Cauchy initial value problem for general linear PDE with analytic coefficients, and analytic data, as long as the initial surface is “noncharacteristic.” The nonlinear case is treated in Chap. 16. Also in Chap. 6 we treat geometrical optics, providing approximations to solutions of wave equations whose initial data either are highly oscillatory or possess simple singularities, such as a jump across a smooth hypersurface. Chapters 1–6, together with Appendix A and B, Manifolds, Vector Bundles, and Lie Groups, make up the first volume of this work. The second volume consists of Chaps. 7–12, covering a selection of more advanced topics in linear PDE, together with Appendix C, Connections and Curvature. Chapter 7 deals with pseudodifferential operators (wDOs). This class of operators includes both differential operators and parametrices of elliptic operators, that is, inverses modulo smoothing operators. There is a “symbol calculus” allowing one to analyze products of wDOs, useful for such a parametrix construction. The L2-boundedness of operators of order zero and the Garding inequality for elliptic wDOs with positive symbol provide very useful tools in linear PDE, which will be used in many subsequent chapters. Chapter 8 is devoted to spectral theory, particularly for self-adjoint elliptic operators. First we give a proof of the spectral theorem for general self-adjoint operators on Hilbert space. Then we discuss conditions under which a differential operator yields a self-adjoint operator. We then discuss the asymptotic distribution of eigenvalues of the Laplace operator on a bounded domain, making use of a construction of a parametrix for the heat equation from Chap. 7. Further material in Chap. 8 includes results on the spectral behavior of various specific differential operators, such as the Laplace operator on a sphere, and on hyperbolic space, the “harmonic oscillator” (5) and the operator

D þ jxj2 ;

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(6)

Preface

D 

K ; jxj

which arises in the simplest quantum mechanical model of the hydrogen atom. We also consider the Laplace operator on cones. In Chap. 9 we study the scattering of waves by a compact obstacle K in R3 . This scattering theory is to some degree an extension of the spectral theory of the Laplace operator on R3 \K, with the Dirichlet boundary condition. In addition to studying how a given obstacle scatters waves, we consider the inverse problem: how to determine an obstacle given data on how it scatters waves. Chapter 10 is devoted to the Atiyah–Singer index theorem. This gives a formula for the index of an elliptic operator D on a compact manifold M, defined by (7)

Index D ¼ dim ker D  dim ker D :

We establish this formula, which is an integral over M of a certain differential form defined by a pair of “curvatures,” when D is a first-order differential operator of “Dirac type,” a class that contains many important operators arising from differential geometry and complex analysis. Special cases of such a formula include the Chern–Gauss–Bonnet formula and the Riemann–Roch formula. We also discuss the significance of the latter formula in the study of Riemann surfaces. In Chap. 11 we study Brownian motion, described mathematically by Wiener measure on the space of continuous paths in Rn . This provides a probabilistic approach to diffusion and it both uses and provides new tools for the analysis of the heat equation and variants, such as (8)

@u ¼ Du þ Vu; @t

where V is a real-valued function. There is an integral formula for solutions to (8), known as the Feynman–Kac formula; it is an integral over path space with respect to the Wiener measure, of a fairly explicit integrand. We also derive an analogous integral formula for solutions to (9)

@u ¼ Du þ Xu; @t

where X is a vector field. In this case, another tool is involved in constructing the integrand, the stochastic integral. We also study stochastic differential equations and applications to more general diffusion equations.  In Chap. 12 we tackle the @-Neumann problem, a boundary problem for an elliptic operator (essentially the Laplace operator) on a domain X  Cn , which is very important in the theory of functions of several complex variables. From a

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technical point of view, it is of particular interest that this boundary problem does not satisfy the regularity criteria investigated in Chap. 5. If X is “strongly pseudo-convex,” one has instead certain “subelliptic estimates,” which are established in Chap. 12. The third and final volume of this work contains Chaps. 13–18. It is here that we study nonlinear PDE. We prepare the way in Chap. 13 with a further development of function space and operator theory, for use in nonlinear analysis. This includes the theory of LpSobolev spaces and Hölder spaces. We derive estimates in these spaces on nonlinear functions F(u), known as “Moser estimates,” which are very useful. We extend the theory of pseudodifferential operators to cases where the symbols have limited smoothness, and also develop a variant of wDO theory, the theory of “paradifferential operators,” which has had a significant impact on nonlinear PDE since about 1980. We also estimate these operators, acting on the function spaces mentioned above. Other topics treated in Chap. 13 include Hardy spaces, compensated compactness, and “fuzzy functions.” Chapter 14 is devoted to nonlinear elliptic PDE, with an emphasis on second-order equations. There are three successive degrees of nonlinearity: semilinear equations, such as Du ¼ F ðx; u; ruÞ;

(10)

quasi-linear equations, such as (11)

X

aj k ðx; u; ruÞ@j @k u ¼ F ðx; u; ruÞ;

and completely nonlinear equations, of the form (12)

Gðx; D2 uÞ ¼ 0:

Differential geometry provides a rich source of such PDE, and Chap. 14 contains a number of geometrical applications. For example, to deform conformally a metric on a surface so its Gauss curvature changes from k(x) to K(x), one needs to solve the semilinear equation (13)

Du ¼ kðxÞ  KðxÞe2u :

As another example, the graph of a function y = u(x) is a minimal submanifold of Euclidean space provided u solves the quasi-linear equation (14)

ð1 þ jruj2 ÞDu þ ðruÞ  HðuÞðruÞ ¼ 0;

xiv

Preface

called the minimal surface equation. Here, HðuÞ ¼ ð@j @k uÞ is the Hessian matrix of u. On the other hand, this graph has Gauss curvature K(x) provided u solves the completely nonlinear equation (15)

det HðuÞ ¼ KðxÞð1 þ jruj2 Þðn þ 2Þ=2 ;

a Monge–Ampère equation. Equations (13)–(15) are all scalar, and the maximum principle plays a useful role in the analysis, together with a number of other tools. Chapter 14 also treats nonlinear systems. Important physical examples arise in studies of elastic bodies, as well as in other areas, such as the theory of liquid crystals. Geometric examples of systems considered in Chap. 14 include equations for harmonic maps and equations for isometric embeddings of a Riemannian manifold in Euclidean space. In Chap. 15, we treat nonlinear parabolic equations. Partly echoing Chap. 14, we progress from a treatment of semilinear equations, (16)

@u ¼ Lu þ F ðx; u; ruÞ; @t

where L is a linear operator, such as L ¼ D, to a treatment of quasi-linear equations, such as (17)

@u X ¼ @j aj k ðt; x; uÞ@k u þ XðuÞ: @t

(We do very little with completely nonlinear equations in this chapter.) We study systems as well as scalar equations. The first application of (16) we consider is to the parabolic equation method of constructing harmonic maps. We also consider “reaction–diffusion” equations, ‘  ‘ systems of the form (16), in which F ðx; u; ruÞ ¼ XðuÞ, where X is a vector field on R‘ , and L is a diagonal operator, with diagonal elements aj D; aj  0: These equations arise in mathematical models in biology and in chemistry. For example, u ¼ ðu1 ;    ; u‘ Þmight represent the population densities of each of ‘ species of living creatures, distributed over an area of land, interacting in a manner described by X and diffusing in a manner described by aj D: If there is a nonlinear (density-dependent) diffusion, one might have a system of the form (17). Another problem considered in Chap. 15 models the melting of ice; one has a linear heat equation in a region (filled with water) whose boundary (where the water touches the ice) is moving (as the ice melts). The nonlinearity in the problem involves the description of the boundary. We confine our analysis to a relatively simple one-dimensional case. Nonlinear hyperbolic equations are studied in Chap. 16. Here continuum mechanics is the major source of examples, and most of them are systems, rather than scalar equations. We establish local existence for solutions to first-order

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xv

hyperbolic systems, which are either “symmetric” or “symmetrizable.” An example of the latter class is the following system describing compressible fluid flow: (18)

@v 1 þ rv v þ grad p ¼ 0; @t q

@q þ rv q þ q div v ¼ 0; @t

for a fluid with velocity v, density q, and pressure p, assumed to satisfy a relation p ¼ pðqÞ, called an “equation of state.” Solutions to such nonlinear systems tend to break down, due to shock formation. We devote a bit of attention to the study of weak solutions to nonlinear hyperbolic systems, with shocks. We also study second-order hyperbolic systems, such as systems for a k-dimensional membrane vibrating in Rn , derived in Chap. 2. Another topic covered in Chap. 16 is the Cauchy–Kowalewsky theorem, in the nonlinear case. We use a method introduced by P. Garabedian to transform the Cauchy problem for an analytic equation into a symmetric hyperbolic system. In Chap. 17 we study incompressible fluid flow. This is governed by the Euler equation (19)

@v þ rv v ¼ grad p; @t

div v ¼ 0;

in the absence of viscosity, and by the Navier–Stokes equation (20)

@v þ rv v ¼ ”Lv  grad p; @t

div v ¼ 0;

in the presence of viscosity. Here L is a second-order operator, the Laplace operator for a flow on flat space; the “viscosity” ” is a positive quantity. Equation (19) shares some features with quasi-linear hyperbolic systems, though there are also significant differences. Similarly, (20) has a lot in common with semilinear parabolic systems. Chapter 18, the last chapter of this work, is devoted to Einstein’s gravitational equations: (21)

Gjk ¼ 8p•Tjk :

Here Gj k is the Einstein tensor, given by Gj k ¼ Ricj k  ð1=2ÞS gj k , where Ricj k is the Ricci tensor and S the scalar curvature, of a Lorentz manifold (or “spacetime”) with metric tensor gj k . On the right side of (21), Tj k is the stress– energy tensor of the matter in the spacetime, and k is a positive constant, which can be identified with the gravitational constant of the Newtonian theory of gravity. In local coordinates, Gj k has a nonlinear expression in terms of gj k and its second-order derivatives. In the empty-space case, where Tj k ¼ 0; (21) is a quasi-linear second-order system for gj k . The freedom to change coordinates

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Preface

provides an obstruction to this equation being hyperbolic, but one can impose the use of “harmonic” coordinates as a constraint and transform (21) into a hyperbolic system. In the presence of matter one couples (21) to other systems, obtaining more elaborate PDE. We treat this in two cases, in the presence of an electromagnetic field, and in the presence of a relativistic fluid. In addition to the 18 chapters just described, there are three appendices, already mentioned above. Appendix A gives definitions and basic properties of Banach and Hilbert spaces (of which Lp-spaces and Sobolev spaces are examples), Fréchet spaces (such as C 1 ðRn Þ), and other locally convex spaces (such as spaces of distributions). It discusses some basic facts about bounded linear operators, including some special properties of compact operators, and also considers certain classes of unbounded linear operators. This functional analytic material plays a major role in the development of PDE from Chap. 3 onward. Appendix B gives definitions and basic properties of manifolds and vector bundles. It also discusses some elementary properties of Lie groups, including a little representation theory, useful in Chap. 8, on spectral theory, as well as in the Chern–Weil construction. Appendix C, Connections and Curvature, contains material of a differential geometric nature, crucial for understanding many things done in Chaps. 10–18. We consider connections on general vector bundles, and their curvature. We discuss in detail the special properties of the primary case: the Levi–Civita connection and Riemann curvature tensor on a Riemannian manifold. We discuss the basic properties of the geometry of submanifolds, relating the second fundamental form to curvature via the Gauss–Codazzi equations. We describe how vector bundles arise from principal bundles, which themselves carry various connections and curvature forms. We then discuss the Chern–Weil construction, yielding certain closed differential forms associated to curvatures of connections on principal bundles. We give several proofs of the classical Gauss–Bonnet theorem and some related results on two-dimensional surfaces, which are useful particularly in Chaps. 10 and 14. We also give a geometrical proof of the Chern– Gauss–Bonnet theorem, which can be contrasted with the proof in Chap. 10, as a consequence of the Atiyah–Singer index theorem. We mention that, in addition to these “global” appendices, there are appendices to some chapters. For example, Chap. 3 has an appendix on the gamma function. Chapter 6 has two appendices; Appendix A has some results on Banach spaces of harmonic functions useful for the proof of the linear Cauchy–Kowalewsky theorem, and Appendix B deals with the stationary phase formula, useful for the study of geometrical optics in Chap. 6 and also for results later, in Chap. 9. There are other chapters with such “local” appendices. Furthermore, there are two sections, both in Chap. 14, with appendices. Section 6, on minimal surfaces, has a companion, Sect. 6B, on the second variation of area and consequences, and Sect. 13, on nonlinear elliptic systems, has a companion, Sect. 12B, with complementary material.

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Having described the scope of this work, we find it necessary to mention a number of topics in PDE that are not covered here or are touched on only very briefly. For example, we devote little attention to the real analytic theory of PDE. We note that harmonic functions on domains in Rn are real analytic, but we do not discuss the analyticity of solutions to more general elliptic equations. We do prove the Cauchy–Kowalewsky theorem, on analytic PDE with analytic Cauchy data. We derive some simple results on unique continuation from these few analyticity results, but there is a large body of lore on unique continuation, for solutions to nonanalytic PDE, neglected here. There is little material on numerical methods. There are a few references to applications of the FFT and of “splitting methods.” Difference schemes for PDE are mentioned just once, in a set of exercises on scalar conservation laws. Finite element methods are neglected, as are many other numerical techniques. There is a large body of work on free boundary problems, but the only one considered here is a simple one-space dimensional problem, in Chap. 15. While we have considered a variety of equations arising from classical physics and from relativity, we have devoted relatively little attention to quantum mechanics. We have considered a few quantum systems in Chap. 8, including models of the hydrogen atom and the deuteron. Also, there are some exercises on potential scattering mentioned in Chap. 9. However, the physical theories behind these equations are not discussed here. There are a number of nonlinear evolution equations, such as the Korteweg– deVries equation, that have been perceived to provide infinite dimensional analogues of completely integrable Hamiltonian systems, and to arise “universally” in asymptotic analyses of solutions to various nonlinear wave equations. They are not here. Nor is there a treatment of the Yang–Mills equations for gauge fields, with their wonderful applications to the geometry and topology of four-dimensional manifolds. Of course, this is not a complete list of omitted material. One can go on and on listing important topics in this vast subject. The author can at best hope that the reader will find it easier to understand many of these topics with this book, than without it.

Acknowledgments I have had the good fortune to teach at least one course relevant to the material of this book, almost every year since 1971. These courses led to many course notes, and I am grateful to many colleagues at Rice University, SUNY at Stony Brook, the California Institute of Technology, and the University of North Carolina, for the supportive atmospheres at these institutions. Also, a number of individuals provided valuable advice on various portions of the manuscript, as it grew over the years. I particularly want to thank Florin David, David Ebin, Frank Jones,

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Anna Mazzucato, Richard Melrose, James Ralston, Jeffrey Rauch, Santiago Simanca, and James York. The final touches were put on the manuscript while I was visiting the Institute for Mathematics and its Applications, at the University of Minnesota, which I thank for its hospitality and excellent facilities. Finally, I would like to acknowledge the impact on my studies of my senior thesis and Ph.D. thesis advisors, Edward Nelson and Heinz Cordes.

Preface to the Second Edition In addition to making numerous small corrections to this work, collected over the past dozen years, I have taken the opportunity to make some very significant changes, some of which broaden the scope of the work, some of which clarify previous presentations, and a few of which correct errors that have come to my attention. There are seven additional sections in this edition, two in Volume 1, two in Volume 2, and three in Volume 3. Chapter 4 has a new section, “Sobolev spaces on rough domains,” which serves to clarify the treatment of the Dirichlet problem on rough domains in Chap. 5. Chapter 6 has a new section, “Boundary layer phenomena for the heat equation,” which will prove useful in one of the new sections in Chap. 17. Chapter 7 has a new section, “Operators of harmonic oscillator type,” and Chap. 10 has a section that presents an index formula for elliptic systems of operators of harmonic oscillator type. Chapter 13 has a new appendix, “Variations on complex interpolation,” which has material that is useful in the study of Zygmund spaces. Finally, Chap. 17 has two new sections, “Vanishing viscosity limits” and “From velocity convergence to flow convergence.” In addition, several other sections have been substantially rewritten, and numerous others polished to reflect insights gained through the use of these books over time.

Preface to the Third Edition I have provided further polishings and supplements for this third edition. New material in Volume 1 includes a section on rigid body motion in Chapter 1, which will tie in to the derivation of the Euler equation of incompressible fluid flow in Chapter 17. Chapter 3 has a new appendix on the central limit theorem, related to a random walk, which will tie in to the treatment of Brownian motion in Chapter 11. In addition there is an expanded treatment of the Poisson integral in Chapter 5, a section on the Schrödinger equation in Chapter 6, and an expanded treatment of holomorphic functional calculus in Appendix A.

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New material in Volume 2 includes sections on a quantum model of the deuteron, a quantum adiabatic theorem, and a quantum ergodic theorem, and appendices on the classical ergodic theorem and on shifted wave equations in Chapter 8, as well as expanded treatments of the spectral theorem and of analysis on hyperbolic space in that chapter. In Chapter 11 I have added a section on diffusion on Riemannian manifolds, with application to models of relativistic diffusion. New material in Volume 3 includes a section on overdetermined elliptic systems in Chapter 14 and a section on Euler flows on rotating surfaces, influenced by the Coriolis force, in Chapter 17. Chapel Hill, USA

Michael E. Taylor

Contents

Contents of Volumes I and III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

7 Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . 1 The Fourier integral representation and symbol classes 2 Schwartz kernels of pseudodifferential operators . . . . . 3 Adjoints and products . . . . . . . . . . . . . . . . . . . . . . . . 4 Elliptic operators and parametrices . . . . . . . . . . . . . . . 5 L2 -estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Gårding’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . 7 Hyperbolic evolution equations . . . . . . . . . . . . . . . . . 8 Egorov’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Microlocal regularity . . . . . . . . . . . . . . . . . . . . . . . . . 10 Operators on manifolds . . . . . . . . . . . . . . . . . . . . . . . 11 The method of layer potentials . . . . . . . . . . . . . . . . . 12 Parametrix for regular elliptic boundary problems . . . . 13 Parametrix for the heat equation . . . . . . . . . . . . . . . . 14 The Weyl calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Operators of harmonic oscillator type . . . . . . . . . . . . . 16 Positive quantization of C 1 ðS  MÞ . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 2 5 12 17 19 23 25 28 31 34 37 52 61 72 84 92 97

8 Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . 1 The spectral theorem . . . . . . . . . . . . . . . . . . . 2 Self-adjoint differential operators . . . . . . . . . . 3 Heat asymptotics and eigenvalue asymptotics . 4 The Laplace operator on S n . . . . . . . . . . . . . 5 The Laplace operator on hyperbolic space . . . 6 The harmonic oscillator . . . . . . . . . . . . . . . . . 7 The quantum Coulomb problem . . . . . . . . . . 8 Potential well–quantum model of a deuteron . 9 The Laplace operator on cones . . . . . . . . . . . 10 Quantum adiabatic limit and parallel transport 11 A quantum ergodic theorem . . . . . . . . . . . . .

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xxi

xxii

Contents

A B

Von Neumann’s mean ergodic theorem . . . . . . . . . . . . . . . . . . . 202 Wave equations and shifted wave equations . . . . . . . . . . . . . . . . 205 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

9 Scattering by Obstacles . . . . . . . . . . . . . . . . . . . . . 1 The scattering problem . . . . . . . . . . . . . . . . . . . 2 Eigenfunction expansions . . . . . . . . . . . . . . . . . 3 The scattering operator . . . . . . . . . . . . . . . . . . . 4 Connections with the wave equation . . . . . . . . . 5 Wave operators . . . . . . . . . . . . . . . . . . . . . . . . 6 Translation representations and the Lax–Phillips semigroup Z(t) . . . . . . . . . . . . . . . . . . . . . . . . . 7 Integral equations and scattering poles . . . . . . . . 8 Trace formulas; the scattering phase . . . . . . . . . 9 Scattering by a sphere . . . . . . . . . . . . . . . . . . . . 10 Inverse problems I . . . . . . . . . . . . . . . . . . . . . . 11 Inverse problems II . . . . . . . . . . . . . . . . . . . . . . 12 Scattering by rough obstacles . . . . . . . . . . . . . . A Lidskii’s trace theorem . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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251 258 272 279 288 294 306 315 318

10 Dirac Operators and Index Theory . . . . . . . . . . 1 Operators of Dirac type . . . . . . . . . . . . . . . . . 2 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . 3 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Weitzenbock formulas . . . . . . . . . . . . . . . . . . 5 Index of Dirac operators . . . . . . . . . . . . . . . . 6 Proof of the local index formula . . . . . . . . . . 7 The Chern–Gauss–Bonnet theorem . . . . . . . . 8 Spinc manifolds . . . . . . . . . . . . . . . . . . . . . . 9 The Riemann–Roch theorem . . . . . . . . . . . . . 10 Direct attack in 2-D . . . . . . . . . . . . . . . . . . . 11 Index of operators of harmonic oscillator type References . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Brownian Motion and Potential Theory . . . . . . . . . . . 1 Brownian motion and Wiener measure . . . . . . . . . . 2 The Feynman–Kac formula . . . . . . . . . . . . . . . . . . 3 The Dirichlet problem and diffusion on domains with boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Martingales, stopping times, and the strong Markov property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 First exit time and the Poisson integral . . . . . . . . .

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Contents

6 7 8 9 10 11 A

xxiii

Newtonian capacity . . . . . . . . . . . . Stochastic integrals . . . . . . . . . . . . . Stochastic integrals, II . . . . . . . . . . . Stochastic differential equations . . . . Application to equations of diffusion Diffusion on Riemannian manifolds . The Trotter product formula . . . . . . References . . . . . . . . . . . . . . . . . . .

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12 The @-Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . A Elliptic complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The @-complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Morrey’s inequality, the Levi form, and strong pseudoconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The 12-estimate and some consequences . . . . . . . . . . . . . . . 4 Higher-order subelliptic estimates . . . . . . . . . . . . . . . . . . . 5 Regularity via elliptic regularization . . . . . . . . . . . . . . . . . . 6 The Hodge decomposition and the @-equation . . . . . . . . . . 7 The Bergman projection and Toeplitz operators . . . . . . . . . 8 The @-Neumann problem on ð0; qÞ-forms . . . . . . . . . . . . . . 9 Reduction to pseudodifferential equations on the boundary . 10 The @-equation on complex manifolds and almost complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Complements on the Levi form . . . . . . . . . . . . . . . . . . . . . C The Neumann operator for the Dirichlet problem . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C Connections and Curvature . . . . . . . . . . . . . . . . . . . . 1 Covariant derivatives and curvature on general vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Second covariant derivatives and covariant-exterior derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The curvature tensor of a Riemannian manifold . . . 4 Geometry of submanifolds and subbundles . . . . . . . 5 The Gauss–Bonnet theorem for surfaces . . . . . . . . 6 The principal bundle picture . . . . . . . . . . . . . . . . . 7 The Chern–Weil construction . . . . . . . . . . . . . . . . 8 The Chern–Gauss–Bonnet theorem . . . . . . . . . . . . 9 Kahler manifolds and their curvature . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

7 Pseudodifferential Operators

Introduction In this chapter we discuss the basic theory of pseudodifferential operators as it has been developed to treat problems in linear PDE. We define pseudodifferential m , introduced by L. H¨ormander. In operators with symbols in classes denoted Sρ,δ §2 we derive some useful properties of their Schwartz kernels. In §3 we discuss adjoints and products of pseudodifferential operators. In §4 we show how the algebraic properties can be used to establish the regularity of solutions to elliptic PDE with smooth coefficients. In §5 we discuss mapping properties on L2 and on the Sobolev spaces H s . In §6 we establish G˚arding’s inequality. In §7 we apply some of the previous material to establish the existence of solutions to hyperbolic equations. In §8 we show that certain important classes of pseudodifferential operators are preserved under the action of conjugation by solution operators to (scalar) hyperbolic equations, a result of Y. Egorov. We introduce the notion of wave front set in §9 and discuss the microlocal regularity of solutions to elliptic equations. We also discuss how solution operators to a class of hyperbolic equations propagate wave front sets. In §10 there is a brief discussion of pseudodifferential operators on manifolds. We give some further applications of pseudodifferential operators in the next three sections. In §11 we discuss, from the perspective of the pseudodifferential operator calculus, the classical method of layer potentials, applied particularly to the Dirichlet and Neumann boundary problems for the Laplace operator. Historically, this sort of application was one of the earliest stimuli for the development of the theory of singular integral equations. One function of §11 is to provide a warm-up for the use of similar integral equations to tackle problems in scattering theory, in §7 of Chap. 9. We also discuss the development of layer potential theory on various classes of domains with rough boundary. Section 12 looks at general regular elliptic boundary problems and includes material complementary to that developed in §11 of Chap. 5. In §13 we construct a parametrix for the heat equation and apply this to obtain an asymptotic expansion of the trace of the solution operator. This expansion will be useful in studies of the spectrum in Chap. 8 and in index theory in Chap. 10. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  M. E. Taylor, Partial Differential Equations II, Applied Mathematical Sciences 116, https://doi.org/10.1007/978-3-031-33700-0 7

1

2

7. Pseudodifferential Operators

In §14 we introduce the Weyl calculus. This can provide a powerful alternative to the operator calculus developed in §§1–6, as can be seen in [Ho4] and in Vol. 3 of [Ho5]. Here we concentrate on identities, tied to symmetries in the Weyl calculus. We show how this leads to a quicker construction of a parametrix for the heat equation than the method used in §13. We will make use of this in §10 of Chap. 10, on a direct attack on the index theorem for elliptic differential operators on two-dimensional manifolds. In §15, we study a class of pseudodifferential operators of “Harmonic oscillator type.” This class contains the Harmonic oscillator, H = −Δ + |x|2 , with symbol |x|2 + |ξ|2 , and results on these operators are interesting variants on m . those with symbols in S1,0 In §16, we return to the issue of defining pseudodifferential operators on a compact manifold M , phrasing it as a quantization of smooth functions on S ∗ M . We discuss the existence of positive quantizations of C ∞ (S ∗ M ). Results here will be of use in the treatment of quantum ergodic theorems, in Chapter 8. Material in §§1–10 is taken from Chap. 0 of [T4], and the author thanks Birkh¨auser Boston for permission to use this material. We also mention some books that take the theory of pseudodifferential operators farther than is done here: [Ho5, Kg, T1], and [Tre].

1. The Fourier integral representation and symbol classes Using a slightly different convention from that established in Chap. 3, we write the Fourier inversion formula as  (1.1) f (x) = fˆ(ξ) eix·ξ dξ,  where fˆ(ξ) = (2π)−n f (x)e−ix·ξ dx is the Fourier transform of a function on Rn . If one differentiates (1.1), one obtains  (1.2) Dα f (x) = ξ α fˆ(ξ)eix·ξ dξ, where Dα = D1α1 · · · Dnαn , Dj = (1/i) ∂/∂xj . Hence, if  aα (x)Dα p(x, D) = |α|≤k

is a differential operator, we have

1. The Fourier integral representation and symbol classes

 (1.3)

p(x, D)f (x) =

where p(x, ξ) =

3

p(x, ξ)fˆ(ξ)eix·ξ dξ 

aα (x)ξ α .

|α|≤k

One uses the Fourier integral representation (1.3) to define pseudodifferential operators, taking the function p(x, ξ) to belong to one of a number of different classes of symbols. In this chapter we consider the following symbol classes, first defined by H¨ormander [Ho2]. m to consist of C ∞ -functions Assuming ρ, δ ∈ [0, 1], m ∈ R, we define Sρ,δ p(x, ξ) satisfying (1.4)

|Dxβ Dξα p(x, ξ)| ≤ Cαβ ξm−ρ|α|+δ|β| ,

for all α, β, where ξ = (1 + |ξ|2 )1/2 . In such a case we say the associated m . We say that p(x, ξ) is the symbol of operator defined by (1.3) belongs to OP Sρ,δ p(x, D). The case of principal interest is ρ = 1, δ = 0. This class is defined by [KN]. Recall that in Chap. 3, §8, we defined P (ξ) ∈ S1m (Rn ) to satisfy (1.4), with m contains S1m (Rn ). ρ = 1, and with no x-derivatives involved. Thus S1,0 If there are smooth pm−j (x, ξ), homogeneous in ξ of degree m − j for |ξ| ≥ 1, that is, pm−j (x, rξ) = rm−j pm−j (x, ξ) for r, |ξ| ≥ 1, and if p(x, ξ) ∼

(1.5)



pm−j (x, ξ)

j≥0

in the sense that (1.6)

p(x, ξ) −

N 

m−N −1 pm−j (x, ξ) ∈ S1,0 ,

j=0 m , or just p(x, ξ) ∈ S m . We call pm (x, ξ) for all N , then we say p(x, ξ) ∈ Scl the principal symbol of p(x, D). We will give a more general definition of the principal symbol in §10. m and ρ, δ ∈ [0, 1], then p(x, D) : S(Rn ) → It is easy to see that if p(x, ξ) ∈ Sρ,δ ∞ n C (R ). In fact, multiplying (1.3) by xα , writing xα eix·ξ = (−Dξ )α eix·ξ , and integrating by parts yield

(1.7)

p(x, D) : S(Rn ) −→ S(Rn ).

Under one restriction, p(x, D) also acts on tempered distributions: Lemma 1.1. If δ < 1, then

4

7. Pseudodifferential Operators

p(x, D) : S  (Rn ) −→ S  (Rn ).

(1.8)

Proof. Given u ∈ S  , v ∈ S, we have (formally) ˆ, v, p(x, D)u = pv , u

(1.9) where

pv (ξ) = (2π)−n



v(x)p(x, ξ)eix·ξ dx.

Now integration by parts gives ξ α pv (ξ) = (2π)−n so



  Dxα v(x)p(x, ξ) eix·ξ dx,

|pv (ξ)| ≤ Cα ξm+δ|α|−|α| .

Thus if δ < 1, we have rapid decrease of pv (ξ). Similarly, we get rapid decrease of derivatives of pv (ξ), so it belongs to S. Thus the right side of (1.9) is well defined. In §5 we will analyze the action of pseudodifferential operators on Sobolev spaces. m have been introduced by R. Beals Classes of symbols more general than Sρ,δ and C. Fefferman [BF, Be], and still more general classes were studied by H¨ormander [Ho4]. These classes have some deep applications, but they will not be used in this book.

Exercises 1. Show that, for a(x, ξ) ∈ S(R2n ),  (1.10) a(x, D)u = a ˆ(q, p) eiq·X eip·D u(x)

dq

dp,

where a ˆ(q, p) is the Fourier transform of a(x, ξ), and the operators eiq·X and eip·D are defined by eiq·X u(x) = eiq·x u(x), eip·D u(x) = u(x + p). 2. Establish the identity eip·D eiq·X = eiq·p eiq·X eip·D .

(1.11)

Deduce that, for (t, q, p) ∈ R × Rn × Rn = Hn , the binary operation (1.12)

(t, q, p) ◦ (t , q  , p ) = (t + t + p · q  , q + q  , p + p )

gives a group and that (1.13)

π ˜ (t, q, p) = eit eiq·X eip·X

2. Schwartz kernels of pseudodifferential operators

5

defines a unitary representation of Hn on L2 (Rn ); in particular, it is a group homomor˜ (z)˜ π (z  ). Hn is called the Heisenberg group. phism: π ˜ (z ◦ z  ) = π 3. Give a definition of a(x − q, D − p), acting on u(x). Show that a(x − q, D − p) = π ˜ (0, q, p) a(x, D) π ˜ (0, q, p)−1 . m and b(x, ξ) ∈ S(Rn × Rn ). Show that c(x, ξ) = (b ∗ a)(x, ξ) 4. Assume a(x, ξ) ∈ Sρ,δ m belongs to Sρ,δ (∗ being convolution on R2n ). Show that  c(x, D)u = b(y, η) a(x − y, D − η) dy dη.

5. Show that the map Ψ(p, u) = p(x, D)u has a unique, continuous, bilinear extension m × S(Rn ) → S(Rn ) to from Sρ,δ Ψ : S  (R2n ) × S(Rn ) −→ S  (Rn ), so that p(x, D) is “well defined” for any p ∈ S  (Rn × Rn ). m , let 6. Let χ(ξ) ∈ C0∞ (Rn ) be 1 for |ξ| ≤ 1, χ (ξ) = χ(ξ). Given p(x, ξ) ∈ Sρ,δ p (x, ξ) = χ (ξ)p(x, ξ). Show that if ρ, δ ∈ [0, 1], then (1.14)

u ∈ S(Rn ) =⇒ p (x, D)u → p(x, D)u in S(Rn ).

If also δ < 1, show that (1.15)

u ∈ S  (Rn ) =⇒ p (x, D)u → p(x, D)u in S  (Rn ),

where we give S  (Rn ) the weak∗ topology. 7. For s ∈ R, define Λs : S  (Rn ) → S  (Rn ) by  ˆ(ξ) eix·ξ dξ, (1.16) Λs u(x) = ξ s u where ξ = (1 + |ξ|2 )1/2 . Show that Λs ∈ OP S s . m 8. Given pj (x, ξ) ∈ Sρ,δj , for j ≥ 0, with ρ, δ ∈ [0, 1] and mj −∞, show that there m0 exists p(x, ξ) ∈ Sρ,δ such that  pj (x, ξ), p(x, ξ) ∼ j≥0

in the sense that, for all k, p(x, ξ) −

k−1 

mk pj (x, ξ) ∈ Sρ,δ .

j=0

2. Schwartz kernels of pseudodifferential operators m To an operator p(x, D) ∈ OP Sρ,δ defined by (1.3) there corresponds a Schwartz  n n kernel K ∈ D (R × R ), satisfying

6

7. Pseudodifferential Operators



u(x)p(x, ξ)ˆ v (ξ)eix·ξ dξ dx  = (2π)−n u(x)p(x, ξ)ei(x−y)·ξ v(y) dy dξ dx.

u(x)v(y), K = (2.1)

Thus, K is given as an “oscillatory integral” (2.2)

−n

K = (2π)



p(x, ξ)ei(x−y)·ξ dξ.

We have the following basic result. Proposition 2.1. If ρ > 0, then K is C ∞ off the diagonal in Rn × Rn . Proof. For given α ≥ 0, (2.3)

α

(x − y) K =



ei(x−y)·ξ Dξα p(x, ξ) dξ.

This integral is clearly absolutely convergent for |α| so large that m − ρ|α| < −n. Similarly, it is seen that applying j derivatives to (2.3) yields an absolutely convergent integral provided m + j − ρ|α| < −n, so in that case (x − y)α K ∈ C j (Rn × Rn ). This gives the proof. Generally, if T has the mapping properties T : C0∞ (Rn ) −→ C ∞ (Rn ),

T : E  (Rn ) −→ D (Rn ),

and its Schwartz kernel K is C ∞ off the diagonal, it follows easily that sing supp T u ⊂ sing supp u, for u ∈ E  (Rn ). m This is called the pseudolocal property. By (1.7)– (1.8) it holds for T ∈ OP Sρ,δ if ρ > 0 and δ < 1. We remark that the proof of Proposition 2.1 leads to the estimate

(2.4)

β K| ≤ C|x − y|−k , |Dx,y

where k ≥ 0 is any integer strictly greater than (1/ρ)(m + n + |β|). In fact, this estimate is rather crude. It is of interest to record a more precise estimate that m . holds when p(x, ξ) ∈ S1,δ m , then the Schwartz kernel K of p(x, D) satisProposition 2.2. If p(x, ξ) ∈ S1,δ fies estimates

(2.5)

β K| ≤ C|x − y|−n−m−|β| |Dx,y

2. Schwartz kernels of pseudodifferential operators

7

provided m + |β| > −n. The result is easily reduced to the case p(x, ξ) = p(ξ), satisfying |Dα p(ξ)| ≤ Cα ξm−|α| , for which p(D) has Schwartz kernel K = pˆ(y − x). It suffices to prove (2.5) for such a case, for β = 0 and m > −n. We make use of the following simple but important characterization of such symbols. m if and only if Assertion. Given p(ξ) ∈ C ∞ (Rn ), it belongs to S1,0

(2.6)

pr (ξ) = r−m p(rξ) is bounded in C ∞ (1 ≤ |ξ| ≤ 2), for r ∈ [1, ∞).

Using this, we establish the following. m Lemma 2.3. Given p(ξ) ∈ S1,0 (Rn ), we can write

 p(ξ) = p0 (ξ) +



0

qτ (e−τ ξ) dτ,

with p0 (ξ) ∈ C ∞ (Rn ) and {e−mτ qτ (ξ) : τ ∈ [0, ∞)} bounded in S(Rn ). m (Rb ). Proof. We describe the following decomposition of a symbol p(ξ) ∈ S1,0 Start with

A ∈ C0∞ (Rn ), radial, supported in {ξ ∈ Rn : 1 ≤ |ξ| ≤ 2}. Set

∞ B(ξ) =

A(e−τ ξ) dτ,

−∞

a radial function, satisfying B(rξ) = B(ξ), for r > 0. Hence B(ξ) is constant. Scaling, we can arange that B(ξ) ≡ 1. Write 1 = B(ξ) = B0 (ξ) + B1 (ξ), 

with B1 (ξ) =



−τ

A(e

 ξ) dτ,

0

B0 (ξ) =



A(eτ ξ) dτ.

0

We have B1 (ξ) = 0 for |ξ| ≤ 1, hence

B0 (ξ) = 0 for |ξ| ≥ 2,

B0 ∈ C0∞ (Rn ).

8

7. Pseudodifferential Operators

Now write p(ξ) = B0 (ξ)p(ξ) + B1 (ξ)p(ξ)  ∞ = p0 (ξ) + p(ξ)A(e−τ ξ) dτ 0  ∞ qτ (e−τ ξ) dτ, = p0 (ξ) + 0

where

qτ (ξ) = p(eτ ξ)A(ξ).

For each τ , qτ is smooth and supported in {ξ ∈ Rn : 1 ≤ |ξ| ≤ 2}. Furthermore, by (2.6), m p(ξ) ∈ S1,0 (Rn ) =⇒ {e−mτ qτ : τ ∈ [0, ∞)} is bounded in S(Rn ). This proves Lemma 2.3. To proceed with the proof of Proposition 2.2, in case p(x, ξ) = p(ξ), we have from Lemma 2.3 that {e−mτ qˆτ (z) : τ ∈ [0, ∞)} is bounded in S(Rn ). qτ (z)| ≤ CN z−N , so In particular, e−mτ |ˆ 

(2.7)



 −N e(n+m)τ 1 + |eτ z| dτ 0  ∞ ≤ C + CN |z|−n−m e(n+m)τ (1 + eτ )−N dτ,

|ˆ p(z)| ≤ |ˆ p0 (z)| + CN

log |z|

which implies (2.5), for β = 0, m > −n. We also see that in the case m + |β| = −n, we obtain a result upon replacing the right side of (2.5) by C log |x − y|−1 , (provided |x − y| < 1/2). We can get a complete characterization of Pˆ (x) ∈ S  (Rn ), given P (ξ) ∈ m S1 (Rn ), provided −n < m < 0. Proposition 2.4. Assume −n < m < 0. Let q ∈ S  (Rn ) be smooth outside the origin and rapidly decreasing as |x| → ∞. Then q = Pˆ for some P (ξ) ∈ S1m (Rn ) if and only if q ∈ L1loc (Rn ) and, for x = 0, (2.8)

|Dxβ q(x)| ≤ Cβ |x|−n−m−|β| .

(2.8) has been established above. For the Proof. That P ∈ S1m (Rn ) implies  converse, write q = q0 (x) + j≥0 ψj (x)q(x), where ψ0 ∈ C0∞ (Rn ) is sup ported in 1/2 < |x| < 2, ψj (x) = ψ0 (2j x), j≥0 ψj (x) = 1 on |x| ≤ 1. Since  |q(x)| ≤ C|x|−n−m , m < 0, it follows that ψj (x)q(x) converges in L1 -norm. Then q0 ∈ S(Rn ). The hypothesis (2.8) implies that 2−nj−mj ψj (2−j x)q(2−j x)

2. Schwartz kernels of pseudodifferential operators

9

n is bounded in S(R similar to that used for Proposition 2.2 ∞), and an argument implies qˆ0 (ξ) + j=0 (ψj q)ˆ(ξ) ∈ S1m (Rn ).

We will deal further with the space of elements of S  (Rn ) that are smooth outside the origin and rapidly decreasing (with all their derivatives) at infinity. We will denote this space by S0 (Rn ). If m ≤ −n, the argument above extends to show that (2.8) is a sufficient condition for q = Pˆ with P ∈ S1m (Rn ), but, as noted above, there exist symbols P ∈ S1m (Rn ) for which q = Pˆ does not satisfy (2.8). Now, given that q ∈ S0 (Rn ), it is easy to see that (2.9)

    ∇q ∈ F S1m+1 (Rn ) ⇐⇒ q ∈ F S1m (Rn ) .

Thus, if −n − 1 < m ≤ −n, then Proposition 2.4 is almost applicable to ∇q, for n ≥ 2. Proposition 2.5. Assume n ≥ 2 and −n − 1 < m ≤ −n. If q ∈ S0 (Rn ) ∩ L1loc , then q = Pˆ for some P ∈ S m (Rn ) if and only if (2.8) holds for |β| ≥ 1. Proof. First note that the hypotheses imply q ∈ L1 (Rn ); thus q˜(ξ) is continuous and vanishes as |ξ| → ∞. In the proposition, we need to prove the “if” part. To use the reasoning behind Proposition 2.4,  we need only deal with the fact1thatn ∇q is 1 . The sum ψj (x)∇q(x) still converges in L (R ), and not assumed to be in L loc  so ∇q − ψj (x)∇q is a sum of an element of S(Rn ) and possibly a distribution (call it ν) supported at 0. Thus νˆ(ξ) is a polynomial. But as noted, qˆ(ξ) is bounded, so νˆ(ξ) can have at most linear growth. Hence ξj q˜(ξ) = Pj (ξ) + j (ξ), where Pj ∈ S1m+1 (Rn ) and j (ξ) is a first-order polynomial in ξ. Since q˜(ξ) → 0 as |ξ| → ∞ and m + 1 ≤ −n + 1 < 0, we deduce that j (ξ) = cj , a constant, that is, (2.10)

ξj q˜(ξ) = Pj (ξ) + cj ,

Pj ∈ S1m+1 (Rn ), m + 1 < 0.

Now the left side vanishes on the hyperplane ξj = 0, which is unbounded if n ≥ 2. This forces cj = 0, and the proof of the proposition is then easily completed. If we take n = 1 and assume −2 < m < −1, the rest of the hypotheses of Proposition 2.5 still yield (2.10), so dq = Pˆ1 + c1 δ. dx If we also assume q is continuous on R, then c1 = 0 and we again conclude that q = Pˆ with P ∈ S1m (R). But if q has a simple jump at x = 0, then this conclusion fails.

10

7. Pseudodifferential Operators

Proposition 2.4 can be given other extensions, which we leave to the reader. We give a few examples that indicate ways in which the result does not extend, making use of results from §8 of Chap. 3. As shown in (8.31) of that chapter, on Rn , (2.11)

v = PF |x|−n =⇒ vˆ(ξ) = Cn log |ξ|.

Now v is not rapidly decreasing at infinity, but if ϕ(x) is a cut-off, belonging to C0∞ (Rn ) and equal to 1 near x = 0, then f = ϕv belongs to S0 (Rn ) and fˆ = cϕˆ ∗ vˆ behaves like log |ξ| as |ξ| → ∞. One can then deduce that, for n = 1, (2.12)

f (x) = ϕ(x) log |x| sgn |x| =⇒ fˆ(ξ) ∼ C ξ −1 log |ξ|,

|ξ| → ∞.

Thus Proposition 2.5 does not extend to the case n = 1, m = −1. However, we note that, in this case, fˆ belongs to S1−1+ε (R), for all ε > 0. In contrast to (2.12), note that, again for n = 1, (2.13)

g(x) = ϕ(x) log |x| =⇒ gˆ(ξ) ∼ C |ξ|−1 ,

|ξ| → ∞.

In this case, (d/dx) log |x| = P V (1/x).  m n  (R ) . When m = −j is a Of considerable utility is the classification of F Scl negative integer, this was effectively solved in §§8 and 9 of Chap. 3. The following result is what follows from the proof of Proposition 9.2 in Chap. 3. Proposition 2.6. Assume q ∈ S0 (Rn ) ∩ L1loc (Rn ). Let j = 1, 2, 3, . . . . Then −j (Rn ) if and only if q = Pˆ for some P ∈ Scl (2.14)

q∼

  q + p (x) log |x| , ≥0

where (2.15)

# (Rn ), q ∈ Hj+ −n

and p (x) is a polynomial homogeneous of degree j + − n; these log coefficients appear only for ≥ n − j. We recall that Hμ# (Rn ) is the space of distributions on Rn , homogeneous of degree μ, which are smooth on Rn \ 0. For μ > −n, Hμ# (Rn ) ⊂ L1loc (Rn ). The meaning of the expansion (2.14) is that, for any k ∈ Z+ , there is an N < ∞ such that the difference between q and the sum over < N belongs to C k (Rn ). Note that, for n = 1, the function g(x) in (2.13) is of the form (2.14), but the function f (x) in (2.12) is not. To go from the proof of Proposition 9.2 of Chap. 3 to the result stated above, it suffices to note explicitly that

Exercises

11

 −n−|α| n  ϕ(x)xα log |x| ∈ F S1 (R ) ,

(2.16)

where ϕ is the cut-off used before. Since F intertwines Dξα and multiplication by xα , it suffices to verify the case α = 0, and this follows from the formula (2.11), with x and ξ interchanged. m m and OP Scl , if We can also classify Schwartz kernels of operators in OP S1,0 we write the kernel K of (2.2) in the form K(x, y) = L(x, x − y),

(2.17) with

L(x, z) = (2π)−n

(2.18)



p(x, ξ)eiz·ξ dξ.

The following two results follow from the arguments given above. Proposition 2.7. Assume −n < m < 0. Let L ∈ S  (Rn × Rn ) be a smooth function of x with values in S0 (Rn ) ∩ L1 (Rn ). Then (2.17) defines the Schwartz m if and only if, for z = 0, kernel of an operator in OP S1,0 |Dxβ Dzγ L(x, z)| ≤ Cβγ |z|−n−m−|γ| .

(2.19)

Proposition 2.8. Assume L ∈ S  (Rn × Rn ) is a smooth function of x with values in S0 (Rn ) ∩ L1 (Rn ). Let j = 1, 2, 3, . . . . Then (2.17) defines the Schwartz kernel −j if and only if of an operator in OP Scl L(x, z) ∼

(2.20)



 q (x, z) + p (x, z) log |z| ,

≥0

where each Dxβ q (x, ·) is a bounded continuous function of x with values in # Hj+ −n , and p (x, z) is a polynomial homogeneous of degree j + − n in z, with coefficients that are bounded, together with all their x-derivatives.

Exercises 1. Using the proof of Proposition 2.2, show that, given p(x, ξ) defined on Rn × Rn , then |Dxβ Dξα p(x, ξ)| ≤ C  ξ −|α|+|β| , for |β| ≤ 1, |α| ≤ n + 1 + |β|, implies |K(x, y)| ≤ C|x − y|−n and |∇x,y K(x, y)| ≤ C|x − y|−n−1 . 2. If the map κ is given by (2.2) (i.e., κ(p) = K) show that we get an isomorphism κ : S  (R2n ) → S  (R2n ). Reconsider Exercise 3 of §1.

12

7. Pseudodifferential Operators

3. Show that κ, defined in Exercise 2, gives an isomorphism (isometric up to a scalar factor) κ : L2 (R2n ) → L2 (R2n ). Deduce that p(x, D) is a Hilbert-Schmidt operator on L2 (Rn ), precisely when p(x, ξ) ∈ L2 (R2n ).

3. Adjoints and products m Given p(x, ξ) ∈ Sρ,δ , we obtain readily from the definition that the adjoint is given by

(3.1)



−n

p(x, D) v = (2π)



p(y, ξ)∗ ei(x−y)·ξ v(y) dy dξ.

This is not quite in the form (1.3), as the amplitude p(y, ξ)∗ is not a function of (x, ξ). We need to transform (3.1) into such a form. Before continuing the analysis of (3.1), we are motivated to look at a general class of operators (3.2)

Au(x) = (2π)−n



a(x, y, ξ)ei(x−y)·ξ u(y) dy dξ.

We assume

(3.3)

|Dyγ Dxβ Dξα a(x, y, ξ)| ≤ Cαβγ ξm−ρ|α|+δ1 |β|+δ2 |γ|

m and then say a(x, y, ξ) ∈ Sρ,δ . A brief calculation transforms (3.2) into 1 ,δ2

(3.4)

−n



(2π)

q(x, ξ)ei(x−y)·ξ u(y) dy dξ,

with

(3.5)

q(x, ξ) = (2π)−n



a(x, y, η)ei(x−y)·(η−ξ) dy dη

= eiDξ ·Dy a(x, y, ξ)|y=x . Note that a formal expansion eiDξ ·Dy = I + iDξ · Dy − (1/2)(Dξ · Dy )2 + · · · gives (3.6)

q(x, ξ) ∼

 i|α|  Dα Dα a(x, y, ξ)y=x . α! ξ y

α≥0

3. Adjoints and products

13

m If a(x, y, ξ) ∈ Sρ,δ , with 0 ≤ δ2 < ρ ≤ 1, then the general term in (3.6) 1 ,δ2 m−(ρ−δ )|α|

2 belongs to Sρ,δ , where δ = max(δ1 , δ2 ), so the sum on the right is formally asymptotic. This suggests the following result:

m Proposition 3.1. If a(x, y, ξ) ∈ Sρ,δ , with 0 ≤ δ2 < ρ ≤ 1, then (3.2) defines 1 ,δ2 an operator m , δ = max(δ1 , δ2 ). A ∈ OP Sρ,δ

Furthermore, A = q(x, D), where q(x, ξ) has the asymptotic expansion (3.6), in the sense that q(x, ξ) −

 i|α|  m−N (ρ−δ2 ) Dξα Dyα a(x, y, ξ)y=x = rN (x, ξ) ∈ Sρ,δ . α!

|α| 0. −L2 (x, ξ) =

5. L2 -estimates Show that



19

−1 −1 iτ − L2 (x, ξ) + 1 = E(t, x, τ, ξ) ∈ S1/2,0 .

Show that E(t, x, D)P = A1 (t, x, D) and P E(t, x, D) = A2 (t, x, D), where 0 are elliptic. Then, using Proposition 4.1, construct a parametrix for Aj ∈ OP S1/2,0 −1 . P , belonging to OP S1/2,0 m 3. Assume −n < m < 0, and suppose P = p(x, D) ∈ OP Scl has Schwartz kernel n K(x, y) = L(x, x − y). Suppose that, at x0 ∈ R , L(x0 , z) ∼ a|z|−m−n + · · · ,

z → 0,

with a = 0, the remainder terms being progressively smoother. Show that pm (x0 , ξ) = b|ξ|m ,

b = 0,

and hence that P is elliptic near x0 . 4. Let P = (Pjk ) be a K × K matrix of operators in OP S ∗ . It is said to be “elliptic in the sense of Douglis and Nirenberg” if there are numbers aj , bj , 1 ≤ j ≤ K, such that the matrix of principal symbols has nonvanishing determinant Pjk ∈ OP S aj +bk and  (homogeneous of order (aj + bj )), for ξ = 0. If Λs is as in (1,17), let A be a K × K diagonal matrix with diagonal entries Λ−aj , and let B be diagonal, with entries Λ−bj . Show that this “DN-ellipticity” of P is equivalent to the ellipticity of AP B in OP S 0 .

5. L2 -estimates Here we want to obtain L2 -estimates for pseudodifferential operators. The following simple basic estimate will get us started. Proposition 5.1. Let (X, μ) be a measure space. Suppose k(x, y) is measurable on X × X and   (5.1) |k(x, y)| dμ(x) ≤ C1 , |k(x, y)| dμ(y) ≤ C2 , X

X

for all y and x, respectively. Then  (5.2)

T u(x) =

k(x, y)u(y) dμ(y)

satisfies (5.3)

1/p

T uLp ≤ C1

1/q

C2

for p ∈ [1, ∞], with (5.4)

1 1 + = 1. p q

uLp ,

20

7. Pseudodifferential Operators

This is proved in Appendix A on functional analysis; see Proposition 5.1 there. To apply this result when X = Rn and k = K is the Schwartz kernel of p(x, D) ∈ m , note from the proof of Proposition 2.1 that OP Sρ,δ (5.5)

|K(x, y)| ≤ CN |x − y|−N , for |x − y| ≥ 1

as long as ρ > 0, while (5.6)

|K(x, y)| ≤ C|x − y|−(n−1) , for |x − y| ≤ 1

as long as m < −n + ρ(n − 1). (Recall that this last estimate is actually rather crude.) Hence we have the following preliminary result. m , ρ > 0, and m < −n + ρ(n − 1), then Lemma 5.2. If p(x, D) ∈ OP Sρ,δ

(5.7)

p(x, D) : Lp (Rn ) −→ Lp (Rn ),

1 ≤ p ≤ ∞.

m , then (5.7) holds for m < 0. If p(x, D) ∈ OP S1,δ

The last observation follows from the improvement of (5.6) given in (2.5). Our main goal in this section is to prove the following. 0 Theorem 5.3. If p(x, D) ∈ OP Sρ,δ and 0 ≤ δ < ρ ≤ 1, then

(5.8)

p(x, D) : L2 (Rn ) −→ L2 (Rn ).

The proof we give, following [Ho5], begins with the following result. −a Lemma 5.4. If p(x, D) ∈ OP Sρ,δ , 0 ≤ δ < ρ ≤ 1, and a > 0, then (5.8) holds.

Proof. Since P u2L2 = (P ∗ P u, u), it suffices to prove that some power of −2ka , so for k large p(x, D)∗ p(x, D) = Q is bounded on L2 . But Qk ∈ OP Sρ,δ enough this follows from Lemma 5.2. To proceed with the proof of Theorem 5.3, set q(x, D) = p(x, D)∗ p(x, D) 0 , and suppose |q(x, ξ)| ≤ M − b, b > 0, so ∈ OP Sρ,δ (5.9)

M − Re q(x, ξ) ≥ b > 0.

  In the matrix case, take Re q(x, ξ) = (1/2) q(x, ξ) + q(x, ξ)∗ . It follows that (5.10)

 1/2 0 A(x, ξ) = M − Re q(x, ξ) ∈ Sρ,δ

and (5.11) A(x, D)∗ A(x, D) = M − q(x, D) + r(x, D),

−(ρ−δ)

r(x, D) ∈ OP Sρ,δ

.

5. L2 -estimates

21

Applying Lemma 5.4 to r(x, D), we have (5.12) M u2L2 − p(x, D)u2L2 = A(x, D)u2L2 − (r(x, D)u, u) ≥ −Cu2L2 , or p(x, D)u2 ≤ (M + C)u2L2 ,

(5.13)

finishing the proof. From these L2 -estimates easily follow L2 -Sobolev space estimates. Recall from Chap. 4 that the Sobolev space H s (Rn ) is defined as (5.14)

ˆ(ξ) ∈ L2 (Rn )}. H s (Rn ) = {u ∈ S  (Rn ) : ξs u

Equivalently, with (5.15)

s



Λ u=

ξs u ˆ(ξ)eix·ξ dξ;

Λs ∈ OP S s ,

we have (5.16)

H s (Rn ) = Λ−s L2 (Rn ).

The operator calculus easily gives the next proposition: m Proposition 5.5. If p(x, D) ∈ OP Sρ,δ , 0 ≤ δ < ρ ≤ 1, m, s ∈ R, then

(5.17)

p(x, D) : H s (Rn ) −→ H s−m (Rm ).

Given Proposition 5.5, one easily obtains the Sobolev regularity of solutions to the elliptic equations studied in §4. Calderon and Vaillancourt sharpened Theorem 5.3, showing that (5.18)

0 , 0 ≤ ρ < 1 =⇒ p(x, D) : L2 (Rn ) −→ L2 (Rn ). p(x, ξ) ∈ Sρ,ρ

This result, particularly for ρ = 1/2, has played an important role in linear PDE, especially in the study of subelliptic operators, but it will not be used in this book. The case ρ = 0 is treated in the exercises below. Another important extension of Theorem 5.3 is that p(x, D) is bounded on 0 . Similarly, Proposition 5.5 extends Lp (Rn ), for 1 < p < ∞, when p(x, ξ) ∈ S1,δ p to a result on L -Sobolev spaces, in the case ρ = 1. This is important for applications to nonlinear PDE, and will be proved in Chap. 13.

22

7. Pseudodifferential Operators

Exercises Exercises 1–7 present an approach to a proof of the Calderon-Vaillancourt theorem, (5.18), in the case ρ = 0. This approach is due to H. O. Cordes [Cor]; see also T. Kato [K] and R. Howe [How]. In these exercises, we assume that U (y) is a (measurable) unitary, operator-valued function on a measure space Y , operating on a Hilbert space H. Assume that, for f, g ∈ V, a dense subset of H,  (U (y)f, g) 2 dm(y) = C0 f 2 g2 . (5.19) Y

1. Let ϕ0 ∈ H be a unit vector, and set ϕy = U (y)ϕ0 . Show that, for any T ∈ L(H),   (5.20) C02 (T f1 , f2 ) = LT (y, y  ) (f1 , ϕy ) (ϕy , f2 ) dm(y) dm(y  ), Y

Y

where LT (y, y  ) = (T ϕy , ϕy ).

(5.21)

(Hint: Start by showing that (f1 , ϕy )(ϕy , f2 ) dm(y) = C0 (f1 , f2 ).) A statement equivalent to (5.20) is  (5.22) T = LT (y, y  ) U (y)Φ0 U (y  ) dm(y) dm(y  ),

where Φ0 is the orthogonal projection of H onto the span of ϕ0 . 2. For a partial converse, suppose L is measurable on Y × Y and   |L(y, y  )| dm(y  ) ≤ C1 . (5.23) |L(y, y  )| dm(y) ≤ C1 , Define (5.24)

 TL =

L(y, y  ) U (y)Φ0 U (y  )∗ dm(y) dm(y  ).

Show that the operator norm of TL on H has the estimate TL  ≤ C02 C1 . 3. If G is a trace class operator, and we set  L(y, y  ) U (y)GU (y  )∗ dm(y) dm(y  ), (5.25) TL,G = show that (5.26)

TL,G  ≤ C02 C1 GTR .

(Hint: In case G = G∗ , diagonalize G and use Exercise 2.) 4. Suppose b ∈ L∞ (Y ) and we set  # = b(y) U (y)GU (y)∗ dm(y). (5.27) Tb,G Show that

6. G˚arding’s inequality (5.28)

23

#  ≤ C0 bL∞ GTR . Tb,G

5. Let Y = R2n , with Lebesgue measure, y = (q, p). Set U (y) = eiq·X eip·D = π ˜ (0, q, p), as in Exercises 1 and 2 of §1. Show that the identity (5.19) holds, for f, g ∈ L2 (Rn ) = H, with C0 = (2π)−n . (Hint: Make use of the Plancherel theorem.) 6. Deduce that if a(x, D) is a trace class operator, (5.29)

(b ∗ a)(x, D)L(L2 ) ≤ CbL∞ a(x, D)TR .

(Hint: Look at Exercises 3–4 of §1.) 0 . Set 7. Suppose p(x, ξ) ∈ S0,0 (5.30)

a(x, ξ) = ψ(x)ψ(ξ),

b(x, ξ) = (1 − Δx )k (1 − Δξ )k p(x, ξ),

ˆ where k is a positive integer, ψ(ξ) = ξ −2k . Show that if k is chosen large enough, 0 . then a(x, D) is trace class. Note that, for all k ∈ Z+ , b ∈ L∞ (R2n ), provided p ∈ S0,0 Show that (5.31)

p(x, D) = (b ∗ a)(x, D),

and deduce the ρ = 0 case of the Calderon-Vaillancourt estimate (5.19). 8. Sharpen the results of problems 3–4 above, showing that (5.32)

TL,G L(H) ≤ C02 LL(L2 (Y )) GTR .

This is stronger than (5.26) in view of Proposition 5.1.

6. G˚arding’s inequality In this section we establish a fundamental estimate, first obtained by L. G˚arding in the case of differential operators. m Theorem 6.1. Assume p(x, D) ∈ OP Sρ,δ , 0 ≤ δ < ρ ≤ 1, and

(6.1)

Re p(x, ξ) ≥ C|ξ|m , for |ξ| large.

Then, for any s ∈ R, there are C0 , C1 such that, for u ∈ H m/2 (Rn ), (6.2)

  Re p(x, D)u, u ≥ C0 u2H m/2 − C1 u2H s .

Proof. Replacing p(x, D) by Λ−m/2 p(x, D)Λ−m/2 , we can suppose without loss of generality that m = 0. Then, as in the proof of Theorem 5.3, take (6.3) so

1 1/2 0 ∈ Sρ,δ , A(x, ξ) = Re p(x, ξ) − C 2

24

7. Pseudodifferential Operators

1 A(x, D)∗ A(x, D) = Re p(x, D) − C + r(x, D), 2 −(ρ−δ) r(x, D) ∈ OP Sρ,δ .

(6.4)

This gives

(6.5)

  1 Re (p(x, D)u, u) = A(x, D)u2L2 + Cu2L2 + r(x, D)u, u 2 1 2 ≥ CuL2 − C1 u2H s 2

with s = −(ρ − δ)/2, so (6.2) holds in this case. If s < −(ρ − δ)/2 = s0 , use the simple estimate (6.6)

u2H s0 ≤ εu2L2 + C(ε)u2H s

to obtain the desired result in this case. This G˚arding inequality has been improved to a sharp G˚arding inequality, of the form (6.7)

  Re p(x, D)u, u ≥ −Cu2L2

when Re p(x, ξ) ≥ 0,

1 , by H¨ormander, then for matrix-valued symbols, first for scalar p(x, ξ) ∈ S1,0   with Re p(x, ξ) standing for (1/2) p(x, ξ)+p(x, ξ)∗ , by P. Lax and L. Nirenberg. Proofs and some implications can be found in Vol. 3 of [Ho5], and in [T1] and [Tre]. A very strong improvement due to C. Fefferman and D. Phong [FP] is that 2 . See also [Ho5] and [F] for further discussion, (6.7) holds for scalar p(x, ξ) ∈ S1,0 and [T6] for an application to wave propagation.

Exercises m 1. Suppose m > 0 and p(x, D) ∈ OP S1,0 has a symbol satisfying (6.1). Examine the solvability of ∂u = p(x, D)u, ∂t s n for u = u(t, x), u(0, x) = f ∈ H (R ). (Hint: Look ahead at §7 for some useful techniques. Solve

∂uε = Jε p(x, D)Jε uε ∂t and estimate (d/dt)Λs u (t)2L2 , making use of G˚arding’s inequality.)

7. Hyperbolic evolution equations

25

7. Hyperbolic evolution equations In this section we examine first-order systems of the form ∂u = L(t, x, Dx )u + g(t, x), ∂t

(7.1)

u(0) = f.

1 , with smooth dependence on t, so We assume L(t, x, ξ) ∈ S1,0

|Dtj Dxβ Dξα L(t, x, ξ)| ≤ Cjαβ ξ1−|α| .

(7.2)

Here L(t, x, ξ) is a K × K matrix-valued function, and we make the hypothesis of symmetric hyperbolicity: 0 . L(t, x, ξ)∗ + L(t, x, ξ) ∈ S1,0

(7.3)

We suppose f ∈ H s (Rn ), s ∈ R, g ∈ C(R, H s (Rn )). Our strategy will be to obtain a solution to (7.1) as a limit of solutions uε to ∂uε = Jε LJε uε + g, ∂t

(7.4)

uε (0) = f,

where Jε = ϕ(εDx ),

(7.5)

for some ϕ(ξ) ∈ S(Rn ), ϕ(0) = 1. The family of operators Jε is called a Friedrichs mollifier. Note that, for any ε > 0, Jε ∈ OP S −∞ , while, for ε ∈ (0, 1], 0 . Jε is bounded in OP S1,0 For any ε > 0, Jε LJε is a bounded linear operator on each H s , and solvability of (7.4) is elementary. Our next task is to obtain estimates on uε , independent of ε ∈ (0, 1]. Use the norm uH s = Λs uL2 . We derive an estimate for (7.6)

d s Λ uε (t)2L2 = 2 Re (Λs Jε LJε uε , Λs uε ) + 2 Re (Λs g, Λs uε ). dt

Write the first two terms on the right as the real part of (7.7)

2(LΛs Jε uε , Λs Jε uε ) + 2([Λs , L]Jε uε , Λs Jε uε ).

0 , so the first term in (7.7) is equal to By (7.3), L + L∗ = B(t, x, D) ∈ OP S1,0

(7.8)



 B(t, x, D)Λs Jε uε , Λs Jε uε ≤ CJε uε 2H s .

s , so the second term in (7.7) is also bounded by the Meanwhile, [Λs , L] ∈ OP S1,0 right side of (7.8). Applying Cauchy’s inequality to 2(Λs g, Λs uε ), we obtain

26

7. Pseudodifferential Operators

(7.9)

d s Λ uε (t)2L2 ≤ CΛs uε (t)2L2 + Cg(t)2H s . dt

Thus Gronwall’s inequality yields an estimate (7.10)



uε (t)2H s ≤ C(t) f 2H s + g2C([0,t],H s ) ,

independent of ε ∈ (0, 1]. We are now prepared to establish the following existence result. Proposition 7.1. If (7.1) is symmetric hyperbolic and f ∈ H s (Rn ),

g ∈ C(R, H s (Rn )),

s ∈ R,

then there is a solution u to (7.1), satisfying (7.11)

s n s−1 (Rn )). u ∈ L∞ loc (R, H (R )) ∩ Lip (R, H

Proof. Take I = [−T, T ]. The bounded family uε ∈ C(I, H s ) ∩ C 1 (I, H s−1 ) will have a weak limit point u satisfying (7.11), and it is easy to verify that such u solves (7.1). As for the bound on [−T, 0], this follows from the invariance of the class of hyperbolic equations under time reversal. Analogous energy estimates can establish the uniqueness of such a solution u and rates of convergence of uε → u as ε → 0. Also, (7.11) can be improved to (7.12)

u ∈ C(R, H s (Rn )) ∩ C 1 (R, H s−1 (Rn )).

To see this, let fj ∈ H s+1 , fj → f in H s , and let uj solve (7.1) with uj (0) = s+1 ) ∩ Lip(R, H s ), so in particular each fj . Then each uj belongs to L∞ loc (R, H s uj ∈ C(R, H ). Now vj = u − uj solves (7.1) with vj (0) = f − fj , and f − fj H s → 0 as j → ∞, so estimates arising in the proof of Proposition 7.1 imply that vj (t)H s → 0 locally uniformly in t, giving u ∈ C(R, H s ). There are other notions of hyperbolicity. In particular, (7.1) is said to be sym0 that metrizable hyperbolic if there is a K × K matrix-valued S(t, x, ξ) ∈ S1,0 ˜ x, ξ) satisfies (7.3). is positive-definite and such that S(t, x, ξ)L(t, x, ξ) = L(t, Proposition 7.1 extends to the case of symmetrizable hyperbolic systems. Again, one obtains u as a limit of solutions u to (7.4). There is one extra ingredient in 0 the energy estimates. In this case, construct S(t) ∈ OP S1,0 , positive-definite, −1 with symbol equal to S(t, x, ξ) mod S1,0 . For the energy estimates, replace the left side of (7.6) by (7.13)

 d s Λ u (t), S(t)Λs u (t) L2 , dt

7. Hyperbolic evolution equations

27

which can be estimated in a fashion similar to (7.7)–(7.9). 1 is said to be strictly A K × K system of the form (7.1) with L(t, x, ξ) ∈ Scl hyperbolic if its principal symbol L1 (t, x, ξ), homogeneous of degree 1 in ξ, has K distinct, purely imaginary eigenvalues, for each x and each ξ = 0. The results above apply in this case, in view of: Proposition 7.2. Whenever (7.1) is strictly hyperbolic, it is symmetrizable. Proof. If we denote the eigenvalues of L1 (t, x, ξ) by iλν (t, x, ξ), ordered so that λ1 (t, x, ξ) < · · · < λK (t, x, ξ), then λν are well-defined C ∞ -functions of (t, x, ξ), homogeneous of degree 1 in ξ. If Pν (t, x, ξ) are the projections onto the iλν -eigenspaces of L1 , (7.14)

1 Pν (t, x, ξ) = 2πi





−1 ζ − L1 (t, x, ξ) dζ,

γν

where γν is a small circle about iλν (t, x, ξ), then Pν is smooth and homogeneous of degree 0 in ξ. Then (7.15)

S(t, x, ξ) =



Pj (t, x, ξ)∗ Pj (t, x, ξ)

j

gives the desired symmetrizer. Higher-order, strictly hyperbolic PDE can be reduced to strictly hyperbolic, first-order systems of this nature. Thus one has an analysis of solutions to such higher-order hyperbolic equations.

Exercises 1. Carry out the reduction of a strictly hyperbolic PDE of order m to a first-order system of the form (7.1). Starting with Lu =

m−1  ∂j u ∂mu + Aj (y, x, Dx ) j , m ∂y ∂y j=0

where Aj (y, x, D) has order ≤ m − j, form v = (v1 , . . . , vm )t with v1 = Λm−1 u, . . . , vj = ∂yj−1 Λm−j u, . . . , vm = ∂ym−1 u, to pass from Lu = f to

∂v = K(y, x, Dx )v + F, ∂y

with F = (0, . . . , 0, f )t . Give an appropriate definition of strict hyperbolicity in this context, and show that this first-order system is strictly hyperbolic provided L is. 2. Fix r > 0. Let γr ∈ E  (R2 ) denote the unit mass density on the circle of radius r:

28

7. Pseudodifferential Operators u, γr =

1 2π



π −π

u(r cos θ, r sin θ) dθ.

Let Γr u = γr ∗ u. Show that there exist Ar (ξ) ∈ S −1/2 (R2 ) and Br (ξ) ∈ S 1/2 (R2 ), such that √ √ sin r −Δ (7.16) Γr = Ar (D) cos r −Δ + Br (D) √ . −Δ (Hint: See Exercise 1 in §7 of Chap. 6.)

8. Egorov’s theorem We want to examine the behavior of operators obtained by conjugating a pseudodm by the solution operator to a scalar hyperbolic ifferential operator P0 ∈ OP S1,0 equation of the form (8.1)

∂u = iA(t, x, Dx )u, ∂t

where we assume A = A1 + A0 with (8.2)

1 real, A1 (t, x, ξ) ∈ Scl

0 A0 (t, x, ξ) ∈ Scl .

We suppose A1 (t, x, ξ) is homogeneous in ξ, for |ξ| ≥ 1. Denote by S(t, s) the solution operator to (8.1), taking u(s) to u(t). This is a bounded operator on each Sobolev space H σ , with inverse S(s, t). Set (8.3)

P (t) = S(t, 0)P0 S(0, t).

We aim to prove the following result of Y. Egorov. m m , then for each t, P (t) ∈ OP S1,0 , Theorem 8.1. If P0 = p0 (x, D) ∈ OP S1,0 m−1 modulo a smoothing operator. The principal symbol of P (t) (mod S1,0 ) at a point (x0 , ξ0 ) is equal to p0 (y0 , η0 ), where (y0 , η0 ) is obtained from (x0 , ξ0 ) by following the flow C(t) generated by the (time-dependent) Hamiltonian vector field n  ∂A1 ∂ ∂A1 ∂ (8.4) HA1 (t,x,ξ) = − . ∂ξj ∂xj ∂xj ∂ξj j=1

To start the proof, differentiating (8.3) with respect to t yields (8.5)

P  (t) = i[A(t, x, D), P (t)],

P (0) = P0 .

We will construct an approximate solution Q(t) to (8.5) and then show that Q(t)− P (t) is a smoothing operator.

8. Egorov’s theorem

29

m So we are looking for Q(t) = q(t, x, D) ∈ OP S1,0 , solving

(8.6)

Q (t) = i[A(t, x, D), Q(t)] + R(t),

Q(0) = P0 ,

where R(t) is a smooth family of operators in OP S −∞ . We do this by constructing the symbol q(t, x, ξ) in the form (8.7)

q(t, x, ξ) ∼ q0 (t, x, ξ) + q1 (t, x, ξ) + · · · .

Now the symbol of i[A, Q(t)] is of the form (8.8)

HA1 q + {A0 , q} + i

 i|α| A(α) q(α) − q (α) A(α) , α!

|α|≥2

where A(α) = Dξα A, A(α) = Dxα A, and so on. Since we want the difference between this and ∂q/∂t to have order −∞, this suggests defining q0 (t, x, ξ) by

∂ − HA1 q0 (t, x, ξ) = 0, q0 (0, x, ξ) = p0 (x, ξ). (8.9) ∂t Thus q0 (t, x0 , ξ0 ) = p0 (y0 , η0 ), as in the statement of the theorem; we have m . Equation (8.9) is called a transport equation. Recursively, we q0 (t, x, ξ) ∈ S1,0 obtain transport equations

∂ − HA1 qj (t, x, ξ) = bj (t, x, ξ), qj (0, x, ξ) = 0, (8.10) ∂t m−j , leading to a solution to (8.6). for j ≥ 1, with solutions in S1,0 Finally, we show that P (t) − Q(t) is a smoothing operator. Equivalently, we show that, for any f ∈ H σ (Rn ),

(8.11)

v(t) − w(t) = S(t, 0)P0 f − Q(t)S(t, 0)f ∈ H ∞ (Rn ),

where H ∞ (Rn ) = ∩s H s (Rn ). Note that (8.12)

∂v = iA(t, x, D)v, ∂t

v(0) = P0 f,

while use of (8.6) gives (8.13)

∂w = iA(t, x, D)w + g, ∂t

w(0) = P0 f,

where (8.14) Hence

g = R(t)S(t, 0)w ∈ C ∞ (R, H ∞ (Rn )).

30

7. Pseudodifferential Operators

(8.15)

∂ (v − w) = iA(t, x, D)(v − w) − g, ∂t

v(0) − w(0) = 0.

Thus energy estimates for hyperbolic equations yield v(t) − w(t) ∈ H ∞ , for any f ∈ H σ (Rn ), completing the proof. A check of the proof shows that (8.16)

m m P0 ∈ OP Scl =⇒ P (t) ∈ OP Scl .

Also, the proof readily extends to yield the following: Proposition 8.2. With A(t, x, D) as before, (8.17)

m m =⇒ P (t) ∈ OP Sρ,δ P0 ∈ OP Sρ,δ

provided (8.18)

ρ>

1 , 2

δ = 1 − ρ.

m One needs δ = 1 − ρ to ensure that p(C(t)(x, ξ)) ∈ Sρ,δ , and one needs ρ > δ to ensure that the transport equations generate qj (t, x, ξ) of progressively lower order.

Exercises 1. Let χ : Rn → Rn be a diffeomorphism that is a linear  map  outside some compact set. Define χ∗ : C ∞ (Rn ) → C ∞ (Rn ) by χ∗ f (x) = f χ(x) . Show that (8.19)

m m =⇒ (χ∗)−1 P χ∗ ∈ OP S1,0 . P ∈ OP S1,0

(Hint: Reduce to the case where χ is homotopic to a linear map through diffeomorphisms, and show that the result in that case is a special case of Theorem 8.1, where A(t, x, D) is a t-dependent family of real vector fields on Rn .) 2. Let a ∈ C0∞ (Rn ), ϕ ∈ C ∞ (Rn ) be real-valued, and ∇ϕ = 0 on supp a. If P ∈ OP S m , show that   (8.20) P a eiλϕ = b(x, λ) eiλϕ(x) , where (8.21)

± −1 + ··· , b(x, λ) ∼ λm b± 0 (x) + b1 (x)λ

λ → ±∞.

(Hint: Using a partition of unity and Exercise 1, reduce to the case ϕ(x) = x · ξ, for some ξ ∈ Rn \ 0.) 3. If a and ϕ are as in Exercise 2 above and Γr is as in Exercise 2 of §7, show that, mod O(λ−∞ ), √ √    sin r −Δ    Br (x, λ)eiλϕ , (8.22) Γr a eiλϕ = cos r −Δ Ar (x, λ)eiλϕ + √ −Δ

9. Microlocal regularity where

31

± −1 + ··· , Ar (x, λ) ∼ λ−1/2 a± 0r (x) + a1r (x)λ ± −1 + ··· , Br (x, λ) ∼ λ1/2 b± 0r (x) + b1r (x)λ

as λ → ±∞.

9. Microlocal regularity We define the notion of wave front set of a distribution u ∈ H −∞ (Rn ) = ∪s H s (Rn ), which refines the notion of singular support. If p(x, ξ) ∈ S m has principal symbol pm (x, ξ), homogeneous in ξ, then the characteristic set of P = p(x, D) is given by (9.1)

Char P = {(x, ξ) ∈ Rn × (Rn \ 0) : pm (x, ξ) = 0}.

If pm (x, ξ) is a K × K matrix, take the determinant. Equivalently, (x0 , ξ0 ) is noncharacteristic for P , or P is elliptic at (x0 , ξ0 ), if |p(x, ξ)−1 | ≤ C|ξ|−m , for (x, ξ) in a small conic neighborhood of (x0 , ξ0 ) and |ξ| large. By definition, a conic set is invariant under the dilations (x, ξ) → (x, rξ), r ∈ (0, ∞). The wave front set is defined by (9.2)

WF(u) =



{Char P : P ∈ OP S 0 , P u ∈ C ∞ }.

Clearly, WF(u) is a closed conic subset of Rn × (Rn \ 0). Proposition 9.1. If π is the projection (x, ξ) → x, then π(WF(u)) = sing supp u. / sing supp u, there is a ϕ ∈ C0∞ (Rn ), ϕ = 1 near x0 , such that Proof. If x0 ∈ ∞ n / Char ϕ for any ξ = 0, so π(W F (u)) ⊂ sing ϕu ∈ C0 (R ). Clearly, (x0 , ξ) ∈ supp u. / π(W F (u)), then for any ξ = 0 there is a Q ∈ OP S 0 such Conversely, if x0 ∈ that (x0 , ξ) ∈ / Char Q and Qu ∈ C ∞ . Thus we can construct finitely many Qj ∈ 0 C ∞ and each (x0 , ξ) (with |ξ| = 1) is noncharacteristic OP S such that Qj u ∈ Q∗j Qj ∈ OP S 0 . Then Q is elliptic near x0 and Qu ∈ for some Qj . Let Q = ∞ ∞ C , so u is C near x0 . We define the associated notion of ES(P ) for a pseudodifferential operator. Let m U be an open conic subset of Rn × (Rn \ 0). We say that p(x, ξ) ∈ Sρ,δ has order −∞ on U if for each closed conic set V of U we have estimates, for each N , (9.3)

|Dxβ Dξα p(x, ξ)| ≤ CαβN V ξ−N ,

(x, ξ) ∈ V.

32

7. Pseudodifferential Operators

m If P = p(x, D) ∈ OP Sρ,δ , we define the essential support of P (and of p(x, ξ)) to be the smallest closed conic set on the complement of which p(x, ξ) has order −∞. We denote this set by ES(P ). From the symbol calculus of §3, it follows easily that

ES(P1 P2 ) ⊂ ES(P1 ) ∩ ES(P2 )

(9.4) m

provided Pj ∈ OP Sρj j,δj and ρ1 > δ2 . To relate WF(P u) to WF(u) and ES(P ), we begin with the following. Lemma 9.2. Let u ∈ H −∞ (Rn ), and suppose that U is a conic open set satisfying WF(u) ∩ U = ∅. m If P ∈ OP Sρ,δ , ρ > 0, δ < 1, and ES(P ) ⊂ U , then P u ∈ C ∞ .

Proof. Taking P0 ∈ OP S 0 with symbol identically 1 on a conic neighborhood of ES(P ), so P = P P0 mod OP S −∞ , it suffices to conclude that P0 u ∈ C ∞ , so we can specialize the hypothesis to P ∈ OP S 0 . C ∞ and each By hypothesis, we can find Qj ∈ OP S 0 such that Qj u ∈  Q∗j Qj , then (x, ξ) ∈ ES(P ) is noncharacteristic for some Qj , and if Q = Qu ∈ C ∞ and Char Q ∩ ES(P ) = ∅. We claim there exists an operator A ∈ ˜ be an elliptic operator OP S 0 such that AQ = P mod OP S −∞ . Indeed, let Q ˜ −1 whose symbol equals that of Q on a conic neighborhood of ES(P ), and let Q ˜ Now simply set A = P Q ˜ −1 . Consequently, (mod C ∞ ) denote a parametrix for Q. P u = AQu ∈ C ∞ , so the lemma is proved. We are ready for the basic result on the preservation of wave front sets by a pseudodifferential operator. m Proposition 9.3. If u ∈ H −∞ and P ∈ OP Sρ,δ , with ρ > 0, δ < 1, then

(9.5)

WF(P u) ⊂ WF(u) ∩ ES(P ).

/ ES(P ), choose Proof. First we show WF(P u) ⊂ ES(P ). Indeed, if (x0 , ξ0 ) ∈ Q = q(x, D) ∈ OP S 0 such that q(x, ξ) = 1 on a conic neighborhood of (x0 , ξ0 ) and ES(Q) ∩ ES(P ) = ∅. Thus QP ∈ OP S −∞ , so QP u ∈ C ∞ . Hence / WF(P u). (x0 , ξ0 ) ∈ In order to show that WF(P u) ⊂ WF(u), let Γ be any conic neighborhood of m , with ES(P1 ) ⊂ Γ and ES(P2 ) ∩ WF(u), and write P = P1 + P2 , Pj ∈ OP Sρ,δ ∞ WF(u) = ∅. By Lemma 9.2, P2 u ∈ C . Thus WF(u) = WF(P1 u) ⊂ Γ, which shows WF(P u) ⊂ WF(u). One says that a pseudodifferential operator of type (ρ, δ), with ρ > 0 and δ < 1, is microlocal. As a corollary, we have the following sharper form of local regularity for elliptic operators, called microlocal regularity.

9. Microlocal regularity

33

m Corollary 9.4. If P ∈ OP Sρ,δ is elliptic, 0 ≤ δ < ρ ≤ 1, then

(9.6)

WF(P u) = WF(u).

Proof. We have seen that WF(P u) ⊂ WF(u). On the other hand, if E ∈ −m is a parametrix for P , we see that WF(u) = WF(EP u) ⊂ WF(P u). In OP Sρ,δ fact, by an argument close to the proof of Lemma 9.2, we have for general P that (9.7)

WF(u) ⊂ WF(P u) ∪ Char P.

We next discuss how the solution operator eitA to a scalar hyperbolic equation 1 , ∂u/∂t = iA(x, D)u propagates the wave front set. We assume A(x, ξ) ∈ Scl with real principal symbol. Suppose WF(u) = Σ. Then there is a countable family of operators pj (x, D) ∈ OP S 0 , each of whose complete symbols vanishes in a neighborhood of Σ, but such that  (9.8) Σ = {(x, ξ) : pj (x, ξ) = 0}. j

We know that pj (x, D)u ∈ C ∞ for each j. Using Egorov’s theorem, we want to construct a family of pseudodifferential operators qj (x, D) ∈ OP S 0 such that qj (x, D)eitA u ∈ C ∞ , this family being rich enough to describe the wave front set of eitA u. Indeed, let qj (x, D) = eitA pj (x, D)e−itA . Egorov’s theorem implies that qj (x, D) ∈ OP S 0 (modulo a smoothing operator) and gives the principal symbol of qj (x, D). Since pj (x, D)u ∈ C ∞ , we have eitA pj (x, D)u ∈ C ∞ , which in turn implies qj (x, D)eitA u ∈ C ∞ . From this it follows that WF(eitA u) is contained in the intersection of the characteristics of the qj (x, D), which is precisely C(t)Σ, the image of Σ under the canonical transformation C(t), generated by HA1 . In other words, WF(eitA u) ⊂ C(t)WF(u). However, our argument is reversible; u = e−itA (eitA u). Consequently, we have the following result: Proposition 9.5. If A = A(x, D) ∈ OP S 1 is scalar with real principal symbol, then, for u ∈ H −∞ , (9.9)

WF(eitA u) = C(t)WF(u).

The same argument works for the solution operator S(t, 0) to a timedependent, scalar, hyperbolic equation.

34

7. Pseudodifferential Operators

Exercises 1. If a ∈ C0∞ (Rn ), ϕ ∈ C ∞ (Rn ) is real-valued, ∇ϕ = 0 on supp a, as in Exercise 2 of §8, and P = p(x, D) ∈ OP S m , so   P a eiλϕ = b(x, λ)eiλϕ(x) , as in (8.20), show that, mod O(|λ|−∞ ), b(x, λ) depends only on the behavior of p(x, ξ) on an arbitrarily small conic neighborhood of

   Cϕ = x, λdϕ(x) : x ∈ supp a, λ = 0 . If Cϕ+ is the subset of Cϕ on which λ > 0, show that the asymptotic behavior of b(x, λ) as λ → +∞ depends only on the behavior of p(x, ξ) on an arbitrarily small conic neighborhood of Cϕ+ . 2. If Γr is as in (8.22), show that, given r > 0, √   (9.10) cos r −Δ (a eiλϕ ) = Γr Qr (a eiλϕ ), mod O(λ−∞ ), λ > 0, for some Qr ∈ OP S 1/2 . Consequently, analyze the behavior of the left side of (9.10), as λ → +∞, in terms of the behavior of Γr analyzed in §7 of Chap. 6.

10. Operators on manifolds Let M be a smooth manifold. It would be natural to say that a continuous linear m (M ) operator P : C0∞ (M ) → D (M ) is a pseudodifferential operator in OP Sρ,δ provided its Schwartz kernel is C ∞ off the diagonal in M × M , and there exists an open cover Ωj of M , a subordinate partition of unity ϕj , and diffeomorphisms Fj : Ωj → Oj ⊂ Rn that transform the operators ϕk P ϕj : C ∞ (Ωj ) → E  (Ωk ) m , as defined in §1. into pseudodifferential operators in OP Sρ,δ m (M ). For example, it poses no This is a rather “liberal” definition of OP Sρ,δ growth restrictions on the Schwartz kernel K ∈ D (M × M ) at infinity. Consem (M ) as defined quently, if M happens to be Rn , the class of operators in OP Sρ,δ m above is a bit larger than the class OP Sρ,δ defined in §1. One negative consequence of this definition is that pseudodifferential operators cannot always be composed. One drastic step to fix this would be to insist that the kernel be properly supported, so P : C0∞ (M ) → C0∞ (M ). If M is compact, these problems do not arise. If M is noncompact, it is often of interest to place specific restrictions on K near infinity, but we won’t go further into this point here. m (M ) given above is liberal is Another way in which the definition of OP Sρ,δ that it requires P to be locally transformed to pseudodifferential operators on Rn by some coordinate cover. One might ask if then P is necessarily so transformed m defined by every coordinate cover. This comes down to asking if the class OP Sρ,δ n n in §1 is invariant under a diffeomorphism F : R → R . It would suffice to establish this for the case where F is the identity outside a compact set. In case ρ ∈ (1/2, 1] and δ = 1 − ρ, this invariance is a special case of the Egorov theorem established in §8. Indeed, one can find a time-dependent vector field X(t) whose flow at t = 1 coincides with F and apply Theorem 8.1 to

10. Operators on manifolds

35

iA(t, x, D) = X(t). Note that the formula for the principal symbol of the conjugated operator given there implies (10.1)

p(1, F (x), ξ) = p0 (x, F  (x)t ξ),

so that the principal symbol is well defined on the cotangent bundle of M . We will therefore generally insist that ρ ∈ (1/2, 1] and δ = 1 − ρ when m (M ) for a manifold M , without a distinguished coordinate talking about OP Sρ,δ chart. In special situations, it might be natural to use coordinate charts with special structure. For instance, for a Cartesian product M = R × Ω, one can stick to product coordinate systems. In such a case, we can construct a parametrix E for the hypoelliptic operator ∂/∂t − Δx , t ∈ R, x ∈ Ω, and unambiguously regard E −1 (R × Ω). as an operator in OP S1/2,0 We make the following comments on the principal symbol of an operator P ∈ m (M ), when ρ ∈ (1/2, 1], δ = 1 − ρ. By the arguments in §8, the principal OP Sρ,δ symbol is well defined, if it is regarded as an element of the quotient space: (10.2)

m−(2ρ−1)

m p(x, ξ) ∈ Sρ,δ (T ∗ M )/Sρ,δ

(T ∗ M ).

m (M ), we have In particular, by Theorem 8.1, in case P ∈ OP S1,0

(10.3)

m−1 m (T ∗ M )/S1,0 (T ∗ M ). p(x, ξ) ∈ S1,0

m (M ), then the principal symbol can be taken to be homogeneous in ξ If P ∈ Scl of degree m, by (8.16). Note that the characterizations of the Schwartz kernels of m m and in OP Scl given in §2 also make clear the invariance of operators in OP S1,0 these classes under coordinate transformations. m (M ), We now discuss some properties of an elliptic operator A ∈ OP S1,0 when M is a compact Riemannian manifold. Denote by B a parametrix, so we have, for each s ∈ R,

(10.4)

A : H s+m (M ) −→ H s (M ),

B : H s (M ) −→ H s+m (M ),

and AB = I + K1 , BA = I + K2 , where Kj : D (M ) → C ∞ (M ). Thus Kj is compact on each Sobolev space H s (M ), so B is a two-sided Fredholm inverse of A in (10.4). In particular, A is a Fredholm operator; ker A = Ks+m ⊂  H s+m (M ) is finite-dimensional, and A H s+m (M ) ⊂ H s (M ) is closed, of finite codimension, so Cs = {v ∈ H −s (M ) : Au, v = 0 for all u ∈ H s+m (M )} is finite-dimensional. Note that Cs is the null space of (10.5)

A∗ : H −s (M ) −→ H −s−m (M ),

36

7. Pseudodifferential Operators

m which is also an elliptic operator in OP S1,0 (M ). Elliptic regularity yields, for all s,

(10.6) Ks+m = {u ∈ C ∞ (M ) : Au = 0},

Cs = {v ∈ C ∞ (M ) : A∗ v = 0}.

Thus these spaces are independent of s. Suppose now that m > 0. We will consider A as an unbounded operator on the Hilbert space L2 (M ), with domain (10.7)

D(A) = {u ∈ L2 (M ) : Au ∈ L2 (M )}.

It is easy to see that A is closed. Also, elliptic regularity implies (10.8)

D(A) = H m (M ).

Since A is closed and densely defined, its Hilbert space adjoint is defined, also as a closed, unbounded operator on L2 (M ), with a dense domain. The symbol A∗ is also our preferred notation for the Hilbert space adjoint. To avoid confusion, we will temporarily use At to denote the adjoint on D (M ), so At ∈ OP S m (M ), At : H s+m (M ) → H s (M ), for all s. Now the unbounded operator A∗ has domain (10.9)

D(A∗ ) = {u ∈ L2 (M ) : |(u, Av)| ≤ c(u)vL2 , ∀ v ∈ D(A)},

and then A∗ u is the unique element of L2 (M ) such that (10.10)

(A∗ u, v) = (u, Av), for all v ∈ D(A).

Recall that D(A) = H m (M ). Since, for any u ∈ H m (M ), v ∈ H m (M ), we have (At u, v) = (u, Av), we see that D(A∗ ) ⊃ H m (M ) and A∗ = At on H m (M ). On the other hand, (u, Av) = (At u, v) holds for all v ∈ H m (M ), u ∈ L2 (M ), the latter inner product being given by the duality of H −m (M ) and H m (M ). Thus it follows that u ∈ D(A∗ ) =⇒ A∗ u = At u ∈ L2 (M ). m (M ) then implies u ∈ H m (M ). Thus But elliptic regularity for At ∈ OP S1,0

(10.11)

D(A∗ ) = H m (M ),

 A∗ = At H m (M ) .

m In particular, if A is elliptic in OP S1,0 (M ), m > 0, and also symmetric (i.e., t A = A ), then the Hilbert space operator is self-adjoint; A = A∗ . For any λ ∈ C \ R, (λI − A)−1 : L2 (M ) → D(A) = H m (M ), so A has compact resolvent. Thus L2 (M ) has an orthonormal basis of eigenfunctions of A, Auj = λj uj , |λj | → ∞, and, by elliptic regularity, each uj belongs to C ∞ (M ).

11. The method of layer potentials

37

Exercises In the following exercises, assume that M is a smooth, compact, Riemannian manifold. Let A ∈ OP S m (M ) be elliptic, positive, and self-adjoint, with m > 0. Let uj be an orthonormal basis of L2 (M ) consisting of eigenfunctions of A, Auj = λj uj . Given f ∈ D (M ), form “Fourier coefficients” fˆ(j) = (f, uj ). Thus f ∈ L2 (M ) implies f=

(10.12)

∞ 

fˆ(j)uj ,

j=0

with convergence in L2 -norm.  ˆ 2 |f (j)| λj 2s/m < ∞. 1. Given s ∈ R, show that f ∈ H s (M ) if and only if s 2. Show that, for any s ∈ R, f ∈ H (M ), (10.12) holds, with convergence in H s -norm. Conclude that if s > n/2 and f ∈ H s (M ), the series converges uniformly to f . 3. If s > n/2 and f ∈ H s (M ), show that (10.12) converges absolutely. (Hint: Fix ˆ x 0 ∈ M and pick cj ∈ C, |cj | = 1, such that cj f (j)uj (x0 ) ≥ 0. Now consider  cj fˆ(j)uj .) 4. Let −L be a second-order, elliptic, positive, self-adjoint differential operator on a compact Riemannian manifold M . Suppose A ∈ OP S 1 (M ) is positive, √ self-adjoint, and A2 = −L + R, where R : D (M ) → C ∞ (M ). Show that A − −L : D (M ) → C ∞ (M ). One approach to Exercise 4 is the following. 5. Given f ∈ H s (M ), form u(y, x) = e−y

√ −L

for (y, x) ∈ [0, ∞) × M . Note that  ∂2  + L u = 0, ∂y 2

f (x),

v(y, x) = e−yA f (x),

 ∂2  + L v = −Rv(y, x). ∂y 2

Use estimates and regularity for the Dirichlet problem for ∂ 2 /∂y 2 + L on [0, ∞) ×M to show that u − v ∈ C ∞ ([0, ∞) × M ). Conclude that ∂u/∂y − ∂v/∂y y=0 = √ (A − −L)f ∈ C ∞ (M ). 6. With L as above, use the symbol calculus of §4 to construct a self-adjoint A ∈ OP S 1 (M ), with positive principal symbol, such that A2 + L ∈ OP S −∞ (M ). Conclude that Exercise 4 applies to A. 0 (M ) has a natural Fr´echet space structure. 7. Show that OP S1,0

11. The method of layer potentials We discuss, in the light of the theory of pseudodifferential operators, the use of “single- and double-layer potentials” to study the Dirichlet and Neumann boundary problems for the Laplace equation. Material developed here will be useful in §7 of Chap. 9, which treats the use of integral equations in scattering theory. Let Ω be a connected, compact Riemannian manifold with nonempty boundary; n = dim Ω. Suppose Ω ⊂ M , a Riemannian manifold of dimension n without

38

7. Pseudodifferential Operators

boundary, on which there is a fundamental solution E(x, y) to the Laplace equation: Δx E(x, y) = δy (x),

(11.1)

where E(x, y) is the Schwartz kernel of an operator E(x, D) ∈ OP S −2 (M ); we have (11.2)

E(x, y) ∼ cn dist(x, y)2−n + · · ·

as x → y, if n ≥ 3, while (11.3)

E(x, y) ∼ c2 log dist(x, y) + · · ·

−1 if n = 2. Here, cn = − (n − 2)Area(S n−1 ) for n ≥ 3, and c2 = 1/2π. The single- and double-layer potentials of a function f on ∂Ω are defined by  (11.4)

S f (x) =

f (y)E(x, y) dS(y), ∂Ω

and  (11.5)

D f (x) =

f (y) ∂Ω

∂E (x, y) dS(y), ∂νy

for x ∈ M \ ∂Ω. Given a function v on M \ ∂Ω, for x ∈ ∂Ω, let v+ (x) and v− (x) denote the limits of v(z) as z → x, from z ∈ Ω and z ∈ M \ Ω = O, respectively, when these limits exist. The following are fundamental properties of these layer potentials. Proposition 11.1. For x ∈ ∂Ω, we have (11.6)

S f+ (x) = S f− (x) = Sf (x)

and (11.7)

1 1 D f± (x) = ± f (x) + N f (x), 2 2

where, for x ∈ ∂Ω,  (11.8)

Sf (x) =

f (y)E(x, y) dS(y) ∂Ω

and

11. The method of layer potentials

 N f (x) = 2

(11.9)

f (y)

∂Ω

39

∂E (x, y) dS(y). ∂νy

 Note that E(x, ·)∂Ω is integrable, uniformly in x, and that the conclusion in (11.6) is elementary, at least for f continuous; the conclusion in (11.7) is a bit more mysterious. To see what is behind such results, let us look at the more general situation of v = p(x, D)(f σ),

(11.10)

where σ ∈ E  (M ) is surface measure on a hypersurface (here ∂Ω), f ∈ D (∂Ω), so f σ ∈ E  (M ). Assume that p(x, D) ∈ OP S m (M ). Make a local coordinate change, straightening out the surface to {xn = 0}. Then, in this coordinate system

(11.11)

v(x , xn ) =



  fˆ(ξ  )eix ·ξ p(x, ξ  , ξn )eixn ξn dξn dξ 

= q(xn , x , Dx )f, for xn = 0, where (11.12)

q(xn , x , ξ  ) =



p(x, ξ  , ξn )eixn ξn dξn .

If p(x, ξ) is homogeneous of degree m in ξ, for |ξ| ≥ 1, then for |ξ  | ≥ 1 we have (11.13)

q(xn , x , ξ  ) = |ξ  |m+1 p˜(x, ω  , xn |ξ  |),

where ω  = ξ  /|ξ  | and p˜(x, ω  , τ ) =



p(x, ω  , ζ)eiζτ dζ.

Now, if m< −1, the integral in (11.12) is absolutely convergent and q(xn , x , ξ  ) is continuous in all arguments, even across xn = 0. On the other hand, if m = −1, then, temporarily neglecting all the arguments of p but the last, we are looking at the Fourier transform of a smooth function of one variable whose asymptotic behavior as ξn → ±∞ is of the form C1± ξn−1 + C2± ξn−2 + · · · . From the results of Chap. 3 we know that the Fourier transform is smooth except at xn = 0, and if C1+ = C1− , then the Fourier transform has a jump across xn = 0; otherwise there may be a logarithmic singularity. It follows that if p(x, D) ∈ OP S m (M ) and m < −1, then (11.10) has a limit on ∂Ω, given by  (11.14) v ∂Ω = Qf, Q ∈ OP S m+1 (∂Ω).

40

7. Pseudodifferential Operators

On the other hand, if m = −1 and the symbol of p(x, D) has the behavior that, for x ∈ ∂Ω, νx normal to ∂Ω at x, (11.15)

p(x, ξ ± τ νx ) = ±C(x, ξ)τ −1 + O(τ −2 ),

τ → +∞,

then (11.10) has a limit from each side of ∂Ω, and (11.16)

v± = Q± f,

Q± ∈ OP S 0 (∂Ω).

To specialize these results to the setting of Proposition 11.1, note that (11.17)

S f = E(x, D)(f σ)

and (11.18)

D f = E(x, D)X ∗ (f σ),

where X is any vector field on M equal to ∂/∂ν on ∂Ω, with formal adjoint X ∗ , given by (11.19)

X ∗ v = −Xv − (div X)v.

The analysis of (11.10) applies directly to (11.17), with m = −2. That the boundary value is given by (11.8) is elementary for f ∈ C(∂Ω), as noted before. Given (11.14), it then follows for more general f . Now (11.18) is also of the form (11.10), with p(x, D) = E(x, D)X ∗ ∈ OP S −1 (M ). Note that the principal symbol at x ∈ ∂Ω is given by (11.20)

p0 (x, ξ) = −|ξ|−2 ν(x), ξ,

which satisfies the condition (11.15), so the conclusion (11.16) applies. Note that p0 (x, ξ ± τ νx ) = −|ξ ± τ νx |−2 νx , ξ ± τ νx , so in this case (11.15) holds with C(x, ξ) = 1. Thus the operators Q± in (11.16) have principal symbols ± const. That the constant is as given in (11.7) follows from keeping careful track of the constants in the calculations (11.11)–(11.13) (cf. Exercise 9 below). Let us take a closer look at the behavior of (∂/∂νy )E(x, y). Note that, for x close to y, if Vx,y denotes the unit vector at y in the direction of the geodesic from x to y, then (for n ≥ 3) (11.21)

∇y E(x, y) ∼ (2 − n)cn dist(x, y)1−n Vx,y + · · · .

If y ∈ ∂Ω and νy is the unit normal to ∂Ω at y, then

11. The method of layer potentials

(11.22)

41

∂ E(x, y) ∼ (2 − n)cn dist(x, y)1−n Vx,y , νy  + · · · . ∂νy

Note that (2−n)cn = −1/Area(S n−1 ). Clearly, the inner product Vx,y , νy  = α(x, y) restricted to (x, y) ∈ ∂Ω × ∂Ω is Lipschitz and vanishes on the diagonal x = y. This vanishing makes (∂E/∂νy )(x, y) integrable on ∂Ω × ∂Ω. It is clear that in the case (11.7), Q± have Schwartz kernels equal to (∂/∂νy )E(x, y) on the complement of the diagonal in ∂Ω × ∂Ω. In light of our analysis above of the principal symbol of Q± , the proof of (11.7) is complete. As a check on the evaluation of the constantc in D f± = ±cf + (1/2)N f , c = 1/2, note that applying Green’s formula to (Δ1) · E(x, y) dy readily gives Ω

 ∂Ω

∂E (x, y) dS(y) = 1, ∂νy

for x ∈ Ω,

0,

for x ∈ O,

as the value of D f± for f = 1. Since D f+ −D f− = 2cf , this forces c = 1/2. The way in which ±(1/2)f (x) arises in (11.7) is captured well by the model case of ∂Ω a hyperplane in Rn , and

(2−n)/2   , E (x , xn ), (y  , 0) = cn (x − y  )2 + x2n when (11.22) becomes

−n/2  ∂   E (x , xn ), (y  , 0) = (2 − n)cn xn (x − y  )2 + x2n , ∂yn though in this example N = 0. The following properties of the operators S and N are fundamental. Proposition 11.2. Given that ∂Ω is smooth, we have (11.23)

S, N ∈ OP S −1 (∂Ω),

S elliptic.

Proof. That S has this behavior follows immediately from (11.2) and (11.3). The ellipticity at x follows from taking normal coordinates at x and using Exercise 3 of §4, for n ≥ 3; for n = 2, the reader can supply an analogous argument. That N also satisfies (11.23) follows from (11.22) and the vanishing of α(x, y) = Vx,y , νy  on the diagonal. An important result complementary to Proposition 11.1 is the following, on the behavior of the normal derivative at ∂Ω of single-layer potentials. Proposition 11.3. For x ∈ ∂Ω, we have

42

7. Pseudodifferential Operators

(11.24)

 ∂ 1 S f± (x) = ∓f + N # f , ∂ν 2

where N # ∈ OP S −1 (∂Ω) is given by (11.25)

N # f (x) = 2

 f (y)

∂Ω

∂E (x, y) dS(y). ∂νx

In case E(x, y) is real valued and E(x, y) = E(y, x), (11.26)

N # = N ∗.

Proof. The proof of (11.24) is directly parallel to that of (11.7). To see on general principles why this should be so, use (11.17) to write (∂/∂ν)S f as the restriction to ∂Ω of (11.27)

XS f = XE(x, D)(f σ).

Using (11.18) and (11.19), we see that (11.28)

D f + XS f = [X, E(x, D)](f σ) − E(x, D)(div X)(f σ) = A(x, D)(f σ),

with A(x, D) ∈ OP S −2 (M ), the same class as E(x, D). Thus the extension of A(x, D)(f σ) to ∂Ω is straightforward, and we have (11.29)

 ∂ S f± = −D f± + A(x, D)(f σ)∂Ω . ∂ν

In particular, the jumps across ∂Ω are related by (11.30)

∂ ∂ S f+ − S f− = D f− − D f+ , ∂ν ∂ν

consistent with the result implied by formulas (11.7) and (11.24). It is also useful to understand the boundary behavior of (∂/∂ν)D f . This is a bit harder since ∂ 2 E/∂νx ∂νy is more highly singular. From here on, assume E(x, y) = E(y, x), so also Δy E(x, y) = δx (y). We define the Neumann operator

(11.31)

N : C ∞ (∂Ω) −→ C ∞ (∂Ω)

as follows. Given f ∈ C ∞ (∂Ω), let u ∈ C ∞ (Ω) be the unique solution to (11.32)

Δu = 0 on Ω,

u = f on ∂Ω,

11. The method of layer potentials

43

and let Nf =

(11.33)

∂u   , ∂ν ∂Ω

the limit taken from within Ω. It is a simple consequence of Green’s formula that if we form    ∂E (11.34) f (y) (x, y) − N f (y)E(x, y) dS(y) = D f (x) − S N f (x), ∂νy ∂Ω

for x ∈ M \ ∂Ω, then (11.35)

D f (x) − S N f (x) = u(x),

x ∈ Ω, x ∈ M \ Ω,

0,

where u is given by (11.32). Note that taking the limit of (11.35) from within Ω, using (11.6) and (11.7), gives f = (1/2)f + (1/2)N f − SN f , which implies the identity (11.36)

1 SN = − (I − N ). 2

Taking the limit in (11.35) from M \ Ω gives the same identity. In view of the behavior (11.23), in particular the ellipticity of S, we conclude that (11.37)

N ∈ OP S 1 (∂Ω),

elliptic.

Now we apply ∂/∂ν to the identity (11.35), evaluating on ∂Ω from both sides. Evaluating from Ω gives (11.38)

∂ ∂ D f+ − S N f+ = N f, ∂ν ∂ν

while evaluating from M \ Ω gives (11.39)

∂ ∂ D f− − S N f− = 0. ∂ν ∂ν

In particular, applying ∂/∂ν to (11.35) shows that (∂/∂ν)D f± exists, by Proposition 11.3. Furthermore, applying (11.24) to (∂/∂ν)S N f± , we have a proof of the following. Proposition 11.4. For x ∈ ∂Ω, we have (11.40)

∂ 1 D f± (x) = (I + N # )N f. ∂ν 2

44

7. Pseudodifferential Operators

In particular, there is no jump across ∂Ω of (∂/∂ν)D f . We have now developed the layer potentials far enough to apply them to the study of the Dirichlet problem. We want an approximate formula for the Poisson integral u = PI f , the unique solution to (11.41)

Δu = 0 in Ω,

 u∂Ω = f.

Motivated by the Poisson integral formula on Rn+ , we look for a solution of the form (11.42)

u(x) = D g(x),

x ∈ Ω,

and try to relate g to f . In view of Proposition 11.1, letting x → z ∈ ∂Ω in (11.42) yields (11.43)

u(z) =

1 (g + N g), for z ∈ ∂Ω. 2

Thus if we define u by (11.42), then (11.41) is equivalent to (11.44)

f=

1 (I + N )g. 2

Alternatively, we can try to solve (11.41) in terms of a single-layer potential: (11.45)

u(x) = S h(x),

x ∈ Ω.

If u is defined by (11.45), then (11.41) is equivalent to (11.46)

f = Sh.

Note that, by (11.23), the operator (1/2)(I + N ) in (11.44) is Fredholm, of index zero, on each space H s (∂Ω). It is not hard to verify that S is elliptic of order −1, with real principal symbol, so for each s, S : H s−1 (∂Ω) −→ H s (∂Ω) is Fredholm, of index zero. One basic case when (11.44) and (11.46) can both be solved is the case of bounded Ω in M = Rn , with the standard flat Laplacian. Proposition 11.5. If Ω is a smooth, bounded subdomain of Rn , with connected complement, then, for all s, (11.47) I + N : H s (∂Ω) −→ H s (∂Ω)

and

S : H s−1 (∂Ω) −→ H s (∂Ω)

11. The method of layer potentials

45

are isomorphisms. Proof. It suffices to show that I + N and S are injective on C ∞ (∂Ω). First, if g ∈ C ∞ (∂Ω) belongs to the null space of I + N , then, by (11.43) and the maximum principle, we have D g = 0 in Ω. By (11.7), the jump of D g across ∂Ω is g, so we have for v = D g|O , where O = Rn \ Ω,  (11.48) Δv = 0 on O, v ∂Ω = −g. Also, v clearly vanishes at infinity. Now, by (11.40), (∂/∂ν)D g does not jump across ∂Ω, so we have ∂v/∂ν = 0 on ∂Ω. But at a point on ∂Ω where −g is maximal, this contradicts Zaremba’s principle, unless g = 0. This proves that I + N is an isomorphism in this case. Next, suppose h ∈ C ∞ (∂Ω) belongs to the null space of S. Then, by (11.46) and the maximum principle, we have S h = 0 on Ω. By (11.24), the jump of (∂/∂ν)S h across ∂Ω is −h, so we have for w = S h|O that ∂w  (11.49) Δw = 0 on O,  = h, ∂ν ∂Ω and w vanishes at infinity. This time, S h does not jump across ∂Ω, so we also have w = 0 on ∂Ω. The maximum principle forces w = 0 on O, so h = 0. This proves that S is an isomorphism in this case. In view of (11.6), we see that (11.45) and (11.46) also give a solution to Δu = 0 on the exterior region Rn \ Ω, satisfying u = f on ∂Ω and u(x) → 0 as |x| → ∞, if n ≥ 3. This solution is unique, by the maximum principle. The topological restriction on Ω in Proposition 11.5 can be circumvented by the device of altering the operator Δ outside Ω. For such an improvement, we return to the folowing setting, (11.50)

Ω ⊂ M, ∂Ω smooth, M compact, connected Riemannian manifold,

and set (11.51)

L = Δ − V, V ∈ C ∞ (M ), V ≥ 0 on M, V = 0 on Ω, V > 0 somewhere on each component of O = M \ Ω.

Then let E = L−1 , with integral kernel E(x, y), and define S and D as before, using this integral kernel. We have the following. Proposition 11.6. in the setting of (11.50)–(11.51), we continue to have that the operators I + N and S in (11.47) are isomorphisms. Proof. As in Proposition 11.5, it suffices to show that I + N and S are injective on C ∞ (∂Ω). We tackle I + N . If g ∈ C ∞ (∂Ω) and (I + N )g = 0, as in the proof

46

7. Pseudodifferential Operators

of Proposition 11.5 we have D g = 0 on Ω. Again, by (11.7), the jump of D g across ∂Ω is g, so we have for v = D g|O ,  v ∂Ω = −g.

(Δ − V )v = 0 on O,

(11.52)

Again, (11.41) implies ∂ν D g does not jump across ∂Ω, so ∂ν v = 0 on ∂Ω. Now (11.52) plus Green’s formula gives  (11.53) O

V |v|2 dV =



 (Δv)v dV = −

O

|∇v|2 dV,

O

hence v = const = 0 on O, so g = 0. Proof that S is injective involves a modification of the proof in Proposition 11.5, and we leave this to the reader. Let us now consider the Neumann problem (11.54)

Δu = 0 on Ω,

∂u = ϕ on ∂Ω. ∂ν

We can relate (11.54) to (11.41) via the Neumann operator: (11.55)

ϕ = N f.

Let us assume that Ω is connected; then (11.56)

Ker N = {f = const. on Ω},

so dim Ker N = 1. Note that, by Green’s theorem, (11.57)

(N f, g)L2 (∂Ω) = −(du, dv)L2 (Ω) = (f, N g)L2 (∂Ω) ,

where u = P I f , v = P I g, so N is symmetric. In particular, (11.58)

(N f, f )L2 (∂Ω) = −du2L2 (Ω) ,

so N is negative-semidefinite. The symmetry of N together with its ellipticity implies that, for each s, (11.59)

N : H s+1 (∂Ω) −→ H s (∂Ω)

is Fredholm, of index zero, with both Ker N and R(N )⊥ of dimension 1, and so    (11.60) R(N ) = ϕ ∈ H s (∂Ω) : ϕ dS = 0 , ∂Ω

11. The method of layer potentials

47

this integral interpreted in the obvious distributional sense when s < 0. By (11.36), whenever S is an isomorphism in (11.47), we can say that (11.55) is equivalent to (11.61)

(I − N )f = −2Sϕ.

We can also represent a solution to (11.54) as a single-layer potential, of the form (11.45). Using (11.24), we see that this works provided h satisfies (11.62)

(I − N # )h = −2ϕ.

In view of the fact that (11.45) solves the Dirichlet problem (11.41) with f = Sh, we deduce the identity ϕ = N Sh, or (11.63)

1 N S = − (I − N # ), 2

complementing (11.36). Comparing these identities, representing SN S in two ways, we obtain the intertwining relation (11.64)

SN # = N S.

Recall that, under the reality and symmetry hypotheses on E(x, y), we have N # = N ∗ , as noted in (11.26). The method of layer potentials is applicable to other boundary problems. An application to the “Stokes system” will be given in Chap. 17, §A. Layer potentials on non-smooth domains The material above dealt with domains Ω with C ∞ boundary. There has been much work on the behavior of layer potentials on rougher domains, and implications for boundary problems in PDE, such as the Dirichlet problem and Neumann problem for the Laplace operator. There are fairly straightforward extensions in the cases where ∂Ω is a C 1+r hypersurface in Rn (or more generally M ). In such a case, operators like N and S can be analyzed in spaces of pseudodifferential operators of limited regularity, such as we will introduce in Chapter 13. Some such results are given in §3.4 of [T5]. It is shown there that if ∂Ω is of class C 1+r , r ∈ (0, 1), then ≈

(11.65)

I + N : H s (∂Ω) −→ H s (∂Ω), ≈

S : H s (∂Ω) −→ H 1+s (∂Ω),

|s| < r.

Furthermore, if ∂Ω is of class C 1,ω , with a modulus of continuity ω satisfying the Dini condition

48

7. Pseudodifferential Operators



1

(11.66) 0

ω(t)t−1 dt < ∞,

then (11.67)



I + N : L2 (∂Ω) −→ L2 (∂Ω),



S : L2 (∂Ω) −→ H 1 (∂Ω).

In fact, as seen in [T5], one has an extension from L2 (∂Ω) to Lp (∂Ω), and from H s (∂Ω) to the Lp -Sobolev space H s,p (∂Ω), for 1 < p < ∞. (In this work, Lp Sobolev spaces are introduced in Chapter 13.) For example, for C 1,ω boundaries satisfying (11.66), we have (11.68)



I + N : Lp (∂Ω) −→ Lp (∂Ω),



S : Lp (∂Ω) −→ H 1,p (∂Ω),

for 1 < p < ∞. In (11.65), (11.66), and (11.67), N is compact, as it is in (11.47), so the Fredholmness of I + N is automatic. For rougher boundaries, matters become more difficult, starting with showing that (11.9), which we rewrite as  (11.69) N f (x) = 2 PV f (y)∂νy E(x, y) dS(y), ∂Ω

is bounded on L2 (∂Ω) and other important function spaces. Breakthroughs initiated in [Ca3] and [CMM] led to such boundedness results when Ω ⊂ Rn is a Lipchitz domain, whose boundary is locally the graph of a Lipschitz function. In such a case, one has (11.70)

N : Lp (∂Ω) −→ Lp (∂Ω),

1 < p < ∞.

When ∂Ω is C 1 , N is even compact, though compactness here is much deeper than in the setting of C 1,ω boundary with (11.66) holding. This led to work on the Dirichlet problem on C 1 domains in Rn in [FJR]. For general Lipschitz domains, N is not compact, and further techniques enter the picture. Isomorphism as in (11.67) was established in [Ver], making use of Rellich identities. (See Exercises 6–7 below.) This allowed for a treatment of the Dirichlet problem (11.41) on Lipschitz domains in Rn . See [DK] for the Neumann problem and [JK] for nonhomogeneous extensions. The theory of layer potentials and applications to such boundary problems was developed in the setting of Lipschitz domains in Riemannian manifolds in [MT1] and [MT2]. To mention just one result, we have the following. Proposition 11.7. If Ω is a Lipschitz domain in a compact Riemannian manifold M , then (11.71)

PI : L2 (∂Ω) −→ H 1/2 (Ω).

11. The method of layer potentials

49

An exposition of some of this work is given in Chapter 4 of [T5]. A further breakthrough, initiated in [D] and [DS], extends the boundedness theory of layer potentials on Lp (∂Ω) from Lipschitz domains to a class of domains known as uniformly rectifiable (UR) domains, defined as follows. Suppose Ω is an open set in a compact, n-dimensional Riemannian manifold M , with nonempty boundary ∂Ω. We assume (11.72)

Hn−1 (∂Ω) < ∞,

Hn−1 (∂Ω \ ∂∗ Ω) = 0,

where Hn−1 denotes (n − 1)-dimensional Hausdorff measure and ∂∗ Ω is the measure-theoretic boundary, consisting of p ∈ ∂Ω such that the densities of both Ω and M \ Ω at p are positive. We next assume that there exist Cj ∈ (0, ∞) such that (11.73)

C1 rn−1 ≤ Hn−1 (Br (q) ∩ ∂Ω) ≤ C2 rn−1 ,

∀ q ∈ ∂Ω, r ∈ (0, 1],

where Br (q) is the ball in M , centered at q, of radius r. (When (11.73) holds, we say ∂Ω is Ahlfors regular.) Finally, we assume ∂Ω has “big pieces of Lipschitz surfaces.” In detail, there exist δ, L ∈ (0, ∞) such that for each x ∈ ∂Ω, R ∈ (0, 1], there is a Lipschitz map (11.74)

n−1 → M, ϕ : BR

n−1 BR = {x ∈ Rn : |x| < R},

such that (11.75)

∇ϕL∞ ≤ L,

n−1 Hn−1 (∂Ω ∩ BR (x) ∩ ϕ(BR )) ≥ δRn−1 .

For one class of examples, if Ω ⊂ M and if ∂Ω is locally the graph of a function (11.76)

A : Rn−1 → M,

∇A ∈ bmo(Rn−1 ),

the space bmo denoting the local John-Nirenberg space (we say Ω is a bmo1 domain), then Ω is a UR domain. Results of [D] yield the Lp -bounds (11.70) when Ω is a UR domain in Rn and Δ the flat Laplacian. Such results in the setting that Ω is a UR domain in a compact Riemannian manifold are established in [HMT], which also proves the boundary trace result (11.7) in that setting. The class of UR domains is a natural setting for another class of layer potentials, called Cauchy transforms. In [MMT] there is a study of Cauchy transforms and Calder´on projectors on UR domains, with applications to Toeplitz operators acting on functions on the boundaries of such domains. This is developed further in [MMT2], into a study of multidimensional Riemann-Hilbert problems on domains with uniformly rectifiable interfaces. An important subclass of bmo1 domains consists of vmo1 domains, whose boundaries are locally graphs of

50

7. Pseudodifferential Operators

(11.77)

A : Rn−1 → M,

∇A ∈ vmo(Rn−1 ),

where vmo(Rn−1 ) is the closure of the class of uniformly continuous functions in bmo(Rn−1 ). It was shown in [Hof] that, if Ω ⊂ Rn is a vmo1 domain and Δ the flat Laplacian, then N in (11.70) is compact, for 1 < p < ∞. The paper [HMT] extends this compactness result to a larger class of UR domains, called there regular SKT domains, named after seminal work of [Sem] and [KT]. The paper [HMT] applied layer potential methods to a number of boundary problems for the Laplace operator and other elliptic operators on regular SKT domains. Further work on this, including natural boundary problems for the Hodge Laplacian, is covered in [M3T]. See also the treatise [MMM] for an expansive treatment of analytical issues related to the theory of layer potentials.

Exercises 1. Let M be a compact, connected Riemannian manifold, with Laplace operator L, and let Ω = [0, 1] × M , with Laplace operator Δ = ∂ 2 /∂y 2 + L, y ∈ [0, 1]. Show that the Dirichlet problem u(0, x) = f0 (x), u(1, x) = f1 (x)

Δu = 0 on Ω, has the solution u(y, x) = e−y





ϕ0 + e−(1−y) −L ϕ1 + κy,

where κ is the constant κ = (vol M )−1 M (f1 − f0 ) dV , and ϕ0 = (1 − e−2

−L

√ −L −1

)

√ −2 −L −1

ϕ1 = (1 − e

)

(f0 − e−

√ −L −

(f1 − κ − e

f1 − √ −L

κ),

f0 ),

√ −2 −L −1

) being well defined on (ker L)⊥ . the operator (1 − e 2. If N f0 (x) = (∂u/∂y)(0, x), where u is as above, with f1 = 0, show that √ N f0 = − −Lf0 + Rf0 , where R is a smoothing operator, R : D (M ) → C ∞ (M ). Using (11.37), deduce that these calculations imply √ −L ∈ OP S 1 (M ). Compare Exercises 4–6 of §10. 3. If PI: C ∞ (∂Ω) → C ∞ (Ω) is the Poisson integral operator solving (11.41), show that, for x ∈ Ω,  k(x, y)f (y) dS(y),

PI f (x) = ∂Ω

with

−(n−1)/2  , |k(x, y)| ≤ C d(x, y)2 + ρ(x)2

Exercises

51

where n = dim Ω, d(x, y) is the distance from x to y, and ρ(x) is the distance from x to ∂Ω. 4. If M is an (n − 1)-dimensional surface with boundary in Ω, intersecting ∂Ω transversally, with ∂M ⊂ ∂Ω, and ρ : C ∞ (Ω) → C ∞ (M ) is restriction to M , show that ρ ◦ PI : L2 (∂Ω) −→ L2 (M ). (Hint: Look at Exercise 2 in §5 of Appendix A on functional analysis.) 5. Given y ∈ Ω, let Gy be the “Green function,” satisfying ΔGy = δy , Show that, for f ∈ C ∞ (∂Ω),

Gy = 0 on ∂Ω.



PI f (y) =

f (x) ∂ν Gy (x) dS(x). ∂Ω

(Hint: Apply Green’s formula to (PI f, ΔGy ) = (PI f, ΔGy ) − (Δ PI f, Gy ).) 6. Assume u is scalar, Δu = f , and w is a vector field on Ω. Show that    ν, w |∇u|2 dS = 2 (∇w u)(∂ν u) dS − 2 (∇w u)f dV (11.78)

∂Ω

Ω

∂Ω



2



(div w)|∇u| dV − 2

+ Ω

(Lw g)(∇u, ∇u) dV, Ω

where g is the metric tensor on Ω.This identity is a “Rellich formula.” (Hint: Compute div ∇u, ∇u w and 2 div(∇w u · ∇u), and apply the divergence theorem to the difference.) 7. In the setting of Exercise 6, assume w is a unit vector field and that ν, w ≥ a > 0 on ∂Ω. Deduce that    2 a |∇u|2 dS ≤ |∂ν u|2 dS + |f |2 dV 2 a Ω ∂Ω ∂Ω   (11.79)  |div w| + 2|Def w| + 1 |∇u|2 dV. + Ω

When Δu = f = 0, compare implications of (11.79) with implications of (11.37). See [Ver] for applications of Rellich’s formula to analysis on Lipschitz domains Ω ⊂ Rn , and [MT1] for Lipschitz domains in a Riemannian manifold. 8. What happens if, in Proposition 11.5, you allow O = Rn \ Ω to have several connected components? Can you show that one of the operators in (11.47) is still an isomorphism? 9. Calculate q(xn , x , ξ  ) in (11.13) when p(x, ξ) = ξj |ξ|−2 . Relate this to the results (11.7) and (11.24) for D f± and ∂ν S f± . (Hint. The calculation involves (1 + ζ 2 )−1 eiζτ dζ = πe−|τ | .)

52

7. Pseudodifferential Operators

12. Parametrix for regular elliptic boundary problems Here we shall complement material on regular boundary problems for elliptic operators developed in §11 of Chap. 5, including in particular results promised after the statement of Proposition 11.16 in that chapter. Suppose P is an elliptic differential operator of order m on a compact manifold M with boundary, with boundary operators Bj of order mj , 1 ≤ j ≤ , satisfying the regularity conditions given in §11 of Chap. 5. In order to construct a parametrix for the solution to P u = f , Bj u|∂M = gj , we will use pseudodifferential operator calculus to manipulate P in ways that constant-coefficient operators P (D) were manipulated in that section. To start, we choose a collar neighborhood C of ∂M , C ≈ [0, 1] × ∂M ; use coordinates (y, x), y ∈ [0, 1], x ∈ ∂M ; and without loss of generality, consider (12.1)

Pu =

m−1 ∂j u ∂mu  + Aj (y, x, Dx ) j , m ∂y ∂y j=0

the order of Aj (y, x, Dx ) being ≤ m − j. We convert P u = f to a first-order system using v = (v1 , . . . , vm )t , with (12.2)

v1 = Λm−1 u, . . . , vj = ∂yj−1 Λm−j u, . . . , vm = ∂ym−1 u,

as in (11.42) of Chap. 5. Here, Λ can be taken to be any elliptic, invertible operator in OP S 1 (∂M ), with principal symbol |ξ| (with respect to some Riemannian metric put on ∂M ). Then P u = f becomes, on C, the system (12.3)

∂v = K(y, x, Dx )v + F, ∂y

where F = (0, . . . , 0, f )t and ⎛ (12.4)

⎜ ⎜ ⎜ K=⎜ ⎜ ⎝

0

C0

Λ 0

C1

⎞ Λ .. .

..

C2

...

. Λ Cm−1

⎟ ⎟ ⎟ ⎟, ⎟ ⎠

where (12.5)

Cj (y, x, Dx ) = −Aj (y, x, Dx )Λ1−(m−j)

is a smooth family of operators in OP S 1 (∂M ), with y as a parameter. As in Lemma 11.3 of Chap. 5, we have that P is elliptic if and only if, for all (x, ξ) ∈ T ∗ ∂M \ 0, the principal symbol K1 (y, x, ξ) has no purely imaginary eigenvalues.

12. Parametrix for regular elliptic boundary problems

53

We also rewrite the boundary conditions Bj u = gj at y = 0. If Bj =

(12.6)



bjk (x, Dx )

k≤mj

∂k ∂y k

at y = 0, then we have for vj the boundary conditions (12.7)



˜bjk (x, Dx )Λk−mj vk+1 (0) = Λm−mj −1 gj = hj ,

1 ≤ j ≤ ,

k≤mj

where ˜bjk (x, D) has the same principal symbol as bjk (x, D). We write this as (12.8)

B(x, Dx ) ∈ OP S 0 (∂M ).

B(x, Dx )v(0) = h,

We will construct a parametrix for the solution of (12.3), (12.8), with F = 0. Generalizing (11.57) of Chap. 5, we construct E0 (y, x, ξ) for (x, ξ) ∈ T ∗ ∂M \0, the projection onto the sum of the generalized eigenspaces of K1 (y, x, ξ) corresponding to eigenvalues of positive real part, annihilating the other generalized eigenspaces, in the form   −1 1 (12.9) E0 (y, x, ξ) = ζ − K1 (y, x, ξ) dζ, 2πi γ

where γ = γ(y, x, ξ) is a curve in the right half-plane of C, encircling all the eigenvalues of K1 (y, x, ξ) of positive real part. Then E0 (y, x, ξ) is homogeneous of degree 0 in ξ, so it is the principal symbol of a family of operators in OP S 0 (∂M ). Recall the statement of Proposition 11.9 of Chap. 5 on the regularity condition for (P, Bj , 1 ≤ j ≤ ). One characterization is that, for (x, ξ) ∈ T ∗ ∂M \ 0, (12.10)

B0 (x, ξ) : V (x, ξ) −→ Cλ isomorphically,

where V (x, ξ) = ker E0 (0, x, ξ), and B0 (x, ξ) : Cν → Cλ is the principal symbol of B(x, Dx ). Another, equivalent characterization is that, for any η ∈ Cλ , (x, ξ) ∈ T ∗ ∂M \ 0, there exists a unique bounded solution on y ∈ [0, ∞) to the ODE (12.11)

∂ϕ − K1 (0, x, ξ)ϕ = 0, ∂y

B0 (x, ξ)ϕ(0) = η.

In that case, of course, ϕ(0) = ϕ(0, x, ξ) belongs to V (x, ξ), so ϕ(y, x, ξ) is actually exponentially decreasing as y → +∞, for fixed (x, ξ), and it is exponentially decreasing as |ξ| → ∞, for fixed y > 0, x ∈ ∂M . On a conic neighborhood Γ of any (x0 , ξ0 ) ∈ T ∗ ∂M \ 0, one can construct U0 (y, x, ξ) smooth and homogeneous of degree 0 in ξ, so that

54

7. Pseudodifferential Operators

U0 K1 U0−1 =

(12.12)

E1 0 , 0 F1

where E1 (y, x, ξ) has eigenvalues all in Re ζ < 0 and F1 has all its eigenvalues in Re ζ > 0. If we set w(0) = U0 (y, x, D)v, then the equation ∂v/∂y = K(y, x, Dx )v is transformed to (12.13)

∂w(0) = ∂y





E F

w(0) + Aw(0) = Gw(0) + Aw(0) ,

where E(y, x, Dx ) and F (y, x, Dx ) have E1 and F1 as their principal symbols, respectively, and A(y, x, Dx ) is a smooth family of operators in the space OP S 0 (∂M ). We want to decouple this equation more completely into two pieces. The next step is to decouple terms of order zero. Let w(1) = (I + V1 )w(0) , with V1 ∈ OP S −1 to be determined. We have (12.14) ∂w(1) = (I + V1 )G(I + V1 )−1 w(1) + (I + V1 )A(I + V1 )−1 w(1) + · · · ∂y = Gw(1) + (V1 G − GV1 + A)w(1) + · · · , where the remainder involves terms of order at most −1 operating on w(1) . We would like to pick V1 so that the off-diagonal terms of V1 G − GV1 + A vanish. We require V1 to be of the form V1 =

0 V12 . V21 0

If A is put into 2 × 2 block form with entries Ajk , we are led to require that (on the symbol level) (12.15)

V12 E1 − F1 V12 = −A12 , V21 F1 − E1 V21 = −A21 .

That we have unique solutions Vjk (y, x, ξ) (homogeneous of degree −1 in ξ) is a consequence of the following lemma. Lemma 12.1. Let F ∈ Mν×ν , the set of ν × ν matrices, and E ∈ Mμ×μ . Define ψ : Mν×μ → Mν×μ by ψ(T ) = T F − ET. Then ψ is bijective, provided E and F have disjoint spectra.

12. Parametrix for regular elliptic boundary problems

55

Proof. In fact, if {fj } are the eigenvalues of F and {ek } those of E, it is easily seen that the eigenvalues of ψ are {fj − ek }. Thus we obtain solutions V12 and V21 to (12.15). With such a choice of the symbol of K1 , we have (12.16)

∂w(1) = Gw(1) + ∂y





A1 A2

w(1) + Bw(1) ,

with B ∈ OP S −1 . To decouple the part of order −1, we try w(2) = (I + V2 )w(1) with V2 ∈ OP S −2 . We get ∂w(2) (12.17) = Gw(2) + ∂y





A1 A2

w(2) + (V2 G − GV2 + B)w(2) + · · · ,

so we want to choose V2 so that, on the symbol level, the off-diagonal terms of V2 G − GV2 + B vanish. This is the problem solved above, so we are in good shape. From here we continue, defining w(j) = (I + Vj )w(j−1) with Vj ∈ OP S −j , decoupling further out along the line. Letting w = (I + V )v, with (12.18)

I + V ∼ · · · (I + V3 )(I + V2 )(I + V1 ),

we have (12.19)

∂w = ∂y





E F

w,

V ∈ OP S −1 ,

mod C ∞ ,

with E  = E, F  = F mod OP S 0 . The system (12.3) is now completely decoupled. We now concentrate on constructing a parametrix for an “elliptic evolution equation” (12.20)

∂u = E(y, x, Dx )u, ∂y

u(0) = f,

where E is a k × k system of first-order pseudodifferential operators, whose principal symbol satisfies (12.21)

spec E1 (y, x, ξ) ⊂ {ζ ∈ C : Re ζ ≤ −C0 |ξ| < 0},

ξ = 0,

for some C0 > 0. We look for the parametrix in the form (in local coordinates on ∂M )  (12.22) u(y) = A(y, x, ξ)eix·ξ fˆ(ξ) dξ,

56

7. Pseudodifferential Operators

with A(y, x, ξ) in the form (12.23)

A(y, x, ξ) ∼



Aj (y, x, ξ),

j≥0

and the Aj (y, x, ξ) constructed inductively. We aim to obtain A(y, x, ξ) bounded 0 , for y ∈ [0, 1], among other things. In such a case, in S1,0 (12.24)





∂ ∂A −n − E u = (2π) − L(y, x, ξ) eix·ξ fˆ(ξ) dξ, ∂y ∂y

where (12.25)

L(y, x, ξ) ∼

 1 E (α) (y, x, ξ)A(α) (y, x, ξ). α!

α≥0

We define A0 (y, x, ξ) by the “transport equation” (12.26)

∂ A0 (y, x, ξ) = E(y, x, ξ)A0 (y, x, ξ), ∂y

A0 (0, x, ξ) = I.

If E is independent of y, the solution is A0 (y, x, ξ) = eyE(x,ξ) . In general, A0 (y, x, ξ) shares with this example the following important properties. Lemma 12.2. For y ∈ [0, 1], k, = 0, 1, 2, . . . , we have (12.27)

−k+ . y k Dy A0 (y, x, ξ) bounded in S1,0

Proof. We can take C2 ∈ (0, C0 ) and M large, so that E(y, x, ξ) has spectrum in the half-space Re ζ < −C2 |ξ|, for |ξ| ≥ M . Fixing K ∈ (0, C2 ), if S(y, σ, x, ξ) is the solution operator to ∂B/∂y = E(y, x, ξ)B, taking B(σ, x, ξ) to B(y, x, ξ), then, for y > σ, (12.28)

|S(y, σ, x, ξ)B| ≤ C e−K(y−σ)|ξ| |B|,

for |ξ| ≥ M.

It follows that, for y ∈ [0, 1], (12.29) which implies

|A0 (y, x, ξ)| ≤ C e−Ky|ξ| , |y k A0 (y, x, ξ)| ≤ Ck ξ−k e−Ky|ξ|/2 .

12. Parametrix for regular elliptic boundary problems

57

Now A0j = ∂A0 /∂ξj satisfies ∂ ∂E A0j = E(y, x, ξ)A0j + (y, x, ξ)A0 , ∂y ∂ξj

A0j (0, x, ξ) = 0,

so  (12.30)

A0j (y, x, ξ) =

y

S(y, σ, x, ξ) 0

∂E (σ, x, ξ)A0 (σ, x, ξ) dσ, ∂ξj

which in concert with (12.28) and (12.29) yields (12.31)

 ∂    A0 (y, x, ξ) ≤ Cye−Ky|ξ| ≤ Cξ−1 e−Ky|ξ|/2 .  ∂ξj

Inductively, one obtains estimates on Dξα Dxβ A0 (y, x, ξ) leading to the = 0 case of (12.27), and then use of (12.26) and induction on give (12.27) in general. For j ≥ 1, we define Aj (y, x, ξ) inductively by (12.32)

∂Aj = E(y, x, ξ)Aj (y, x, ξ) + Rj (y, x, ξ), ∂y

Aj (0, x, ξ) = 0,

where (12.33)



Rj (y, x, ξ) =

0 and all N < ∞, (12.50)

|Dy Dxβ Dξα A0 (y, x, ξ)| ≤ Cαβ e−y|ξ| ξ m+−|α| + CN αβ ξ −N .

4. If A(y, x, ξ) ∈ Pe−j , show that, for some κ > 0, you can write (12.51)

A(y, x, D) = e−κyΛ B(y, x, D),

B(y, x, ξ) ∈ P −j , y ∈ [0, 1],

modulo a smooth family of smoothing operators. 5. If u = PI f is the solution to Δu = 0, u ∂Ω = f , use Proposition 12.4 and Theorem 12.6 to show that (12.52)

PI : H s (∂Ω) −→ H s+1/2 (Ω),

1 ∀s≥− . 2

13. Parametrix for the heat equation

61

Compare the regularity result of Propositions 11.14–11.15 in Chap. 5.

13. Parametrix for the heat equation Let L = L(x, D) be a second-order, elliptic differential operator, whose principal symbol L2 (x, ξ) is a positive scalar function, though lower-order terms need not be scalar. We want to construct an approximate solution to the initial-value problem ∂u = −Lu, ∂t

(13.1)

u(0) = f,

in the form  (13.2)

u(t, x) =

a(t, x, ξ)eix·ξ fˆ(ξ) dξ,

for f supported in a coordinate patch. The amplitude a(t, x, ξ) will have an asymptotic expansion of the form a(t, x, ξ) ∼

(13.3)



aj (t, x, ξ),

j≥0

and the aj (t, x, ξ) will be defined recursively, as follows. By the Leibniz formula, write L(a eix·ξ ) = eix·ξ (13.4)

 i|α| L(α) (x, ξ)Dxα a(t, x, ξ) α!

|α|≤2

2    = eix·ξ L2 (x, ξ)a(t, x, ξ) + B2− (x, ξ, Dx )a(t, x, ξ) , =1

where B2− (x, ξ, Dx ) is a differential operator (of order ) whose coefficients are polynomials in ξ, homogeneous of degree 2 − in ξ. Thus, we want the amplitude a(t, x, ξ) in (13.2) to satisfy (formally) 2

 ∂a ∼ −L2 a − B2− (x, ξ, Dx )a. ∂t =1

If a is taken to have the form (13.3), we obtain the following equations, called “transport equations,” for aj : (13.5)

∂a0 = −L2 (x, ξ)a0 (t, x, ξ) ∂t

62

7. Pseudodifferential Operators

and, for j ≥ 1, (13.6)

∂aj = −L2 (x, ξ)aj (t, x, ξ) + Ωj (t, x, ξ), ∂t

where (13.7)

Ωj (t, x, ξ) = −

2 

B2− (x, ξ, Dx )aj− (t, x, ξ).

=1

By convention we set a−1 = 0. So that (6.15) reduces to Fourier inversion at t = 0, we set (13.8)

aj (0, x, ξ) = 0, for j ≥ 1.

a0 (0, x, ξ) = 1,

Then we have a0 (t, x, ξ) = e−tL2 (x,ξ) ,

(13.9) and the solution to (13.6) is

 (13.10)

aj (t, x, ξ) =

0

t

e(s−t)L2 (x,ξ) Ωj (s, x, ξ) ds.

In view of (13.7), this defines aj (t, x, ξ) inductively in terms of aj−1 (t, x, ξ) and aj−2 (t, x, ξ). We now make a closer analysis of these terms. Define Aj (t, x, ξ) by (13.11)

aj (t, x, ξ) = Aj (t, x, ξ)e−tL2 (x,ξ) .

The following result is useful; it applies to Aj for all j ≥ 1. Lemma 13.1. If μ = 0, 1, 2, . . . , ν ∈ {1, 2}, then A2μ+ν can be written in the form (13.12)

1/2 ξ. A2μ+ν (t, x, ξ) = tμ+1 A# 2μ+ν (x, ω, ξ), with ω = t

The factor A# 2μ+ν (x, ω, ξ) is a polynomial in both ω and ξ. It is homogeneous of degree 2 − ν in ξ (i.e., either linear or constant). Furthermore, as a polynomial = in ω, each monomial has even order; equivalently, A# 2μ+ν (x, −ω, ξ) (x, ω, ξ). A# 2μ+ν To prove the lemma, we begin by recasting (13.10). Let Γj (t, x, ξ) be defined by

13. Parametrix for the heat equation

63

Ωj (t, x, ξ) = Γj (t, x, ξ)e−tL2 (x,ξ) .

(13.13)

Then the recursion (13.7) yields (13.14)

Γj e−tL2 = −

2 

  B2− (x, ξ, Dx ) Aj− e−tL2 .

=1

Applying the Leibniz formula gives (13.15)

Γj = −

2  

[γ]

Λ (x, ω)B2− (x, ξ, Dx )Aj− (t, x, ξ),

=1 |γ|≤

evaluated at ω = t1/2 ξ, where (13.16)

etL2 (x,ξ) Dxγ e−tL2 (x,ξ) = Λγ (x, t1/2 ξ).

Clearly, Λγ (x, t1/2 ξ) is a polynomial in ξ and also a polynomial in t; hence Λγ (x, ω) is an even polynomial in ω. Note also that the differential operator [γ] B2− (x, ξ, Dx ) is of order −|γ|, and its coefficients are polynomials in ξ, homogeneous of degree 2 − , as were those of B2− (x, ξ, Dx ). The factor Aj is given by  Aj (t, x, ξ) =

(13.17)

0

t

Γj (s, x, ξ) ds.

The recursion (13.15)–(13.17) will provide an inductive proof of Lemma 13.1. To carry this out, assume the lemma true for Aj , for all j < 2μ + ν. We then have Γ2μ+ν (t, x, ξ) =

 

[γ]

μ+1 Λ (x, ω)B2− (x, ξ, Dx )A# 2μ+ν− (x, ω, ξ)t

1≤ 0,

is the integral kernel of an operator that is regularizing, and if one defines  t (13.45)

H0 #Q(t, x, y) =

0

H0 (t − s, x, z)Q(s, z, y) dV (z) ds, M

then a parametrix that is as good as (13.39) can be obtained in the form (13.46)

∼ H0 − H0 #Q + H0 #Q#Q − · · · .

This approach, one of several alternatives to that used above, is taken in [MS]. One can also look at (13.43)–(13.46) from a pseudodifferential operator perspective, as done in [Gr]. The symbol of ∂/∂t + L is iτ + L(x, ξ), and (13.47)



−1 −1 ∈ S1/2,0 (R × M ). H0 (x, τ, ξ) = iτ + L2 (x, ξ)

The operator with integral kernel H0 (t − s, x, y) given by (13.43) belongs to −1 (R × M ) and has (13.47) as its principal symbol. This operator has two OP S1/2,0 additional properties; it is causal, that is, if v vanishes for t < T , so does H0 v, for any T , and it commutes with translations. Denote by C m the class of operators m (R × M ) with these two properties. One easily has Pj ∈ C mj ⇒ in OP S1/2,0 m1 +m2 P1 P 2 ∈ C . The symbol computation gives (13.48)

∂ + L H0 = I + Q, ∂t

Q ∈ C −1 ,

and from here one obtains a parametrix (13.49)

H ∈ C −1 ,

H ∼ H0 − H0 Q + H0 Q2 − · · · .

The formulas (13.46) and (13.49) agree, via the correspondence of operators and their integral kernels.

68

7. Pseudodifferential Operators

One can proceed to construct a parametrix for the heat equation on a manifold with boundary. We sketch an approach, using a variant of the double-layerpotential method described for elliptic boundary problems in §11. Let Ω be an open domain, with smooth boundary, in M , a compact Riemannian manifold without boundary. We construct an approximate solution to ∂u = −Lu, ∂t

(13.50) for (t, x) ∈ R+ × Ω, satisfying

u(t, x) = h(t, x), for x ∈ ∂Ω,

u(0, x) = 0,

(13.51) in the form

 (13.52)

u = D g(t, x) =



 g(s, y)

0

∂Ω

∂H (t − s, x, y) dS(y) ds, ∂νy

where H(t, x, y) is the heat kernel on R+ × M studied above. For x ∈ ∂Ω, denote by D g+ (t, x) the limit of D g from within R+ × Ω. As in (11.7), one can establish the identity (13.53)

D g+ =

1 (I + N )g, 2

where (1/2)N g is given by the double integral on the right side of (13.52), with y and x both in ∂Ω. In analogy with (11.23), we have −1/2

N ∈ OP S1/2,0 (R+ × ∂Ω). For u to solve (13.50)–(13.51), we need (13.54)

h=

1 (I + N )g. 2

Thus we have a parametrix for (13.50)–(13.51) in the form (13.52) with (13.55)

g ∼ 2(I − N + N 2 − · · · )h.

We can use the analysis of (13.50)–(13.55) to construct a parametrix for the solution operator to (13.56)

∂u = Δu, for x ∈ Ω, ∂t

u(0, x) = f (x),

u(t, x) = 0, for x ∈ ∂Ω.

13. Parametrix for the heat equation

69

To begin, let v solve (13.57)

∂v = Δv on R+ × M, ∂t

v(0) = f,

where f(x) = f (x),

for x ∈ Ω,

(13.58)

for x ∈ M \ Ω.

0,

One way to obtain u would be to subtract a solution to (13.50)–(13.51), with  −L = Δ, h = v R+ ×∂Ω . This leads to a parametrix for the solution operator for (13.56) of the form  p(t, x, y) = H(t, x, y) − (13.59)



 h(s, z, y)

0

∂Ω

∂H (t − s, x, z) dS(z) ds, ∂νz

h(s, z, y) ∼ 2H(s, z, y) + · · · , where, as above, H(t, x, y) is the heat kernel on R+ × M . We mention an alternative treatment of (13.56) that has some advantages. We will apply a reflection to v. To do this, assume that Ω is contained in a compact Riemannian manifold M , diffeomorphic to the double of Ω, and let R : M → M be a smooth involution of M , fixing ∂Ω, which near ∂Ω is a reflection of each geodesic normal to ∂Ω, about the point where the geodesic intersects ∂Ω. Pulling back the metric tensor on M by R yields a metric tensor that agrees with the original on ∂Ω. Now set (13.60)

  u1 (t, x) = v(t, x) − v t, R(x) ,

x ∈ Ω.

We see that u1 satisfies (13.61)

∂u1 = Δu1 + g, ∂t

u1 (0, x) = f,

u1 (t, x) = 0, for x ∈ ∂Ω,

where (13.62)

 g = Lb vR+ ×Ω ,

  v(t, x) = v t, R(x) ,

and where Lb is a second-order differential operator, with smooth coefficients, whose principal symbol vanishes on ∂Ω. Thus the difference u − u1 = w solves (13.63)

∂w = Δw − g, ∂t

w(0) = 0,

w(t, x) = 0, for x ∈ ∂Ω.

70

7. Pseudodifferential Operators

Next let v2 solve (13.64)

∂v2 = Δv2 − g on R+ × M, ∂t

v2 (0) = 0,

where (13.65)

g(t, x) = g(t, x), 0,

for x ∈ Ω, for x ∈ M \ Ω,

and set (13.66)

 u2 = v2 R+ ×Ω .

It follows that w2 = u − (u1 + u2 ) satisfies (13.67)

∂w2 = Δw2 on R+ × Ω, ∂t

w2 (0) = 0,

  w2 R+ ×∂Ω = −v2 R+ ×∂Ω .

Now we can obtain w2 by the construction (13.52)–(13.55), with  h = −v2 R+ ×∂Ω . To illustrate the effect of this construction using reflection, suppose that, in (13.56), (13.68)

f ∈ H01 (Ω).

  Then, in (13.57)–(13.58), f ∈ H 1 (M ), so v ∈ C R+ , H 1 (M ) , and hence (13.69)

  u1 ∈ C R+ , H01 (Ω) .

Furthermore, given the nature of Lb and that of the heat kernel on R+ × M × M , one can show that, in (13.62), (13.70)

  g ∈ C R+ , L2 (Ω) ,

operator on v, when one restricts to that is, Lb effectively acts   like a first-order Ω. It follows that g ∈ C R+ , L2 (M ) and hence,  via Duhamel’s formula for the solution to (13.64), that v2 ∈ C R+ , H 2− (M ) , ∀  > 0. Therefore, (13.71)

 u2 ∈ C R+ , H 2− (Ω)),

and,  in (13.67), we have a PDE of the form (13.50)–(13.51), with h ∈ C R+ , H 3/2− (∂Ω) , for all  > 0. One can deduce from (13.52)–(13.55) that w2 has as much regularity as that given for u2 in (13.71). It also follows directly from Duhamel’s principle, applied to (13.63), that

13. Parametrix for the heat equation

(13.72)

71

  w ∈ C R+ , H 2− (Ω) ,

so we can see without analyzing (13.52)–(13.55) that w2 has as much regularity as mentioned above. Either way, we see that when f satisfies (13.68), the principal singularities of the solution u to (13.56) are captured by u1 , defined by (13.60). Constructions of u2 and, via (13.52)–(13.55), of w2 yield smoother corrections, at least when smoothness is measured in the spaces used above. The construction (13.56)–(13.67) can be compared with constructions in §7 of Chap. 13.

Exercises 1. Let L be a positive, self-adjoint, elliptic differential operator of order 2k > 0 on a compact manifold M , with scalar principal symbol L2k (x, ξ). Show that a parametrix for ∂u/∂t = −Lu can be constructed in the form (13.2)–(13.3), with aj (t, x, ξ) of the following form, generalizing (13.11)–(13.12): aj (t, x, ξ) = Aj (t, x, ξ)e−tL2k (x,ξ) , where A0 (t, x, ξ) = 1 and if μ = 0, 1, 2, . . . and ν ∈ {1, . . . , 2k}, then A2kμ+ν (t, x, ξ) = tμ+1 A# 2kμ+ν (x, ω, ξ),

ω = t1/2k ξ,

where A# 2kμ+ν (x, ω, ξ) is a polynomial in ξ, homogeneous of degree 2k − ν, whose coefficients are polynomials in ω, each monomial of which has degree (in ω) that is an πi/k ω, ξ) = A# integral multiple of 2k, so A# 2kμ+ν (x, e 2kμ+ν (x, ω, ξ). 2. In the setting of Exercise 1, show that   Tr e−tL ∼ t−n/2k a0 + a1 t1/k + a2 t2/k + · · · , generalizing (13.41). 3. Let gjk (y, x) denote the components of the metric tensor at x in a normal coordinate system centered at y. Suppose −Lu(x) = Δu(x) = g jk (y, x) ∂j ∂k u(x) + bj (y, x) ∂j u(x) in this coordinate system. With H0 (t, x, y) given by (13.43), show that   ∂ + Lx H0 (t, x, y) ∂t  = H0 (t, x, y) (2t)−2 g jk (x, x) − g jk (y, x) (xj − yj )(xk − yk )  − (2t)−1 g j j (x, x) − g j j (y, x) − bj (y, x)(xj − yj )      |x − y|4 |x − y|2 = H0 (t, x, y) O + O . t2 t Compare formula (2.10) in Chap. 5. Note that gjk (y, y) = δjk , ∂ gjk (y, y) = 0, and bj (y, y) = 0. Relate this calculation to the discussion involving (13.43)–(13.49). 4. Using the parametrix, especially (13.39), show that if M is a smooth, compact Riemannian manifold, without boundary, then

72

7. Pseudodifferential Operators etΔ : C k (M ) −→ C k (M ) is a strongly continuous semigroup, for each k ∈ Z+ .

14. The Weyl calculus To define the Weyl calculus, we begin with a modification of the formula (1.10) for a(x, D). Namely, we replace eiq·X eip·D by ei(q·X+p·D) , and set  (14.1) a(X, D)u = a ˆ(q, p)ei(q·X+p·D) u dq dp, initially for a(x, ξ) ∈ S(R2n ). Note that v(t, x) = eit(q·X+p·D) u(x) solves the PDE (14.2)

∂v  ∂v = pj + i(q · x)v, ∂t ∂xj j

v(0, x) = u(x),

and the  solution is readily obtained by integrating along the integral curves of ∂/∂t − pj ∂/∂xj , which are straight lines. We get (14.3)

ei(q·X+p·D) u(x) = eiq·x+iq·p/2 u(x + p).

Note that this is equivalent to the identity (14.4)

ei(q·X+p·D) = eiq·p/2 eiq·X eip·D .

If we plug (14.3) into (14.1), a few manipulations using the Fourier inversion formula yield (14.5)

a(X, D)u(x) = (2π)−n



x + y

, ξ ei(x−y)·ξ u(y) dy dξ, a 2

which can be compared with the formula  (1.3) for a(x,  D). Note that a(X, D) is of the form (3.2) with a(x, y, ξ) = a (x + y)/2, ξ , while a(x, D) is of the form (3.2) with a(x, y, ξ) = a(x, ξ). In particular, Proposition 3.1 is applicable; we have (14.6)

a(X, D) = b(x, D),

where (14.7)

x + y   ,ξ  = e(i/2)Dξ ·Dx a(x, ξ). b(x, ξ) = eiDξ ·Dy a 2 y=x

m m , with 0 ≤ δ < ρ ≤ 1, then b(x, ξ) also belongs to Sρ,δ and, If a(x, ξ) ∈ Sρ,δ by (3.6),

14. The Weyl calculus

(14.8)

b(x, ξ) ∼

73

 i|α| 2−|α| Dξα Dxα a(x, ξ). α!

α≥0

Of course this relation is invertible; we have a(x, ξ) = e−(i/2)Dξ ·Dx b(x, ξ) and a corresponding asymptotic expansion. Thus, at least on a basic level, the two methods of assigning an operator, either a(x, D) or a(X, D), to a symbol a(x, ξ) lead to equivalent operator calculi. However, they are not identical, and the differences sometimes lead to subtle advantages for the Weyl calculus. One difference is that since the adjoint of ei(q·X+p·D) is e−i(q·X+p·D) , we have the formula (14.9)

a(X, D)∗ = b(X, D),

b(x, ξ) = a(x, ξ)∗ ,

which is somewhat simpler than the formula (3.13)–(3.14) for a(x, D)∗ . Other differences can be traced to the fact that the Weyl calculus exhibits certain symmetries rather clearly. To explain this, we recall, from the exercises after §1, that the set of operators (14.10)

˜ (t, q, p) eit eiq·X eip·D = π

form a unitary group of operators on L2 (Rn ), a representation of the group Hn , with group law (14.11)

(t, q, p) ◦ (t , q  , p ) = (t + t + p · q  , q + q  , p + p ).

Now, using (14.4), one easily computes that (14.12)







ei(t+q·X+p·D) ei(t +q ·X+p ·D) = ei(s+u·X+v·D) ,

with u = q + q  , v = p + p , and (14.13)

 1 1  s = t + t + (p · q  − q · p ) = t + t + σ (p, q), (p , q  ) , 2 2

where σ is the natural symplectic form on Rn × Rn . Thus (14.14)

π(t, q, p) = ei(t+q·X+p·D)

defines a unitary representation of a group we’ll denote Hn , which is R × R2n with group law (14.15)



1 (t, w) · (t , w ) = t + t + σ(w, w ), w + w , 2

74

7. Pseudodifferential Operators

where we have set w = (q, p). Of course, the groups Hn and Hn are isomorphic; both are called the Heisenberg group. The advantage of using the group law (14.15) rather than (14.11) is that it makes transparent the existence of the action of the group Sp(n, R) of linear symplectic maps on R2n , as a group of automorphisms of Hn . Namely, if g : R2n → R2n is a linear map preserving the symplectic form, so σ(gw, gv) = σ(w, v) for v, w ∈ R2n , then (14.16)

α(g) : Hn → Hn ,

α(g)(t, w) = (t, gw)

defines an automorphism of Hn , so (14.17)

(t, w) · (t , w ) = (s, v) ⇒ (t, gw) · (t , gw ) = (s, gv)

and α(gg  ) = α(g)α(g  ). The associated action of Sp(n, R) on Hn has a formula that is less clean. This leads to an action of Sp(n, R) on operators in the Weyl calculus. Setting   (14.18) ag (x, ξ) = a g −1 (x, ξ) , we have (14.19)

a(X, D)b(X, D) = c(X, D) ⇒ ag (X, D)bg (X, D) = cg (X, D),

for g ∈ Sp(n, R). In fact, let us rewrite (14.1) as  a(X, D) =

a ˆ(w)π(0, w) dw.

Then

(14.20)

a(X, D)b(X, D)  = a ˆ(w)ˆb(w )π(0, w)π(0, w ) dw dw   = a ˆ(w)ˆb(w )eσ(w,w )/2 π(0, w + w ) dw dw ,

so c(X, D) in (14.19) has symbol satisfying (14.21)

cˆ(w) = (2π)−n



 a ˆ(w − w )ˆb(w )eiσ(w,w )/2 dw .

The implication in (14.19) follows immediately from this formula. Let us write c(x, ξ) = (a ◦ b)(x, ξ) when this relation holds. From (14.21), one easily obtains the product formula

14. The Weyl calculus

(14.22)

  (a ◦ b)(x, ξ) = e(i/2)(Dy ·Dξ −Dx ·Dη ) a(x, ξ)b(y, η)

y=x,η=ξ

75

.

μ m If a ∈ Sρ,δ , b ∈ Sρ,δ , 0 ≤ δ < ρ ≤ 1, we have the following asymptotic expansion:

(14.23)

(a ◦ b)(x, ξ) ∼ ab +

 1 {a, b}j (x, ξ), j! j≥1

where (14.24)

 i j  j  ∂y · ∂ξ − ∂x · ∂η a(x, ξ)b(y, η) . {a, b}j (x, ξ) = − 2 y=x,η=ξ

For comparison, recall the formula for a(x, D)b(x, D) = (a#b)(x, D)

(14.25) given by (3.16)–(3.20):

  (a#b)(x, ξ) = eiDη ·Dy a(x, η)b(y, ξ) (14.26)

y=x,η=ξ

 (−i)|α| ∂ξα a(x, ξ)∂xα b(x, ξ). ∼ ab + α! α>0

In the respective cases, (a ◦ b)(x, ξ) differs from the sum over j < N by an m+μ−N (ρ−δ) element of Sρ,δ and (a#b)(x, ξ) differs from the sum over |α| < N by an element of the same symbol class. In particular, for ρ = 1, δ = 0, we have (14.27)

i m+μ−2 (a ◦ b)(x, ξ) = a(x, ξ)b(x, ξ) + {a, b}(x, ξ) mod S1,0 , 2

where {a, b} is the Poisson bracket, while (14.28)

(a#b)(x, ξ) = a(x, ξ)b(x, ξ) − i

 ∂a ∂b m+μ−2 mod S1,0 . ∂ξj ∂xj

Consequently, in the scalar case, (14.29)

[a(X, D), b(X, D)] = [a(x, D), b(x, D)] m+μ−2 = e(x, D) = e(X, D) mod OP S1,0 ,

with (14.30)

e(x, ξ) = i{a, b}(x, ξ).

76

7. Pseudodifferential Operators

Now we point out one of the most useful aspects of the difference between (14.27) and (14.28). Namely, one starts with an operator A = a(X, D) = a1 (x, D), maybe a differential operator, and perhaps one wants to construct a parametrix for A, or perhaps a “heat semigroup” e−tA , under appropriate hypotheses. In such a case, the leading term in the symbol of the operator b(X, D) = b1 (x, D) used in (14.20) or (14.25) is a function of a(x, ξ), for example, a(x, ξ)−1 , or e−ta(x,ξ) . But then, at least when a(x, ξ) is scalar, the last term in (14.27) vanishes! On the other hand, the last term in (14.28) generally does not vanish. From this it follows that, with a given amount of work, one can often construct a more accurate approximation to a parametrix using the Weyl calculus, instead of using the constructions of the previous sections. In the remainder of this section, we illustrate this point by reconsidering the parametrix construction for the heat equation, made in §13. Thus, we look again at ∂u = −Lu, ∂t

(14.31)

u(0) = f.

This time, set Lu = a(X, D)u + b(x)u,

(14.32) where

a(x, ξ) =



(14.33)

g jk (x)ξj ξk +



j (x)ξj

= g(x, ξ) + (x, ξ). We assume g(x, ξ) is scalar, while (x, ξ) and b(x) can be K × K matrix-valued. As the notation indicates, we assume (g jk ) is positive-definite, defining an inner product on cotangent vectors, corresponding to a Riemannian metric (gjk ). We note that a symbol that is a polynomial in ξ also defines a differential operator in the Weyl calculus. For example, (x, D)u = (14.34) (X, D)u =

 

j (x) ∂j u =⇒ j (x) ∂j u +

1 (∂j j )u 2

and

(14.35)



ajk (x)∂j ∂k u =⇒    1 a(X, D)u = ajk (x)∂j ∂k u + (∂j ajk )∂k u + (∂j ∂k ajk )u 4   1 ∂j (ajk ∂k u) + (∂j ∂k ajk )u . = 4 a(x, D) =

14. The Weyl calculus

77

We use the Weyl calculus to construct a parametrix for (14.31). We will begin by treating the case when all the terms in (14.33) are scalar, and then we will discuss the case when only g(x, ξ) is assumed to be scalar. We want to write an approximate solution to (14.31) as u = E(t, X, D)f.

(14.36) We write (14.37)

E(t, x, ξ) ∼ E0 (t, x, ξ) + E1 (t, x, ξ) + · · ·

and obtain the various terms recursively. The PDE (14.31) requires (14.38)

∂ E(t, X, D) = −LE(t, X, D) = −(L ◦ E)(t, X, D), ∂t

where, by the Weyl calculus, (14.39)

(L ◦ E)(t, x, ξ) ∼ L(x, ξ)E(t, x, ξ) +

 1 {L, E}j (t, x, ξ). j! j≥1

It is natural to set (14.40)

E0 (t, x, ξ) = e−ta(x,ξ) ,

as in (13.9). Note that the Weyl calculus applied to this term provides a better approximation than the previous calculus, because {a, e−ta }1 = 0.

(14.41)

If we plug (14.37) into (14.39) and collect the highest order nonvanishing terms, we are led to define E1 (t, x, ξ) as the solution to the “transport equation” (14.42)

∂E1 1 = −aE1 − {a, E0 }2 − b(x)E0 , ∂t 2

E1 (0, x, ξ) = 0.

Let us set (14.43)

1 Ω1 (t, x, ξ) = − {a, e−ta }2 − b(x)e−ta(x,ξ) . 2

Then the solution to (14.42) is  (14.44)

E1 (t, x, ξ) =

0

t

e(s−t)a(x,ξ) Ω1 (s, x, ξ) ds.

78

7. Pseudodifferential Operators

Higher terms Ej (t, x, ξ) are then obtained in a straightforward fashion. This construction is similar to (13.6)–(13.10), but there is the following important difference. Once you have E0 (t, x, ξ) and E1 (t, x, ξ) here, you have the first two terms in the expansion of the integral kernel of e−tL on the diagonal: K(t, x, x) ∼ c0 (x)t−n/2 + c1 (x)t−n/2+1 + · · · .

(14.45)

To get so far using the method of §13, it is necessary to go further and compute the solution a2 (t, x, ξ) to the next transport equation. Since the formulas become rapidly more complicated, the advantage is with the method of this section. We proceed with an explicit determination of the first two terms in (14.45). Thus we now evaluate the integral in (14.44). Clearly, 

t

(14.46)

e(s−t)a(x,ξ) b(x)e−sa(x,ξ) ds = tb(x)e−ta(x,ξ) .

0

Now, a straightforward calculation yields {a, e−sa }2 =

(14.47)

s s2 Q(∇2 a)e−sa − T (∇a, ∇2 a)e−sa , 2 4

where Q(∇2 a) =

(14.48)

  (∂ξk ∂ξ a)(∂xk ∂x a) − (∂ξk ∂x a)(∂xk ∂ξ a) k,

and T (∇a, ∇2 a)  (∂ξk ∂ξ a)(∂xk a)(∂x a) =

(14.49)

k,

 + (∂xk ∂x a)(∂ξk a)(∂ξ a) − 2(∂ξk ∂x a)(∂xk a)(∂ξ a) . Therefore,  (14.50) 0

t

e(s−t)a {a, e−sa }2 ds =

t2 t3 Q(∇2 a)e−ta − T (∇a, ∇2 a)e−ta . 4 12

We get E1 (t, x, ξ) in (14.44) from (14.46) and (14.50). Suppose for the moment that (x, ξ) = 0 in (14.33), that is, a(X, D) = g(X, D). Suppose also that, for some point x0 , (14.51)

∇x g jk (x0 ) = 0,

g jk (x0 ) = δjk .

14. The Weyl calculus

79

Then, at x0 , Q(∇2 a) =

   ∂ξk ∂ξ a ∂xk ∂x a k,

(14.52)

 ∂ 2 g jk (x0 )ξj ξk ∂x2

=2

j,k,

and T (∇a, ∇2 a) =

    ∂xk ∂x a ∂ξk a ∂ξ a k,

(14.53) =4

 j,k, ,m

∂ 2 g jk (x0 )ξj ξk ξ ξm . ∂x ∂xm

Such a situation as (14.51) arises if g jk (x) comes from a metric tensor gjk (x), and one uses geodesic normal coordinates centered at x0 . Now the LaplaceBeltrami operator is given by Δu = g −1/2

(14.54)



∂j g jk g 1/2 ∂k u,

where g = det(gjk ). This is symmetric when one uses the Riemannian volume √ element dV = g dx1 · · · dxn . To use the Weyl calculus, we want an operator that is symmetric with respect to the Euclidean volume element dx1 · · · dxn , so we conjugate Δ by multiplication by g 1/4 : (14.55)

     −Lu = g 1/4 Δ g −1/4 u = g −1/4 ∂j g jk g 1/2 ∂k g −1/4 u .

t t Note that the integral kernel kL (x, y) of etL is g 1/4 (x)kΔ (x, y)g −1/4 (y); in particular, of course, the two kernels coincide on the diagonal x = y. To compare L with g(X, D), note that

−Lu =

(14.56)



∂j g jk ∂k u + Φ(x)u,

where (14.57) Φ(x) =



   jk 1/2  −1/4   g g ∂j g ∂k g −1/4 . ∂j g jk g 1/2 ∂k g −1/4 −

If g jk (x) satisfies (14.51), we see that (14.58)

Φ(x0 ) =

 j

∂j2 g −1/4 (x0 ) = −

1 2 ∂ g(x0 ). 4

80

7. Pseudodifferential Operators

  Since g(x0 + he ) = det δjk + (1/2)h2 ∂ 2 gjk + O(h3 ), we have (14.59)

Φ(x0 ) = −

1 2 ∂ gjj (x0 ). 4 j,

By comparison, note that, by (14.35),

(14.60)



∂j g jk ∂k u + Ψ(x)u, 1 ∂j ∂k g jk (x). Ψ(x) = − 4

g(X, D)u = −

If x0 is the center of a normal coordinate system, we can express these results in terms of curvature, using (14.61)

∂ ∂m g jk (x0 ) =

1 1 Rj km (x0 ) + Rjmk (x0 ), 3 3

in terms of the components of the Riemann curvature tensor, which follows from formula (3.51) of Appendix C. Thus we get 1 2 1 Φ(x0 ) = − · Rj j (x0 ) = − S(x0 ), 4 3 6 (14.62)

j,

 1 1  1 S(x0 ). Rjjkk (x0 ) + Rjkkj (x0 ) = Ψ(x0 ) = − · 4 3 12 j,k

Here S is the scalar curvature of the metric gjk . When a(X, D) = g(X, D), we can express the quantities (14.52) and (14.53) in terms of curvature: 2 4 Rj k (x0 )ξj ξk = Ricjk (x0 )ξj ξk , (14.63) Q(∇2 g) = 2 · 3 3 j,k,

j,k

where Ricjk denotes the components of the Ricci tensor, and (14.64)

T (∇g, ∇2 g) = 4 ·

2  Rj km (x0 )ξj ξk ξ ξm = 0, 3 j,k, ,m

the cancelation here resulting from the antisymmetry of Rj km in (j, ) and in (k, m). Thus the heat kernel for (14.31) with (14.65)

Lu = g(X, D)u + b(x)u

is of the form (14.36)–(14.37), with E0 = e−tg(x,ξ) and

14. The Weyl calculus

(14.66)

81



t2 t3 E1 (t, x, ξ) = −tb(x) − Q(∇2 g) + T (∇g, ∇2 g) e−tg 8 24

2 t = − tb(x) + Ric(ξ, ξ) e−tg(x,ξ) , 6

at x = x0 . Note that g(x0 , ξ) = |ξ|2 . Now the integral kernel of Ej (t, X, D) is (14.67)

Kj (t, x, y) = (2π)−n



x+y

, ξ ei(x−y)·ξ dξ. Ej t, 2

In particular, on the diagonal we have 

−n

Kj (t, x, x) = (2π)

(14.68)

Ej (t, x, ξ) dξ.

We want to compute these quantities, for j = 0, 1, and at x = x0 . First, −n



K0 (t, x0 , x0 ) = (2π)

(14.69)

2

e−t|ξ| dξ = (4πt)−n/2 ,

since, as we know, the Gaussian integral in (14.69) is equal to (π/t)n/2 . Next,

(14.70)

(2π)n K1 (t, x0 , x0 )   2 2 t2  Ricjk (x0 ) ξj ξk e−t|ξ| dξ. = −tb(x0 ) e−t|ξ| dξ − 6

We need to compute more Gaussian integrals. If j = k, the integrand is an odd function of ξj , so the integral vanishes. On the other hand, 

2

ξj2 e−t|ξ| dξ =

1 n



(14.71) =−

2

|ξ|2 e−t|ξ| dξ

1 d n dt



2

e−t|ξ| dξ =

1 n/2 −n/2−1 π t . 2

Thus (14.72)



t K1 (t, x0 , x0 ) = −(4πt)−n/2 tb(x0 ) + S(x0 ) , 12

 since Ricjj (x) = S(x). As noted above, the Laplace operator Δ on scalar functions, when conjugated by g 1/4 , has the form (14.65), with b(x0 ) = Φ(x0 ) − Ψ(x0 ) = −S(x0 )/4. Thus, for the keat kernel etΔ on scalars, we have

82

7. Pseudodifferential Operators

(14.73)

t K1 (t, x0 , x0 ) = (4πt)−n/2 S(x0 ). 6

We now generalize this, setting (14.74)

a(x, ξ) = g(x, ξ) + (x, ξ),

(x, ξ) =



j (x)ξj .

Continue to assume that a(x, ξ) is scalar and consider L = a(X, D) + b(x). We have (14.75)

E0 (t, x, ξ) = e−ta(x,ξ) = e−t (x,ξ) e−tg(x,ξ) ,

and E1 (t, x, ξ) is still given by (14.42)–(14.50). A point to keep in mind is that we can drop (x, ξ) from the computation involving {a, e−ta }2 , altering K1 (t, x, x) only by o(t−n/2+1 ) as t  0. Thus, mod o(t−n/2+1 ), K1 (t, x0 , x0 ) is still given by (14.73). To get K0 (t, x0 , x0 ), expand e−t (x,ξ) in (14.75) in powers of t: (14.76)

  t2 E0 (t, x, ξ) ∼ 1 − t (x, ξ) + (x, ξ)2 + · · · e−tg(x,ξ) . 2

When doing the ξ-integral, the term t (x, ξ) is obliterated, of course, while, by (14.71),   2 t2 1 (14.77) j (x0 )2 . (x0 , ξ)2 e−t|ξ| dξ = π n/2 t−n/2+1 2 4 Hence, in this situation,

(14.78)

K0 (t, x0 , x0 ) + K1 (t, x0 , x0 )    1 −n/2 2 j (x0 ) − b(x0 ) − S(x0 ) + O(t2 ) . 1+t = (4πt) 12

Finally, we drop the assumption that (x, ξ) in (14.74) be scalar. We still assume that g(x, ξ) defines the metric tensor. There are several changes whose effects on (14.78) need to be investigated. In the first place, (14.41) is no longer quite true. We have (14.79)

{a, e−ta }1 =

i 2



 ∂a ∂ −ta ∂a ∂ −ta e − e . ∂xj ∂ξj ∂ξj ∂xj

In this case, with a(x, ξ) matrix-valued, we have

(14.80)

  ∂a ∂ −ta e = −te−ta Ξ ad(−ta) ∂xj ∂xj   ∂a = −te−ta Ξ ad(−t ) , ∂xj

14. The Weyl calculus

83

where Ξ(z) = (1 − e−z )/z, so   ∂a ∂ −ta t ∂ e = te−ta + , + ··· ∂xj ∂xj 2 ∂xj ∂a = −t + O(t2 |ξ|)e−ta + · · · , ∂xj

(14.81)

and so forth. Hence   i  ∂ ∂ −ta {a, e−ta }1 = − t , + ··· . e 2 ∂xj ∂ξj

(14.82)

This is smaller than any of the terms in the transport equation (14.42) for E1 , so it could be put in a higher transport equation. It does not affect (14.78). Another change comes from the following modification of (14.46): 

t

0

(14.83)

e(s−t)a(x,ξ) b(x)e−sa(x,ξ) ds   t e(s−t) (x,ξ) b(x)e−s (x,ξ) ds · e−tg(x,ξ) . = 0

This time, b(x) and (x, ξ) may not commute. We can write the right side as  (14.84)

0

t

 es ad (x,ξ) b(x) ds e−t (x,ξ) e−tg(x,ξ)    t = t b(x) − (x, ξ)b(x) + b(x) (x, ξ) + · · · e−tg(x,ξ) . 2

Due to the extra power of t with the anticommutator, this does not lead to a change in (14.78). 2  The other change in letting (x, ξ) be nonscalar is that the quantity (x, ξ) = j (x) k (x)ξj ξk generally has noncommuting factors, but this also does not affect (14.78). Consequently, allowing (x, ξ) to be nonscalar does not change (14.78). We state our conclusion: Theorem 14.1. If Lu = a(X, D)u + b(x)u, with (14.85)

a(x, ξ) =



g jk (x)ξj ξk +



j (x)ξj ,

where (g jk ) is the inverse of a metric tensor (gjk ), and j (x) and b(x) are matrixvalued, and if gjk (x0 ) = δjk , ∇gjk (x0 ) = 0, then the integral kernel K(t, x, y) of e−tL has the property (14.86)

 

 1 j (x0 )2 − b(x0 ) − S(x0 ) + O(t2 ) . K(t, x0 , x0 ) = (4πt)−n/2 1 + t 12

84

7. Pseudodifferential Operators

Exercises  1. If a(x, ξ) = aα (x)ξ α is a polynomial in ξ, so that a(x, D) is a differential operator, show that a(X, D) is also a differential operator, given by   α  x + y  u(y) Dy aα a(X, D)u(x) = 2 y=x α     α −|γ| γ = D aα (x) Dβ u(x). 2 β α β+γ=α

Verify the formulas (14.34) and (14.35) as special cases. μ m and q ∈ S1,0 are scalar symbols and p ◦ q is defined so that the product 2. If p ∈ S1,0 p(X, D)q(X, D) = (p ◦ q)(X, D), as in (14.22)–(14.23), show that m+2μ−2 q ◦ p ◦ q = q 2 p mod S1,0 μ m More generally, if pjk ∈ S1,0 , pjk = pkj , and qj ∈ S1,0 , show tha

 j,k

qj ◦ pjk ◦ qk =



m+2μ−2 qj pjk qk mod S1,0 .

j,k

Relate this to the last identity in (14.35), comparing a second-order differential operator in the Weyl calculus and in divergence form.

15. Operators of harmonic oscillator type In this section we study operators with symbols in S1m (Rn ), defined to consist of functions p(x, ξ), smooth on R2n and satisfying (15.1)

|Dxβ Dξα p(x, ξ)| ≤ Cαβ (1 + |x| + |ξ|)m−|α|−|β| .

This class has the property of treating x and ξ on the same footing. We define OP S1m (Rn ) to consist of operators p(X, D) with p(x, ξ) ∈ S1m (Rn ). Here we use the Weyl calculus, (14.5). In this setting, the Sp(n, R) action (14.18)–(14.19) m (Rn ), but it does preserve can be well exploited. This action does not preserve S1,0 m n m n S1 (R ). The class OP S1 (R ) has been studied in [GLS, Ho4], and [V], and played a role in microlocal analysis on the Heisenberg group in [T2]. Note that (15.2)

0 (Rn ), S10 (Rn ) ⊂ S1,0

so it follows from Theorem 6.3, plus (14.6)–(14.8), that (15.3)

P ∈ OP S10 (Rn ) =⇒ P : L2 (Rn ) → L2 (Rn ).

15. Operators of harmonic oscillator type

85

If a ∈ S1m and b ∈ S1μ , variants of methods of §3 and (14.22)–(14.24) give a(X, D)b(X, D) = (a ◦ b)(X, D) ∈ OP S1m+μ (Rn ),

(15.4) with

(a ◦ b)(x, ξ) ∼ ab +

(15.5)

 1 {a, b}j (x, ξ), j! j≥1

where {a, b}j is given by (14.24). Note that {a, b}j ∈ S1m+μ−2j (Rn ).

(15.6)

We mention that if either a(x, ξ) or b(x, ξ) is a polynomial in (x, ξ), then the sum in (15.5) is finite and provides an exact formula for (a ◦ b)(x, ξ). The set of “classical” symbols in S1m (Rn ), denoted S m (Rn ), is defined to consist of all p(x, ξ) ∈ S1m (Rn ) such that p(x, ξ) ∼

(15.7)



pj (x, ξ),

j≥0

with pj (x, ξ) smooth and, for |x|2 + |ξ|2 ≥ 1, homogeneous of degree m − 2j in (x, ξ). The meaning of (15.7) is that for each N , p(x, ξ) −

(15.8)

N −1 

pj (x, ξ) ∈ S1m−2N (Rn ).

j=0

It follows from (15.4)–(15.6) that a ∈ S m (Rn ), b ∈ S μ (Rn ) =⇒ a(X, D)b(X, D) (15.9)

= (a ◦ b)(X, D), a ◦ b ∈ S m+μ (Rn ).

Sobolev spaces tailored to these operator classes are defined as follows, for k ∈ Z+ . (15.10)

Hk (Rn ) = {u ∈ L2 (Rn ) : P u ∈ L2 (Rn ), ∀ P ∈ Dk (Rn )}, Dk (Rn ) = span of xβ Dxα , |α| + |β| ≤ k.

Note that Dk (Rn ) ⊂ OP S k (Rn ). The following Rellich type theorem is straightforward: (15.11)

The natural inclusion Hk (Rn ) → L2 (Rn ) is compact, ∀ k ≥ 1.

86

7. Pseudodifferential Operators

The results (15.4) and (15.3) yield, for k ∈ Z+ , (15.12)

A ∈ OP S1−k (Rn ) =⇒ A : L2 (Rn ) → H−k (Rn ).

We will obtain other Sobolev mapping properties below. These spaces will be seen to be natural settings for elliptic regularity results. An operator P = p(X, D) ∈ OP S1m (Rn ) is said to be elliptic provided (15.13)

|p(x, ξ)−1 | ≤ C(1 + |x| + |ξ|)−m ,

for |x|2 + |ξ|2 sufficiently large. With the results (15.4)–(15.6) in hand, natural variants of the parametrix construction of §4 yield for such elliptic P , Q ∈ OP S1−m (Rn ), (15.14)

P Q = I + R1 , QP = I + R2 ,  OP S1−k (Rn ). Rj ∈ OP S1−∞ (Rn ) = k≥1

Clearly, for each m ∈ R, (15.15)

Am (x, ξ) = (1 + |x|2 + |ξ|2 )m/2

is the symbol of an elliptic operator in OP S1m (Rn ). We have (15.16)

Am (X, D)A−m (X, D) = I + Rm ,

Rm ∈ OP S1−4 (Rn ).

In this situation, (15.5) applies, and {Am , A−m }1 = 0. We now introduce the central operator in this class, the harmonic oscillator, (15.17)

H = −Δ + |x|2 =

n  j=1



∂2 + x2j . 2 ∂xj

This is an elliptic element of OP S 2 (Rn ), with symbol |x|2 + |ξ|2 . It defines a positive, self adjoint operator on L2 (Rn ). Note that Lj = ∂j + xj =⇒ L∗j = −∂j + xj (15.18)

=⇒ L∗j Lj = −∂j2 + x2j − 1  =⇒ H = L∗j Lj + n,

so H is positive definite, with H −1 bounded on L2 (Rn ). The following result will be very useful. Theorem 15.1. For all s ∈ (0, ∞), H −s ∈ OP S −2s (Rn ). With Am (x, ξ) as in (15.15),

15. Operators of harmonic oscillator type

87

H −s − A−2s (X, D) ∈ OP S −2s−2 (Rn ).

(15.19)

We postpone the proof of Theorem 15.1 and observe some of its consequences. Proposition 15.2. For k ∈ Z+ , H −k/2 : L2 (Rn ) −→ Hk (Rn )

(15.20) is an isomorphism.

Proof. The mapping property (15.20) follows from Theorem 15.1 and (15.12). If k = 2 is even, the two sided inverse to (15.20) is H : H2 (Rn ) −→ L2 (Rn ).

(15.21)

We need to show that if k = 2 − 1 is odd, (15.22)

H k/2 = H −1/2 : H2 −1 (Rn ) −→ L2 (Rn ).

Indeed, take u ∈ H2 −1 (Rn ). Then (15.23)

H u =



Xj u j ,

uj ∈ L2 (Rn ), Xj ∈ D1 (Rn ),

and hence (15.24)

H −1/2 u =



H −1/2 Xj uj ,

which belongs to L2 (Rn ) since H −1/2 Xj ∈ OP S 0 (Rn ). Given Proposition 15.2, it is natural to set (15.25)

Hs (Rn ) = H −s/2 L2 (Rn ),

for s ∈ R, and we have that this space agrees with (15.10) for s = k ∈ Z+ . For s > 0, this says (15.26)

Hs (Rn ) = D(H s/2 ).

Thus, by Proposition 2.2 of Chap. 4, we can identify Hs (Rn ) with the complex interpolation space: (15.27) Also note that

Hs (Rn ) = [L2 (Rn ), Hk (Rn )]θ ,

s = kθ, θ ∈ (0, 1).

88

7. Pseudodifferential Operators

(15.28)





Hs (Rn ) = S(Rn ),

s−∞

In fact, (15.10) gives ∩k∈Z+ Hk (Rn ) = S(Rn ), and (15.25) gives Hs (Rn ) = H−s (Rn ). Given Theorem 15.1, it easily follows that H s ∈ OP S 2s (Rn ),

(15.29)

∀ s ∈ R.

In fact, given s > 0, take an integer k > s and write H s = H k H s−k . Also, given (15.25), we have, for all m, s ∈ R, (15.30)

P ∈ OP S1m (Rn ) =⇒ P : Hs (Rn ) → Hs−m (Rn ).

Indeed, P = H −(s−m)/2 (H (s−m)/2 P H −s/2 )H s/2 , and H (s−m)/2 P H −s/2 ∈ OP S10 (Rn ) is bounded on L2 (Rn ). We will approach the proof of Theorem 15.1 via the identity (15.31)

H

−s

1 = Γ(s)





e−tH ts−1 dt,

s > 0.

0

Thus we have the task of writing (15.32)

e−tH = ht (X, D)

and computing ht (x, ξ). We need to solve (15.33)

∂ ht (X, D) = −Hht (X, D), ∂t

h0 (x, ξ) = 1.

Taking (15.34) bt (X, D) = Hht (X, D),

H = Q(X, D),

Q(x, ξ) = |x|2 + |ξ|2 ,

since Q(x, ξ) is a polynomial, the formula (15.5) for composition is a finite sum, and it is exact: (15.35)

bt (x, ξ) = Q(x, ξ)ht (x, ξ) +

2  1 {Q, ht }j (x, ξ). j! j=1

Now we make the “guess” that for each t > 0, ht (x, ξ) is a function of |x|2 + |ξ|2 = Q, (15.36)

ht (x, ξ) = g(t, Q).

15. Operators of harmonic oscillator type

89

In that case, {Q, ht }1 = 0, and (15.33)–(15.35) lead to the equation (15.37)

1  ∂ 2 ∂2

∂ht (x, ξ) = −(|x|2 + |ξ|2 )ht (x, ξ) + + 2 ht (x, ξ), 2 ∂t 4 ∂xk ∂ξk k

with initial condition h0 (x, ξ) = 1, or equivalently to solve (15.38)

∂2g ∂g ∂g = −Qg + Q 2 + n , ∂t ∂Q ∂Q

g(0, Q) = 1.

We now guess that (15.38) has a solution of the form g(t, Q) = a(t)eb(t)Q . Then the left side of (15.38) is (a /a + b Q)g and the right side is (−Q + Qb2 + nb)g, so (15.38) is equivalent to (15.39)

a (t) = nb(t), a(t)

b (t) = b(t)2 − 1.

We can solve the second equation for b(t) by separation of variables. Since g(0, Q) = 1, we need b(0) = 0, and the unique solution is b(t) = − tanh t.

(15.40)

Then the equation a /a = −n tanh t with a(0) = 1 gives a(t) = (cosh t)−n .

(15.41) We have our desired formula (15.42)

ht (x, ξ) = (cosh t)−n e−(tanh t)(|x|

2

+|ξ|2 )

.

We discuss briefly why the “guess” that ht (x, ξ) is a function of |x|2 + |ξ|2 was bound to succeed. It is related to the identity (14.19) for the composition of operators transformed by ag (x, ξ) = a(g −1 (x, ξ)), g ∈ Sp(n, R). If we identify R2n with Cn and (x, ξ) with x + iξ, then the unitary group U (n) acts on Cn = R2n , as a subgroup of Sp(n, R), preserving |x|2 + |ξ|2 = |x + iξ|2 . It follows from (14.9) that the set of operators whose symbols are invariant under this U (n) action forms an algebra. From there, it is a short step to guess that e−tH belongs to this algebra. For more details, see Chap. 1, §7 of [T3]. We return to the identity (15.31), which implies (15.43)

H −s = Q−s (X, D)

90

7. Pseudodifferential Operators

with (15.44)

1 Q−s (x, ξ) = Γ(s)





ts−1 (cosh t)−n e−(tanh t)(|x|

2

+|ξ|2 )

dt.

0

To complete the proof of Theorem 15.1, it remains to show that, whenever s > 0, (15.45) Q−s (x, ξ) ∈ S −2s (Rn ), and Q−s (x, ξ)−A−2s (x, ξ) ∈ S −2s−2 (Rn ). To begin, it is clear by inspection that Q−s ∈ C ∞ (R2n ) whenever s > 0. Also, if we set (15.46)

Qb−s (x, ξ) =

1 Γ(s)



1

ts−1 (cosh t)−n e−(tanh t)(|x|

2

+|ξ|2 )

dt,

0

we easily get (15.47)

Q−s (x, ξ) − Qb−s (x, ξ) ∈ S1−∞ (Rn ).

We can set τ = tanh t and write (15.48)

Qb−s (x, ξ) =

1 Γ(s)



a

τ s−1 ϕ(τ )e−τ (|x|

2

+|ξ|2 )

dτ,

0

with a = tanh 1 and ϕ ∈ C ∞ ([0, a]), with power series (15.49)

ϕ(τ ) ∼ 1 + b1 τ 2 + b2 τ 4 + · · · .

Thus, as |x|2 + |ξ|2 → ∞, we have (15.50)

Q−s (x, ξ) ∼



q−s,j (x, ξ),

j≥0

with  ∞ 2 2 bj e−τ (|x| +|ξ| ) τ s+2j−1 dτ Γ(s) 0 −s−2j Γ(s + 2j)  2 |x| + |ξ|2 = bj , Γ(s)

q−s,j (x, ξ) = (15.51)

and b0 = 1. This proves Theorem 15.1. −2s (x, ξ), Remark: We can sharpen (15.45) as follows. Replace A−2s (x, ξ) by A smooth on R2n and equal to (|x|2 + |ξ|2 )−2s for |x|2 + |ξ|2 ≥ 1. Then (15.52)

−2s (x, ξ) ∈ S −2s−4 (Rn ). Q−s (x, ξ) − A

Exercises

91

We make a further specific study of the harmonic oscillator H in §6 of Chap. 8, including results on the eigenvalues and eigenfunctions of H, and an alternative approach to the analysis of the semigroup e−tH . We can extend the Rellich type result (15.11) as follows. By (15.11) and (15.20), we have H −1 compact on L2 (Rn ), so H −1 has a discrete set of eigenvalues, tending to 0, and hence so does H −σ for all σ > 0. Thus H −σ is compact on L2 (Rn ), and, by (15.26), also compact on Hs (Rn ), for all s ∈ R. This gives the following. Proposition 15.3. Given r < s ∈ R, the natural inclusion Hs (Rn ) → Hr (Rn )

(15.53) is compact.

If P ∈ OP S1m (Rn ) is elliptic (say a k × k system), and Q ∈ OP S1−m (Rn ) is a parametrix, as in (15.14), we see that the operators Rj are compact on Hs (Rn ) for all s, so we have the following. Proposition 15.4. If P ∈ OP S1m (Rn ) is elliptic, then, for all s ∈ R, (15.54)

P : Hs (Rn ) −→ Hs−m (Rn ) is Fredholm.

Also Ker P, Ker P ∗ ⊂ S(Rn ),

(15.55)

and the index of P is independent of s. Material on the index of elliptic operators in OP S m (Rn ) will be covered in §11 of Chap. 10. See the exercises below for some preliminary results.

Exercises 1. In case n = 1, consider D1 = ∂1 + x1 . 1

Show that D1 ∈ OP S (R) is elliptic, and that Index D1 = 1. 2. In case n = 2, consider

  ∂1 + x1 ∂2 − x2 D2 = . ∂2 + x2 −∂1 + x1

Show that D2 ∈ OP S 1 (R2 ) is elliptic and that Index D2 = 1.

92

7. Pseudodifferential Operators

3. In the setting of Exercises 1–2, compute Dj∗ Dj and Dj Dj∗ , and compare with H. This should help to compute the kernels of Dj and Dj∗ .

16. Positive quantization of C ∞ (S ∗ M ) Let M be a compact Riemannian manifold. A quantization of C ∞ (S ∗ M ) is a continuous linear map (16.1)

op : C ∞ (S ∗ M ) −→ L(L2 (M ))

with the property that for each a ∈ C ∞ (S ∗ M ), op(a) is a pseudodifferential operator of order 0, whose principal symbol is a. One such quantization, called the Kohn-Nirenberg quantization, (16.2)

0 (M ), opKN : C ∞ (S ∗ M ) −→ OP Scl

is obtained via a partition of unity and local coordinate charts, specializing (10.3). Here we desire to modify this to produce a positive quantization, i.e., one satisfying opF (a)∗ = opF (a) for real valued a, and (16.3)

a ≥ 0 on S ∗ M =⇒ (opF (a)u, u) ≥ 0,

∀ u ∈ L2 (M ).

We use a symmetrization procedure, introduced by K. Friedrichs in his approach to a result known as the sharp G˚arding inequality. We begin the construction in the Euclidean space setting, where opKN (a) can be written  ˆ(ξ) dξ, (16.4) b(x, D) = (2π)−n/2 b(x, ξ)eix·ξ u with (16.5)

b(x, ξ) = (1 − χ(ξ))a(x, ξ/|ξ|),

where χ ∈ C0∞ (Rn ) is radial, 0 ≤ χ ≤ 1, χ(ξ) = 1 for |ξ| ≤ 1, 0 for |ξ| ≥ 2. We look at symmetrizations of the form  (16.6)

opF (a) =

ϕω (D)∗ Ma(·,ω) ϕω (D) dS(ω),

S n−1

where (16.7)

Ma(·,ω) u(x) = a(x, ω)u(x),

16. Positive quantization of C ∞ (S ∗ M )

93

and ϕω (D)u(x) = (2π)−n/2

(16.8)



ϕω (ξ)ˆ u(ξ)eix·ξ dξ.

Assume a ∈ C0∞ (Rn × S n−1 ). We set bω (x, D) = ϕω (D)∗ Ma(·,ω) ϕω (D),

(16.9)

and note that, if ϕω (ξ) satisfies mild bounds,  (16.10)

a(x, ω)|ϕω (D)u(x)|2 dx,

(bω (x, D)u, u) =

∀ u ∈ S(Rn ).

The functions ϕω (ξ) we use arise as follows. Pick ρ ∈ (0, 1), take (16.11)

ψ ∈ C0∞ (Rn ), radial, real valued,

and set (16.12)

ϕ0ω (ξ) = (1 − χ(ξ))ψ(|ξ|−ρ Pω⊥ ξ),

where Pω is the orthogonal projection of Rn onto the span of ω, and Pω⊥ = I −Pω , so Pω⊥ ξ = ξ − (ω · ξ)ω.

(16.13) We take χ as in (16.5). Then

ϕ0ω ∈ Sρ0 (Rn ).

(16.14) Finally, we set

ϕω (ξ) = |ξ|γ ϕ0ω (ξ),

(16.15)

with γ > 0 to be selected shortly. This construction yields (16.16)

ϕω ∈ Sργ (Rn ),

2γ bω (x, D) ∈ OP Sρ,0 (Rn ),

with (16.17)

bω (x, ξ) ∼

 i|α| Dα ϕω (ξ) Dxα a(x, ω)ϕω (ξ), α! ξ

α≥0

94

7. Pseudodifferential Operators

hence (16.18) 



ϕω (ξ)2 a(x, ω) dS(ω)

bω (x, ξ) dS(ω) ∼ S n−1

S n−1

+

 i|α|  α!

|α|≥1

ϕ(α) ω (ξ)a(α) (x, ω)ϕω (ξ) dS(ω).

S n−1

There is a unique value of γ such that  (16.19)

ϕω (ξ)2 a(x, ω) dS(ω) −→ Ca(x, ζ), as |ξ| → ∞, ζ =

S n−1

To obtain γ, note that 

ϕ0ω (rζ)2 dS(ω),

(16.20)

ζ ∈ S n−1 ,

S n−1

is independent of ζ ∈ S n−1 , and is equal to 

ϕ0ω (rζ)2 dS(ζ),

∀ ω ∈ S n−1

S n−1

(16.21)



=

ψ(r1−ρ Pω⊥ ζ)2 dS(ζ),

S n−1 −(1−ρ)(n−1)

∼ Cr

(for r ≥ 2)

as r → ∞.

,

Thus we take γ=

(16.22)

1 (1 − ρ)(n − 1), 2

and have that, for ξ = rζ, r = |ξ| ≥ 1, 

ϕω (ξ)2 a(x, ω) dS(ω)

S n−1

(16.23)

= r(1−ρ)(n−1)



ψ(r1−ρ Pω⊥ ζ)2 a(x, ω) dS(ω)

S n−1

∼ a(x, ζ) +



k≥1

r−k(1−ρ) ak (x, ζ),

ξ . |ξ|

16. Positive quantization of C ∞ (S ∗ M )

95

as r → ∞, provided ψ is multiplied by a constant so that C in (16.21) is 1. (In fact, one has ak = 0 for k odd.) Rather than applying a similar argument to the other terms on the right side of (16.18), we will simplify our task by requiring that ρ is sufficiently close to 1 that 2γ < ρ,

(16.24)

equivalently , ρ > 1 −

1 . n

Thus, in (16.17), 2γ−ρ , bω (x, ξ) = ϕω (ξ)2 a(x, ω) mod Sρ,0

(16.25) hence 

bω (x, ξ) dS(ω) = a(x, ζ),

(16.26)

−(1−ρ)

2γ−ρ mod Sρ,0 + Sρ,0

.

S n−1

We conclude that, with opF (a) defined by (16.6), (16.27)

−δ , opF (a) = opKN (a) mod OP Sρ,0

with δ = min(ρ − 2γ, 1 − ρ),

(16.28) provided (16.24) holds, and (16.29)

a ≥ 0 =⇒ (opF (a)u, u) ≥ 0,

∀ u ∈ L2 (Rn ).

Also a real valued ⇒ opF (a)∗ = opF (a). From here, we can use a partition of unity and local coordinate charts to obtain the following. Proposition 16.1. Let M be a compact, n-dimensional Riemannian manifold. Pick ρ ∈ (1 − 1/n, 1), and set γ = (1 − ρ)(n − 1)/2. Take δ as in (16.28). Then there is a quantization (16.30)

opF : C ∞ (S ∗ M ) −→ L(L2 (M )),

satisfying (16.31) and

−δ (M ), opF (a) = opKN (a) mod OP Sρ,1−ρ

∀ a ∈ C ∞ (S ∗ M ),

96

7. Pseudodifferential Operators

(16.32)

a ≥ 0 =⇒ opF (a) ≥ 0.

The following result is a useful complement to Proposition 16.1 Proposition 16.2. In the setting of Proposition 16.1, one can also arrange that (16.33)

opF (1) = I.

Proof. Suppose we have opF (1) = J. Then J = J ∗ ≥ 0 and J − I is compact, so {1} is the only accumulation point of the spectrum of J. We first alter opF slightly to guarantee that J = opF (1) is injective. To do this, replace opF (a) by opF (a) + aeΔ , where a is the mean value of a. We know that J is Fredholm of index 0, so having J injective, we now have J −δ . (See Exercise invertible. Furthermore, J, J 1/2 , J −1/2 are all = I mod OP Sρ,1−ρ 1 below.) Thus, if we make the adjustment (16.34)

opF (a) → J −1/2 opF (a)J −1/2 ,

the modified quantization opF satisfies all the desired criteria. Note that (16.32)–(16.33) imply (16.35)

 opF (a)L(L2 ) ≤ sup |a|, S∗ M

∀ a ∈ C ∞ (S ∗ M ).

Hence opF has a unique extension to C(S ∗ M ): Proposition 16.3. Let M be a compact, n-dimensional Riemannian manifold. Then the quantization opF in (16.30)–(16.33) has a unique extension to (16.36)

opF : C(S ∗ M ) −→ L(L2 (M )),

satisfying (16.32), (16.33), and (16.35).

R EMARK . As indicated above, a more careful treatment of the remainders in (16.23) and (16.18) leads to a sharper result on the remainder in (16.31), known as the sharp G˚arding inequality. For more on this, we refer back to the end of §6 of this chapter. Propositions 16.1–16.3 suffice for use in Chapter 8 of this book.

Exercises 1. Take a < 0 < b, and assume f ∈ C ∞ ((a, b)). Assume −δ (M ), T ∈ OP Sρ,1−ρ

T = T ∗,

Spec T ⊂ (a, b),

References

97

with δ > 0, ρ ∈ (1/2, 1], and define f (T ) by the spectral theorem. Note that, for each N ∈ N, we can write f (x) =

N −1  k=0

f (k) (0) k x + xN gN (x), k!

Hence f (T ) =

N −1  k=0

gN ∈ C ∞ ((a, b)).

f (k) (0) k T + T N gN (T ), k!

2

and gN (T ) is bounded on L (M ) and commutes with T . Deduce that T N gN (T ) : H s (M ) −→ H s+N δ (M ), Conclude that

for − N δ ≤ s ≤ 0.

−δ (M ). f (T ) − f (0)I ∈ OP Sρ,1−ρ

2. In the setting of Proposition 16.3, show that a, b ∈ C(S ∗ M ) =⇒ opF (ab) − opF (a) opF (b) ∈ K(L2 (M )), where K(L2 (M )) denotes the space of compact operators on L2 (M ). Deduce that a, a−1 ∈ C(S ∗ M ) =⇒ opF (a) is Fredholm on L2 (M ).

References [ADN] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic differential equations satisfying general boundary conditions, CPAM 12(1959), 623–727; II, CPAM 17(1964), 35–92 [Be] R. Beals, A general calculus of pseudo-differential operators, Duke Math. J. 42(1975), 1–42 [BF] R. Beals and C. Fefferman, Spatially inhomogeneous pseudodifferential operators, Comm. Pure Appl. Math. 27(1974), 1–24 [BGM] M. Berger, P. Gauduchon, and E. Mazet, Le Spectre d’une Vari´et´e Riemannienne, LNM #194, Springer, New York, 1971 [BJS] L. Bers, F. John, and M. Schechter, Partial Differential Equations, Wiley, New York, 1964 [Ca1] A. P. Calderon, Uniqueness in the Cauchy problem of partial differential equations, Am. J. Math. 80(1958), 16–36 [Ca2] A. P. Calderon, Singular integrals, Bull. AMS 72(1966), 427–465 [Ca3] A. P. Calderon, Cauchy integrals on Lipschitz curves and related operators, Proc. NAS, USA 74(1977), 1324–1327 [CV] A. P. Calderon and R. Vaillancourt, A class of bounded pseudodifferential operators, Proc. NAS, USA 69(1972), 1185–1187 [CZ] A. P. Calderon and A. Zygmund, Singular integral operators and differential equations, Am. J. Math. 79(1957), 901–921

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[CMM] R. Coifman, A. McIntosh, and Y. Meyer, L’integrale de Cauchy definit un operateur borne sur L2 pour les courbes Lipschitziennes, Ann. Math. 116(1982), 361–388 [Cor] H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudo-differential operators, J. Funct. Anal. 18(1975), 115–131 [Cor2] H. O. Cordes, Elliptic Pseudodifferential Operators—An Abstract Theory, LNM #756, Springer, New York, 1979 [Cor3] H. O. Cordes, Spectral Theory of Linear Differential Operators and Comparison Algebras, London Math. Soc. Lecture Notes #70, Cambridge University Press, London, 1987 [CH] H. O. Cordes and E. Herman, Gelfand theory of pseudo-differential operators, Am. J. Math. 90(1968), 681–717 [DK] B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in Lp for Laplace’s equation in Lipschitz domains, Ann. Math. 125(1987), 437–465 [D] G. David, Op´erateurs d’int´egrale singuli`eres sur les surfaces r´eguli`eres, Ann. Scient. Ecole Norm. Sup. 21(1988), 225–258 [DS] G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets, American Math. Soc., Providence, RI, 1993 [Dui] J. J. Duistermaat, Fourier Integral Operators, Courant Institute Lecture Notes, New York, 1974 [Eg] Y. Egorov, On canonical transformations of pseudo-differential operators, Uspehi Mat. Nauk. 24(1969), 235–236 [FJR] E. Fabes, M. Jodeit, and N. Riviere, Potential techniques for boundary problems in C 1 domains, Acta Math. 141(1978), 165–186 [F] C. Fefferman, The Uncertainty Principle, Bull. AMS 9(1983), 129–266 [FP] C. Fefferman and D. Phong, On positivity of pseudo-differential operators, Proc. NAS, USA 75(1978), 4673–4674 [Fo2] G. Folland, Harmonic Analysis on Phase Space, Princeton University Press, Princeton, NJ, 1989 [G˚a] L. G˚arding, Dirichlet’s problem for linear elliptic partial differential equations, Math. Scand. 1(1953), 55–72 [GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, New York, 1983 [Gr] P. Greiner, An asymptotic expansion for the heat equation, Arch. Rat. Mech. Anal. 41(1971), 163–218 [GLS] A. Grossman, G. Loupias, and E. Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space, Ann. Inst. Fourier 18(1969), 343–368 [Hof] S. Hofmann, On singular integral operators of Calder´on’s type in Rn , and BMO, Revista Math. Iberoam. 10 (1994), 467–505. [HMT] S. Hofmann, M. Mitrea, and M. Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains, International Math. Research Notices 2010 (2010), 2567–2865. [HMT2] S. Hofmann, M. Mitrea, and M. Taylor, Symbol calculus for operators of layer potential type on Lipschitz surfaces with vmo normals, and related pseudodifferential operator calculus, Anal. and PDE 8 (2015), 115–181. [Ho1] L. H¨ormander, Pseudo-differential operators, Comm. Pure Appl. Math. 18(1965), 501–517

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[Ho2] L. H¨ormander, Pseudodifferential operators and hypoelliptic equations, Proc. Symp. Pure Math. 10(1967), 138–183 [Ho3] L. H¨ormander, Fourier integral operators I, Acta Math. 127(1971), 79–183 [Ho4] L. H¨ormander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32(1979), 355–443 [Ho5] L. H¨ormander, The Analysis of Linear Partial Differential Operators, Vols. 3 and 4, Springer, New York, 1985 [How] R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal. 38(1980), 188–254 [JK] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130(1995), 161–219 [K] T. Kato, Boundedness of some pseudo-differential operators, Osaka J. Math. 13(1976), 1–9 [Keg] O. Kellogg, Foundations of Potential Theory, Dover, New York, 1954 [KT] C. Kenig and T. Toro, Harmonic measure on locally flat domains, Duke Math. J. 87 (1997), 509–551. [KN] J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18(1965), 269–305 [Kg] H. Kumano-go, Pseudodifferential Operators, MIT, Cambridge, MA, 1981 [LM] J. Lions and E. Magenes, Non-homogeneous Boundary Problems and Applications I, II, Springer, New York, 1972 [MS] H. McKean and I. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom. 1(1967), 43–69 [Mik] S. Mikhlin, Multidimensional Singular Integral Equations, Pergammon, New York, 1965 [MMM] D. Mitrea, I. Mitrea, and M. Mitrea, Geometric Harmonic Analysis, Springer, New York, 2023. [M3T] D. Mitrea, I. Mitrea, M. Mitrea, and M. Taylor, The Hodge Laplacian: Boundary Value Problems on Riemannian Manifolds, W. de Gruyter, New York, 2016. [MMT] I. Mitrea, M. Mitrea, and M. Taylor, Cauchy integrals, Calder´on projectors, and Toeplitz operators on uniformly rectifiable domains, Adv. in Math. 268 (2015), 666–757. [MMT2] I. Mitrea, M. Mitrea, and M. Taylor, Multidimensional Riemann-Hilbert problems on domains with uniformly rectifiable interfaces, Preprint. [MT1] M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds, J. Funct. Anal. 163(1999), 181–251 [MT2] M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem, J. Funct. Anal. 176(2000), 1–79 [Miz] S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, Cambridge, 1973 [Mus] N. Muskhelishvilli, Singular Integral Equations, P. Nordhoff, Groningen, 1953 [Ni] L. Nirenberg, Lectures on Linear Partial Differential Equations, Reg. Conf. Ser. in Math., no. 17, AMS, Providence, RI, 1972 [Pal] R. Palais, Seminar on the Atiyah-Singer Index Theorem, Princeton University Press, Princeton, NJ, 1965 [Po] J. Polking, Boundary value problems for parabolic systems of partial differential equations, Proc. Symp. Pure Math. 10(1967), 243–274 [RS] M. Reed and B. Simon, Methods of Mathematical Physics, Academic, New York, Vols. 1,2, 1975; Vols. 3,4, 1978

100 7. Pseudodifferential Operators [Se1] R. Seeley, Refinement of the functional calculus of Calderon and Zygmund, Proc. Acad. Wet. Ned. Ser. A 68(1965), 521–531 [Se2] R. Seeley, Singular integrals and boundary value problems, Am. J. Math. 88(1966), 781–809 [Se3] R. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math 10(1967), 288–307 [Sem] S. Semmes, Chord arc surfaces with small constant, Adv. Math. 85 (1991), 198– 223. [So] S. Sobolev, Partial Differential Equations of Mathematical Physics, Dover, New York, 1964 [St] E. Stein, Singular Integrals and the Differentiability of Functions, Princeton University Press, Princeton, NJ, 1972 [St2] E. Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ, 1993 [SW] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Space, Princeton University Press, Princeton, NJ, 1971 [T1] M. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981 [T2] M. Taylor, Noncommutative Microlocal Analysis, Memoirs AMS, no. 313, Providence, RI, 1984 [T3] M. Taylor, Noncommutative Harmonic Analysis, AMS, Providence, RI, 1986 [T4] M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhauser, Boston, 1991 [T5] M. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, AMS, Providence RI, 2000. [T6] M. Taylor, Fefferman-Phong inequalities in diffraction theory, Proc. Symp. Pure Math. 43 (1985), 261–300. [Tre] F. Treves, Introduction to Pseudodifferential Operators and Fourier Integral Operators, Plenum, New York, 1980 [Ver] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 39(1984), 572–611 [V] A. Voros, An algebra of pseudodifferential operators and asymptotics of quantum mechanics, J. Funct. Anal. 29(1978), 104–132 [Wey] H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, New York, 1931 [Zw] M. Zworski, Semiclassical Analysis, GSM #138, AMS, Providence RI, 2012.

8 Spectral Theory

Introduction This chapter is devoted to the spectral theory of self-adjoint, differential operators. We cover a number of different topics, beginning in §1 with a proof of the spectral theorem. After some consideration, we decided to put that material here, rather than in Appendix A, Outline of Functional Analysis. The main motivation for putting it here is to begin a line of reasoning that will be continued in subsequent sections, using the great power of studying unitary groups as a tool in spectral theory. After we show how easily this study leads to a proof of the spectral theorem in §1, in later sections we use it in various ways: as a tool to establish self-adjointness, as a tool for obtaining specific formulas, including basic identities among special functions, and in other capacities. Sections 2 and 3 deal with some general questions in spectral theory, such as when does a differential operator define a self-adjoint operator, when does it have a compact resolvent, and what asymptotic properties does its spectrum have? We tackle the latter question, for the Laplace operator Δ, by examining the asymptotic behavior of the trace of the solution operator etΔ for the heat equation, showing that (0.1)

Tr etΔ = (4πt)−n/2 vol Ω + o(t−n/2 ),

t  0,

when Ω is either a compact Riemannian manifold or a bounded domain in Rn (and has the Dirichlet boundary condition). We bring in a Tauberian theorem to show that (0.1) leads to (0.2)

N (λ) =

(4π)−n/2 vol Ω n/2 λ + o(λn/2 ), Γ( n2 + 1)

where N (λ) is the spectral counting function, N (λ) = #{j : λj ≤ λ}. Using techniques developed in §13 of Chap. 7, we could extend (0.1) to general compact Riemannian manifolds with smooth boundary and to other boundary conditions, such as the Neumann boundary condition. We use instead a different method here c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  M. E. Taylor, Partial Differential Equations II, Applied Mathematical Sciences 116, https://doi.org/10.1007/978-3-031-33700-0 8

101

102 8. Spectral Theory

in §3, one that works without any regularity hypotheses on ∂Ω. In such generality, (0.1) does not necessarily hold for the Neumann boundary problem. The study of (0.1) and refinements got a big push from [Kac]. As pursued in [MS], it led to developments that we will discuss in Chap. 10. The problem of to what extent a Riemannian manifold is determined by the spectrum of its Laplace operator has led to much work, which we do not include here. Some is discussed in [Ber,Br,BGM,Cha], and [NRR]. We mention particularly some distinct regions in R2 whose Laplace operators have the same spectra, given in [GWW]. We have not included general results on the spectral behavior of Δ obtained via geometrical optics, its refinement, the theory of Fourier integral operators, and other microlocal techniques. Results of this nature can be found in Volume 3 of [Ho], and in [Shu], [T1], [Iv], and [Zw]. One direction is to improve the remainder estimate in (0.2) from o(λn/2 ) to O(λ(n−1)/2 ). Refinements, initiated in [DG], lead to a sharper remainder estimate, o(λ(n−1)/2 ), under additional constraints on the dynamics of the geodesic flow, and under further constraints, to O(λ(n−1)/2 / log λ), as in [Be], and (in much greater generality) in [CaG]. See these works for details. Sections 4–9 are devoted to specific examples. In §4 we study the Laplace operator on the unit spheres S n . We specify precisely the spectrum of Δ and discuss explicit formulas for certain functions of Δ, particularly (0.3)

A−1 sin tA,

1/2  K A = −Δ + (n − 1)2 . 4

with K = 1, the sectional curvature of S n . In §5 we obtain an explicit formula for (0.3), with K = −1, on hyperbolic space. In §6 we study the spectral theory of the harmonic oscillator (0.4)

H = −Δ + |x|2 .

We obtain an explicit formula for e−tH , an analogue of which will be useful in Chap. 10. In §7 we study the operator (0.5)

H = −Δ − K|x|−1

on R3 , obtaining in particular all the eigenvalues of this operator. This operator arises in the simplest quantum mechanical model of the hydrogen atom. In §8 we look at a quantum mechanical model for a deuteron given by (0.6)

H = −Δ − V0 χa ,

where V0 is a positive constant and χa (x) = 1 for |x| ≤ a, 0 otherwise. We focus on values of V0 and a experimentally seen to be relevant for the deuteron. In §9 we study the Laplace operator on a cone. Studies done in these sections bring in a number of special functions, including Legendre functions, Bessel functions, and

1. The spectral theorem

103

hypergeometric functions. We have included two auxiliary problem sets, one on confluent hypergeometric functions and one on hypergeometric functions. There follow sections on two general phenomena related to quantum mechanics. Section 10 considers an adiabatic limit of slowly varying self-adjoint operators with discrete spectra. Section 11 presents a quantum ergodic theorem. This chapter ends with two appendices. Appendix A discusses the classical ergodic theorem behind the quantum result of §11. Appendix B derives formulas relating the solution operators to wave equations and shifted wave equations, of relevance to the results of §§4–5.

1. The spectral theorem Appendix A, Outline of Functional Analysis, contains a proof of the spectral theorem for a compact, self-adjoint operator A on a Hilbert space H. In that case, H has an orthonormal basis {uj } such that Auj = λj uj , λj being real numbers having only 0 as an accumulation point. The vectors uj are eigenvectors. A general bounded, self-adjoint operator A may not have any eigenvectors, and the statement of the spectral theorem is somewhat more subtle. The following is a useful version. Theorem 1.1. If A is a bounded, self-adjoint operator on a separable Hilbert space H, then there is a σ-compact space Ω, a Borel measure μ, a unitary map (1.1)

W : L2 (Ω, dμ) −→ H,

and a bounded, Borel function a : Ω → R such that (1.2)

W −1 AW f (x) = a(x)f (x),

∀ f ∈ L2 (Ω, dμ).

The map (1.2) is called a spectral representation of A. Note that when A is compact, the eigenvector decomposition described above yields (1.1) and (1.2) with (Ω, μ) a purely atomic measure space. Later in this section we will extend Theorem 1.1 to the case of unbounded, self-adjoint operators. In order to prove Theorem 1.1, we will work with the operators U (t) = eitA ,

(1.3)

defined by the power-series expansion (1.4)

itA

e

∞  (it)n n A . = n! n=0

104 8. Spectral Theory

This is a special case of a construction made in §4 of Chap. 1. U (t) is uniquely characterized as the solution to the differential equation (1.5)

d U (t) = iAU (t), dt

U (0) = I.

We have the group property (1.6)

U (s + t) = U (s)U (t),

which follows since both sides satisfy the ODE (d/ds)Z(s) = iAZ(s), Z(0) = U (t). If A = A∗ , then applying the adjoint to (1.4) gives (1.7)

U (t)∗ = U (−t),

which is the inverse of U (t) in view of (1.6). Thus {U (t) : t ∈ R} is a group of unitary operators. For a given v ∈ H, let Hv be the closed linear span of {U (t)v : t ∈ R}; we say Hv is the cyclic space generated by v. We say v is a cyclic vector for H if H = Hv . If Hv is not all of H, note that Hv⊥ is invariant under U (t), that is, U (t)Hv⊥ ⊂ Hv⊥ for all t, since for a linear subspace V of H, generally (1.8)

U (t)V ⊂ V =⇒ U (t)∗ V ⊥ ⊂ V ⊥ .

Using this observation, we can prove the next result. Lemma 1.2. If U (t) is a unitary group on a separable Hilbert space H, then H is an orthogonal direct sum of cyclic subspaces. Proof. Let {wj } be a countable, dense subset of H. Take v1 = w1 and H1 = Hv1 . If H1 = H, let v2 be the first nonzero element P1 wj , j ≥ 2, where P1 is the orthogonal projection of H onto H1⊥ , and let H2 = Hv2 . Continue. In view of this, Theorem 1.1 is a consequence of the following: Proposition 1.3. If A = A∗ ∈ L(H), U (t) = eitA , and H has a cyclic vector v, then we can take Ω = R, and there exists a positive Borel measure μ on R and a unitary map W : L2 (R, d μ) → H such that (1.9)

W −1 U (t)W f (x) = eitx f (x),

∀ f ∈ L2 (R, dμ).

One ingredient in the proof of Proposition 1.3 will be the following functional calculus. Given f ∈ S(R), the Schwartz space, we define (1.10)

f (A) = (2π)−1/2





−∞

fˆ(t)eitA dt.

1. The spectral theorem

105

Here fˆ is the Fourier transform of f , and (1.10) formally reflects the Fourier inversion formula. Note that f (A) ≤ (2π)−1/2 fˆ L1 . We record the following basic results. Lemma 1.4. Given f, g ∈ S(Rn ), (1.11)

(1.12)

fs (A) = eisA f (A),

with fs (x) = eisx f (x),

g(A)f (A) = (gf )(A),

and f (A) = f (A)∗ .

(1.13)

Proof. Since fˆs (t) = fˆ(t − s), we have fs (A) = (2π)−1/2 (1.14) −1/2

 

= (2π)

fˆ(t − s)eitA dt fˆ(t)ei(t+s)A dt,

which gives (1.11). From here, (gf )(A) = (2π)−1/2 (1.15) = (2π)−1/2



  gˆ(s) eisx f (A) ds

 gˆ(s)eisA f (A) ds,

by (1.11), and this gives (1.12). The proof of (1.13) is an exercise. We now tackle the proof of Proposition 1.3. To define the map W , we first define (1.16)

W : S(R) −→ H,

by (1.17)

W (f ) = f (A)v,

with f (A) defined by (1.10). We set (1.18)

(f (A)v, v) = f, μ ,

106 8. Spectral Theory

defining the tempered distribution μ ∈ S  (R). We will show below that μ is actually a positive measure. Making use of (1.12)–(1.13), we have the following key identity: (f (A)v, g(A)v) = (g(A)∗ f (A)v, v) = (f g(A)v, v)

(1.19)

= f g, μ . Now, if g = f , the left side of (1.19) is f (A)v 2 , which is ≥ 0. Hence 

(1.20)

 |f |2 , μ ≥ 0, for all f ∈ S(R).

With this, we can establish: Lemma 1.5. The tempered distribution μ, defined by (1.18), is a positive measure on R. Proof. It suffices to show that g, μ ≥ 0 when g ∈ C0∞ (R) and g ≥ 0. So take such g, and then take ψ ∈ C0∞ (R) such that ψ = 1 on supp g. We have g, μ = gψ 2 , μ = lim (g + ε)ψ 2 , μ ε0

= lim ((g + ε)1/2 ψ)2 , μ ≥ 0, ε0

as desired. Now we can finish the proof of Proposition 1.3. Since f g, μ = see from (1.19) that W has a unique continuous extension



f g dμ, we

W : L2 (R, d μ) −→ H,

(1.21)

and W is an isometry. Since v is assumed to be cyclic, the range of W must be dense in H, so W must be unitary. We have from Lemma 1.4 that if f ∈ S(R), then (1.22)

eisA f (A) = fs (A), with fs (τ ) = eisτ f (τ ).

Hence, for f ∈ S(R), (1.23)

W −1 eisA W f = W −1 fs (A)v = eisτ f (τ ).

Since S(R) is dense in L2 (R, d μ), this gives (1.9). Thus the spectral theorem for bounded, self-adjoint operators is proved. Given (1.9), we have from (1.10) that

1. The spectral theorem

(1.24)

W −1 f (A)W g(x) = f (x)g(x),

107

f ∈ S(R), g ∈ L2 (R, d μ),

which justifies the notation f (A) in (1.10). Note that (1.9) implies (1.25)

W −1 AW f (x) = x f (x),

f ∈ L2 (R, d μ),

since (d/dt)U (t) = iAU (t). The essential supremum of x on (R, μ) is equal to A . Thus μ has compact support in R if A is bounded. If a self-adjoint operator A has the representation (1.25), one says A has simple spectrum. It follows from Proposition 1.3 that A has simple spectrum if and only if it has a cyclic vector. One can generalize the results above to a k-tuple of commuting, bounded, self-adjoint operators A = (A1 , . . . , Ak ). In that case, for t = (t1 , . . . , tk ) ∈ Rk , set (1.26)

U (t) = eit·A ,

t · A = t1 A1 + · · · + tk Ak .

The hypothesis that the Aj all commute implies U (t) = U1 (t1 ) · · · Uk (tk ), where Uj (s) = eisAj . U (t) in (1.26) continues to satisfy the properties (1.6) and (1.7); we have a k-parameter unitary group. As above, for v ∈ H, we set Hv equal to the closed linear span of {U (t)v : t ∈ Rk }, and we say v is a cyclic vector provided Hv = H. Lemma 1.2 goes through in this case. Furthermore, for f ∈ S(Rk ), we can define  −k/2 fˆ(t)eit·A dt, (1.27) f (A) = (2π) and if H has a cyclic vector v, the proof of Proposition 1.3 generalizes, giving a unitary map W : L2 (Rk , d μ) → H such that (1.28)

W −1 U (t)W f (x) = eit·x f (x),

f ∈ L2 (Rk , d μ),

t ∈ Rk .

Therefore, Theorem 1.1 has the following extension Proposition 1.6. If A = (A1 , . . . , Ak ) is a k-tuple of commuting, bounded, selfadjoint operators on H, there is a σ-compact space Ω, a Borel measure μ, a unitary map W :L2 (Ω, d μ) → H, and bounded Borel functions aj : Ω → R such that (1.29)

W −1 Aj W f (x) = aj (x)f (x),

f ∈ L2 (Ω, d μ), 1 ≤ j ≤ k.

A bounded operator B ∈ L(H) is said to be normal provided B and B ∗ commute. Equivalently, if we set (1.30)

A1 =

1 B + B∗ , 2

A2 =

1 B − B∗ , 2i

108 8. Spectral Theory

then B = A1 + iA2 , and (A1 , A2 ) is a 2-tuple of commuting, self-adjoint operators. Applying Proposition 1.6 and setting b(x) = a1 (x) + ia2 (x), we have: Corollary 1.7. If B ∈ L(H) is a normal operator, there is a unitary map W : L2 (Ω, d μ) → H and a (complex-valued) b ∈ L∞ (Ω, d μ) such that (1.31)

W −1 BWf (x) = b(x)f (x),

f ∈ L2 (Ω, dμ).

In particular, Corollary 1.7 holds when B = U is unitary. We next extend the spectral theorem to an unbounded, self-adjoint operator A on a Hilbert space H, whose domain D(A) is a dense linear subspace of H. This extension, due to von Neumann, uses von Neumann’s unitary trick, described in (8.18)–(8.19) of Appendix A. We recall that, for such A, the following three properties hold: A ± i : D(A) −→ H bijectively, (1.32)

U = (A − i)(A + i)−1 is unitary on H, A = i(I + U )(I − U )−1 ,

where the range of I − U = 2i(A + i)−1 is D(A). Applying Corollary 1.7 to B = U , we have the following theorem: Theorem 1.8. If A is an unbounded, self-adjoint operator on a separable Hilbert space H, there is a σ-compact space Ω, a Borel measure μ, a unitary map W : L2 (Ω, dμ) → H, and a real-valued Borel function a on Ω such that (1.33)

W −1 AW f (x) = a(x)f (x),

W f ∈ D(A).

In this situation, given f ∈ L2 (Ω, d μ), W f belongs to D(A) if and only if the right side of (1.33) belongs to L2 (Ω, d μ). The formula (1.33) is called the “spectral representation” of a self-adjoint operator A. Using it, we can extend the functional calculus defined by (1.10) as follows. For a Borel function f : R → C, define f (A) by (1.34)

W −1 f (A)W g(x) = f (a(x))g(x).

If f is a bounded Borel function, this is defined for all g ∈ L2 (Ω, dμ) and provides a bounded operator f (A) on H. More generally,

(1.35) D f (A) = W g ∈ H : g ∈ L2 (Ω, dμ) and f (a(x))g ∈ L2 (Ω, dμ) . In particular, we can define eitA , for unbounded, self-adjoint A, by W −1 eitA W g = eita(x) g(x)

1. The spectral theorem

109

Then eitA is a strongly continuous unitary group, and we have the following result, known as Stone’s theorem (stated as Proposition 9.5 in Appendix A): Proposition 1.9. If A is self-adjoint, then iA generates a strongly continuous, unitary group, U (t) = eitA . Note that Lemma 1.2 and Proposition 1.3 are proved for a strongly continuous, unitary group U (t) = eitA , without the hypothesis that A be bounded. This yields the following analogue of (1.2): (1.36)

W −1 U (t)Wf (x) = eita(x) f (x),

f ∈ L2 (Ω, d μ),

for this more general class of unitary groups. Sometimes a direct construction, such as by PDE methods, of U (t) is fairly easy. In such a case, the use of U (t) can be a more convenient tool than the unitary trick involving (1.32). We say a self-adjoint operator A is positive, A ≥ 0, provided (Au, u) ≥ 0, for all u ∈ D(A). In terms of the spectral representation, this says we have (1.33) with a(x) ≥ 0 on Ω. In such a case, e−tA is bounded for t ≥ 0, even for complex t with Re t ≥ 0, and also defines a strongly continuous semigroup. This proves Proposition 9.4 of Appendix A. Given a self-adjoint operator A and a Borel set S ⊂ R, define P (S) = χS (A), that is, using (1.33), (1.37)

W −1 P (S)W g = χS (a(x))g(x),

g ∈ L2 (Ω, dμ),

function of S. Then each P (S) is an orthogonal where χS is the characteristic

projection. Also, if S = j≥1 Sj is a countable union of disjoint Borel sets Sj , then, for each u ∈ H, (1.38)

lim

n→∞

n 

P (Sj )u = P (S)u,

j=1

with convergence in the H-norm. This is equivalent to the statement that n 

χSj (a(x))g → χS (a(x))g in L2 -norm,

for each g ∈ L2 (Ω, dμ),

j=1

which in turn follows from Lebesgue’s dominated convergence theorem. By (1.38), P (·) is a strongly countably additive, projection-valued measure. Then (1.34) yields  (1.39)

f (A) =

f (λ) P (dλ).

P (·) is called the spectral measure of A.

110 8. Spectral Theory

One useful formula for the spectral measure is given in terms of the jump of the resolvent Rλ = (λ − A)−1 , across the real axis. We have the following Proposition 1.10. For bounded, continuous f : R → C, 1 (1.40) f (A)u = lim ε0 2πi





−∞

  f (λ) (λ − iε − A)−1 − (λ + iε − A)−1 u d λ.

Proof. Since W −1 f (A)W is multiplication by f (a(x)), (1.40) follows from the fact that  1 ∞ εf (λ) (1.41) dλ −→ f (a(x)), π −∞ (λ − a(x))2 + ε2 pointwise and boundedly, as ε  0. An important class of operators f (A) are the fractional powers f (A) = Aα , α ∈ (0, ∞), defined by (1.34)–(1.35), with f (λ) = λα , provided A ≥ 0. Note that if g ∈ C([0, ∞)) satisfies g(0) = 1, g(λ) = O(λ−α ) as λ → ∞, then, for u ∈ H, (1.42)

u ∈ D(Aα ) ⇐⇒ Aα g(εA)u H is bounded, for ε ∈ (0, 1],

as follows easily from the characterization (1.35) and Fatou’s lemma. We note that Proposition 2.2 of Chap. 4 applies to D(Aα ), describing it as an interpolation space. We particularly want to identify D(A1/2 ), when A is a positive, self-adjoint operator on a Hilbert space H constructed by the Friedrichs method, as described in Proposition 8.7 of Appendix A. Recall that this arises as follows. One has a Hilbert space H1 , a continuous injection J : H1 → H with dense range, and one defines A by (1.43)



A(Ju), Jv H = (u, v)H1 ,

with (1.44)

D(A) = Ju ∈ JH1 ⊂ H : v → (u, v)H1 is continuous in Jv, in the H-norm .

We establish the following. Proposition 1.11. If A is obtained by the Friedrichs extension method (1.43)– (1.44), then (1.45)

D(A1/2 ) = J(H1 ) ⊂ H.

1. The spectral theorem

111

Proof. D(A1/2 ) consists of elements of H that are limits of sequences in D(A), in the norm A1/2 u H + u H . As shown in the proof of Proposition 8.7 in Appendix A, D(A) = R(JJ ∗ ) and A−1 = JJ ∗ . Now (1.46)

A1/2 JJ ∗ f 2H = (AJJ ∗ f, JJ ∗ f )H = J ∗ f 2H1 .

Thus a sequence (JJ ∗ fn ) converges in the D(A1/2 )-norm (to an element g) if and only if (J ∗ fn ) converges in the H1 -norm (to an element u), in which case g = Ju. Since J ∗ : H → H1 has dense range, precisely all u ∈ H1 arise as limits of such (J ∗ fn ), so the proposition is proved.

Exercises 1. The map f → f (A) given by (1.10) defines a linear transformation α : S(R) → L(H). Show that f ∈ S(R), f ≥ 0 =⇒ f (A) ≥ 0. Deduce that f ∈ S(R) =⇒ f (A) ≤ sup |f |. 2. Let I = [−A, A]. Show that f = 0 on I =⇒ f (A) = 0. Deduce that α has a unique continuous linear extension α : C(I) → L(H), satisfying f ∈ C(I) =⇒ f (A) ≤ sup |f |, I

f ≥ 0 on I =⇒ f (A) ≥ 0. 3. Let g : R → R be a bounded function and suppose there exist fk ∈ C(R) such that fk (λ) → g(λ), ∀ λ ∈ R,

|fk (λ)| ≤ M, ∀ k, λ.

Show that fk (A) −→ g(A), strongly, on H, where g(A) is defined via a spectral representation of A. Deduce that g(A) is well defined, independently of the spectral representation chosen. 4. Let K ⊂ R be compact. Show that Exercise 3 applies to g(λ) = χK (λ), and hence P (K) is well defined independent of the spectral representation of A, where P is the spectral measure, defined in (1.37). 5. Given v ∈ H, consider the positive Borel measure μv on R, defined by μv (S) = (P (S)v, v). Since each finite Borel measure on R is regular, deduce that μv (S) = sup{μv (K) : K ⊂ S, K compact} is independent of the spectral representation of A.

112 8. Spectral Theory 6. Now set μu,v (S) = (P (S)u, v), for u, v ∈ H, S ⊂ R a Borel set. Show that μu,v is a complex measure. Show that μu+v,u+v = μu,u + μv,v + 2 Re μu,v , and derive a similar formula for Im μu,v . Deduce that P (S) is well defined, independent of the choice of spectral representation of A. 7. Given a bounded Borel function F on R, and a spectral representation as in Theorem 1.1, define F (A) ∈ L(H) by F (A)u = W F (a(x))W −1 u, Show that, with μu,v as in Exercise 6,  (F (A)u, v) = F dμu,v ,

u ∈ H.

∀ u, v ∈ H,

R

and deduce that F (A) is well defined, independent of the choice of spectral representation of A 8. Take H = L2 (R), D = (1/i)d/dx, so eitD u(x) = u(x + t). Show that F eitD u(ξ) = eitξ F u(ξ),

f (D) = F ∗ Mf F ,

where F is the Fourier transform. Then show that if we pick v ∈ L2 (R), the formula (1.17) for W f becomes W f = f (D)v = F ∗ (f vˆ). Deduce directly that g(D) = F ∗ Mg F =⇒ g(D)W (f ) = W (gf ). Show that

Hv = L2 (R) ⇐⇒ vˆ(ξ) = 0, for a.e. ξ ∈ R.

2. Self-adjoint differential operators In this section we present some examples of differential operators on a manifold Ω which, with appropriately specified domains, give unbounded, self-adjoint operators on L2 (Ω, dV ), dV typically being the volume element determined by a Riemannian metric on Ω. We begin with self-adjoint operators arising from the Laplacian, making use of material developed in Chap. 5. Let Ω be a smooth, compact Riemannian manifold with boundary, or more generally the closure of an open subset Ω of a compact manifold M without boundary. Then, as shown in Chap. 5, (2.1)

I − Δ : H01 (Ω) −→ H01 (Ω)∗

2. Self-adjoint differential operators

113

is bijective, with inverse we denote T ; if we restrict T to L2 (Ω), (2.2)

T : L2 (Ω) −→ L2 (Ω) is compact and self-adjoint.

Denote by R(T ) the image of L2 (Ω) under T . We can apply Proposition 8.2 of Appendix A to deduce the following Proposition 2.1. If Ω is a region in a compact Riemannian manifold M , then Δ is self-adjoint on L2 (Ω), with domain D(Δ) = R(T ) ⊂ H01 (Ω) described above. For a further description of D(Δ), note that (2.3)

D(Δ) = {u ∈ H01 (Ω) : Δu ∈ L2 (Ω)}.

If ∂Ω is smooth, we can apply the regularity theory of Chap. 5 to obtain (2.4)

D(Δ) = H01 (Ω) ∩ H 2 (Ω).

Instead of relying on Proposition 8.2, we could use the Friedrichs construction, given in Proposition 9.7 of Appendix A. This construction can be applied more generally. Let Ω be any Riemannian manifold, with Laplace operator Δ. We can define H01 (Ω) to be the closure of C0∞ (Ω) in the space {u ∈ L2 (Ω) : du ∈ L2 (Ω, Λ1 )}. The inner product on H01 (Ω) is (2.5)

(u, v)1 = (u, v)L2 + (du, dv)L2 .

We have a natural inclusion H01 (Ω) → L2 (Ω), and the Friedrichs method gives a self-adjoint operator A on L2 (Ω) such that (2.6)

(Au, v)L2 = (u, v)1 , for u ∈ D(A), v ∈ H01 (Ω),

with (2.7)

D(A) = u ∈ H01 (Ω) : v → (u, v)1 extends from H01 (Ω) → C to a continuous linear functional L2 (Ω) → C ,

that is, (2.8)

D(A) = u ∈ H01 (Ω) : ∃f ∈ L2 (Ω) such that (u, v)1 = (f, v)L2 , ∀v ∈ H01 (Ω) .

Integrating (2.5) by parts for v ∈ C0∞ (Ω), we see that A = I − Δ on D(A), so we have a self-adjoint extension of Δ in this general setting, with domain again described by (2.3). The process above gives one self-adjoint extension of Δ, initially defined on C0∞ (Ω). It is not always the only self-adjoint extension. For example, suppose Ω

114 8. Spectral Theory

is compact with smooth boundary; consider H 1 (Ω), with inner product (2.5), and apply the Friedrichs extension procedure. Again we have a self-adjoint operator A, extending I − Δ, with (2.8) replaced by (2.9)

D(A) = u ∈ H 1 (Ω) : ∃f ∈ L2 (Ω) such that (u, v)1 = (f, v)L2 , ∀v ∈ H 1 (Ω) .

In this case, Proposition 7.2 of Chap. 5 yields the following Proposition 2.2. If Ω is a smooth, compact manifold with boundary and Δ the self-adjoint extension just described, then (2.10)

D(Δ) = {u ∈ H 2 (Ω) : ∂ν u = 0 on ∂Ω}.

In case (2.10), we say D(Δ) is given by the Neumann boundary condition, while in case (2.4) we say D(Δ) is given by the Dirichlet boundary condition. In both cases covered by Propositions 2.1 and 2.2, (−Δ)1/2 is defined as a self-adjoint operator. We can specify its domain using Proposition 1.11, obtaining the next result: Proposition 2.3. In case (2.3), D((−Δ)1/2 ) D((−Δ)1/2 ) = H 1 (Ω).

=

H01 (Ω); in case (2.10),

Though Δ on C0∞ (Ω) has several self-adjoint extensions when Ω has a boundary, it has only one when Ω is a complete Riemannian manifold. This is a classical result, due to Roelcke; we present an elegant proof due to Chernoff [Chn]. When an unbounded operator A0 on a Hilbert space H, with domain D0 , has exactly one self-adjoint extension, namely the closure of A0 , we say A0 is essentially self-adjoint on D0 . Proposition 2.4. If Ω is a complete Riemannian manifold, then Δ is essentially self-adjoint on C0∞ (Ω). Thus the self-adjoint extension with domain given by (2.3) is the closure of Δ on C0∞ (Ω). Proof. We will obtain this as a consequence of Proposition 9.6 of Appendix A, which states the following. Let U (t) = eitA be a unitary group on a Hilbert space H which leaves invariant a dense linear space D; U (t)D ⊂ D. If A is an extension of A0 and A0 : D → D, then A0 and all its powers are essentially self-adjoint on D. In this case, U (t) will be the solution operator for a wave equation, and we will exploit finite propagation speed. Set   0 I (2.11) iA0 = , D(A0 ) = C0∞ (Ω) ⊕ C0∞ (Ω). Δ−I 0 The group U (t) will be the solution operator for the wave equation

2. Self-adjoint differential operators

U (t)

(2.12)

115

    f u(t) , = g ut (t)

where u(t, x) is determined by ∂2u − (Δ − 1)u = 0; ∂t2

u(0, x) = f, ut (0, x) = g

It was shown in §2 of Chap. 6 that U (t) is a unitary group on H = H01 (Ω) ⊕ L2 (Ω); its generator is an extension of (2.11), and finite propagation speed implies that U (t) preserves C0∞ (Ω)⊕C0∞ (Ω) for all t, provided Ω is complete. Thus each Ak0 is essentially self-adjoint on this space. Since   Δ−I 0 2 (2.13) −A0 = , 0 Δ−I we have the proof of Proposition 2.3. Considering A2k 0 , we deduce furthermore that each power Δk is essentially self-adjoint on C0∞ (Ω), when Ω is complete. Though Δ is not essentially self-adjoint on C0∞ (Ω) when Ω is compact, we do have such results as the following: Proposition 2.5. If Ω is a smooth, compact manifold with boundary, then Δ is essentially self-adjoint on {u ∈ C ∞ (Ω) : u = 0 on ∂Ω},

(2.14)

its closure having domain described by (2.3). Also, Δ is essentially self-adjoint on {u ∈ C ∞ (Ω) : ∂ν u = 0 on ∂Ω},

(2.15)

its closure having domain described by (2.10). Proof. It suffices to note the simple facts that the closure of (2.14) in H 2 (Ω) is (2.3) and the closure of (2.15) in H 2 (Ω) is (2.10). We note that when Ω is a smooth, compact Riemannian manifold with boundary, and D(Δ) is given by the Dirichlet boundary condition, then (2.16)

∞ 

D(Δj ) = {u ∈ C ∞ (Ω) : Δk u = 0 on ∂Ω, k = 0, 1, 2, . . . },

j=1

and when D(Δ) is given by the Neumann boundary condition, then (2.17)

∞  j=1

D(Δj ) = {u ∈ C ∞ (Ω) : ∂ν (Δk u) = 0 on ∂Ω, k ≥ 0}.

116 8. Spectral Theory

We now derive a result that to some degree amalgamates Propositions 2.4 and 2.5. Let Ω be a smooth Riemannian manifold with boundary, and set (2.18)

Cc∞ (Ω) = {u ∈ C ∞ (Ω) : supp u is compact in Ω};

we do not require elements of this space to vanish on ∂Ω. We say that Ω is complete if it is complete as a metric space. Proposition 2.6. If Ω is a smooth Riemannian manifold with boundary which is complete, then Δ is essentially self-adjoint on {u ∈ Cc∞ (Ω) : u = 0 on ∂Ω}.

(2.19)

In this case, the closure has domain given by (2.3). Proof. Consider the following linear subspace of (2.19): (2.20)

D0 = {u ∈ Cc∞ (Ω) : Δj u = 0 on ∂Ω for j = 0, 1, 2, . . . }.

Let U (t) be the unitary group on H01 (Ω) ⊕ L2 (Ω) defined as in (2.12), with u also satisfying the Dirichlet boundary condition, u(t, x) = 0 for x ∈ ∂Ω. Then, by finite propagation speed, U (t) preserves D0 ⊕ D0 , provided Ω is complete, so as in the proof of Proposition 2.4, we deduce that Δ is essentially self-adjoint on D0 ; a fortiori it is essentially self-adjoint on the space (2.19). By similar reasoning, we can show that if Ω is complete, then Δ is essentially self-adjoint on {u ∈ Cc∞ (Ω) : ∂ν u = 0 on ∂Ω}.

(2.21)

The results of this section so far have involved only the Laplace operator Δ. It is also of interest to look at Schr¨odinger operators, of the form −Δ+V , where the “potential” V (x) is a real-valued function. In this section we will restrict attention to the case V ∈ C ∞ (Ω) and we will also suppose that V is bounded from below. By adding a constant to −Δ + V , we may as well suppose V (x) ≥ 1 on Ω.

(2.22)

We can define a Hilbert space HV1 0 (Ω) to be the closure of C0∞ (Ω) in the space (2.23)

HV1 (Ω) = {u ∈ L2 (Ω) : du ∈ L2 (Ω, Λ1 ), V 1/2 u ∈ L2 (Ω)},

with inner product (2.24)

(u, v)1,V = (du, dv )L2 + (V u, v)L2 .

2. Self-adjoint differential operators

117

Then there is a natural injection HV1 0 (Ω) → L2 (Ω), and the Friedrichs extension method provides a self-adjoint operator A. Integration by parts in (2.24), with v ∈ C0∞ (Ω), shows that such A is an extension of −Δ + V . For this self-adjoint extension, we have (2.25)

D(A1/2 ) = HV1 0 (Ω).

In case Ω is a smooth, compact Riemannian manifold with boundary and V ∈ C ∞ (Ω), one clearly has HV1 0 (Ω) = H01 (Ω). In such a case, we have an immediate extension of Proposition 2.1, including the characterization (2.4) of D(−Δ + V ). One can also easily extend Proposition 2.2 to −Δ + V in this case. It is of substantial interest that Proposition 2.4 also extends, as follows: Proposition 2.7. If Ω is a complete Riemannian manifold and the function V ∈ C ∞ (Ω) satisfies V ≥ 1, then −Δ + V is essentially self-adjoint on C0∞ (Ω). Proof. We can modify the proof of Proposition 2.4; replace Δ − 1 by Δ − V in (2.11) and (2.12). Then U (t) gives a unitary group on HV1 0 (Ω) ⊕ L2 (Ω), and the finite propagation speed argument given there goes through. As before, all powers of −Δ + V are essentially self-adjoint on C0∞ (Ω). Some important classes of potentials V have singularities and are not bounded below. In §7 we return to this, in a study of the quantum mechanical Coulomb problem. We record here an important compactness property when V ∈ C ∞ (Ω) tends to +∞ at infinity in Ω Proposition 2.8. If the Friedrichs extension method described above is used to construct the self-adjoint operator −Δ + V for smooth V ≥ 1, as above, and if V → +∞ at infinity (i.e., for each N < ∞, ΩN = {x ∈ Ω : V (x) ≤ N } is compact), then −Δ + V has compact resolvent. Proof. Given (2.25), it suffices to prove that the injection HV1 0 (Ω) → L2 (Ω) is compact, under the current hypotheses on V . Indeed, if {un } is bounded in HV1 0 (Ω), with inner product (2.24), then {du n } and {V 1/2 un } are bounded in L2 (Ω). By Rellich’s theorem and a diagonal argument, one has a subsequence {unk } whose restriction to each ΩN converges in L2 (ΩN )-norm. The boundedness of {V 1/2 un } in L2 (Ω) then gives convergence of this subsequence in L2 (Ω)-norm, proving the proposition. The following result extends Proposition 2.4 of Chap. 5 Proposition 2.9. Assume that Ω is connected and that either Ω is compact or V → +∞ at infinity. Denote by λ0 the first eigenvalue of −Δ + V . Then a λ0 -eigenfunction of −Δ + V is nowhere vanishing on Ω. Consequently, the λ0 -eigenspace is one-dimensional.

118 8. Spectral Theory

Proof. Let u be a λ0 -eigenfunction of −Δ+V . As in the proof of Proposition 2.4 of Chap. 5, we can write u = u+ + u− , where u+ (x) = u(x) for u(x) > 0 and u− (x) = u(x) for u(x) ≤ 0, and the variational characterization of the λ0 -eigenspace implies that u± are eigenfunctions (if nonzero). Hence it suffices to prove that if u is a λ0 -eigenfunction and u(x) ≥ 0 on Ω, then u(x) > 0 on Ω. To this end, write  t(Δ−V +λ0 ) u(x) = pt (x, y)u(y) dV (y) u(x) = e Ω

We see that this forces pt (x, y) = 0 for all t > 0, when x ∈ Σ = {x : u(x) = 0},

y ∈ O,

O = {x : u(x) > 0},

since pt (x, y) is smooth and ≥ 0. The strong maximum principle (see Exercise 3 in §1 of Chap. 6 forces Σ = ∅.

Exercises 1. Let HV1 (Ω) be the space (2.23). If V ≥ 1 belongs to C ∞ (Ω), show that the Friedrichs extension also defines a self-adjoint operator A1 , equal to −Δ + V on C0∞ (Ω), such 1/2 that D(A1 ) = HV1 (Ω). If Ω is complete, show that this operator coincides with the extension A defined in (2.25). Conclude that, in this case, HV1 (Ω) = HV1 0 (Ω). 2. Let Ω be complete, V ≥ 1 smooth. Show that if A is the self-adjoint extension of −Δ + V described in Proposition 2.7, then (2.26)

D(A) = {u ∈ L2 (Ω) : −Δu + V u ∈ L2 (Ω)},

where a priori we regard −Δu + V u as an element of D (Ω). 3. Define T : L2 (Ω) → L2 (Ω, Λ1 ) ⊕ L2 (Ω) by D(T ) = HV1 0 (Ω), T u = (du, V 1/2 u). Show that (2.27)

D(T ∗ ) = {(v1 , v2 )∈L2 (Ω, Λ1 ) ⊕ L2 (Ω) : δv1 ∈ L2 (Ω), V 1/2 v2 ∈ L2 (Ω)}.

Show that T ∗ T is equal to the self-adjoint extension A of −Δ + V defined by the Friedrichs extension, as in (2.25). 4. If Ω is complete, show that the self-adjoint extension A of −Δ + V in Proposition 2.7 satisfies (2.28)

D(A) = {u ∈ L2 (Ω) : Δu ∈ L2 (Ω), V u ∈ L2 (Ω)}.

(Hint: Denote the right side by W. Use Exercise 3 and A = T ∗ T to show that D(A) ⊂ W. Use Exercise 2 to show that W ⊂ D(A).) 5. Let D = −i d/dx on C ∞ (R), and let B(x) ∈ C ∞ (R) be real-valued. Define the unbounded operator L on L2 (R) by (2.29) D(L) = {u ∈ L2 (R) : Du ∈ L2 (R), Bu ∈ L2 (R)},

Lu = Du + iB(x)u.

3. Heat asymptotics and eigenvalue asymptotics

119

Show that L∗ = D − iB, with D(L∗ ) = {u ∈ L2 (R) : Du − iBu ∈ L2 (R)} Deduce that A0 = L∗ L is given by A0 u = D2 u + B 2 u + B  (x)u on D(A0 ) = {u ∈ L2 (R) : Du ∈ L2 (R), Bu ∈ L2 (R), D2 u+B 2 u+B  (x)u ∈ L2 (R)} 6. Suppose that |B  (x)| ≤ ϑB(x)2 + C, for some ϑ < 1, C < ∞. Show that D(A0 ) = {u ∈ L2 (R) : D2 u + (B 2 + B  )u ∈ L2 (R)} (Hint: Apply Exercise 2 to D2 + (B 2 + B  ) = A, and show that D(A1/2 ) is given by D(L), defined in (2.29).) 7. In the setting of Exercise 6, show that the operator L of Exercise 5 is closed. 1/2 (Hint: L∗ L = A is a self-adjoint extension of D2 + (B 2 + B  ). Show that D(A1 ) ∗ = D(L) and also = D(L).) Also show that D(L ) = D(L) in this case.

3. Heat asymptotics and eigenvalue asymptotics In this section we will study the asymptotic behavior of the eigenvalues of the Laplace operator on a compact Riemannian manifold, with or without boundary. We begin with the boundaryless case. Let M be a compact Riemannian manifold without boundary, of dimension n. In §13 of Chap. 7 we have constructed a parametrix for the solution operator etΔ of the heat equation (3.1)

 ∂ − Δ u = 0 on R+ × M, ∂t

u(0, x) = f (x)

and deduced that (3.2)

Tr etΔ ∼ t−n/2 a0 + a1 t + a2 t2 + · · · ),

t  0,

for certain constants aj . In particular, a0 = (4π)−n/2 vol M.

(3.3)

This is related to the behavior of the eigenvalues of Δ as follows. Let the eigenvalues of −Δ be 0 = λ0 ≤ λ1 ≤ λ2 ≤ · · ·  ∞. Then (3.2) is equivalent to (3.4)

∞ 

e−tλj ∼ t−n/2 a0 + a1 t + a2 t2 + · · · ),

j=0

We will relate this to the counting function

t  0.

120 8. Spectral Theory

N (λ) ∼ #{λj : λj ≤ λ},

(3.5) establishing the following:

Theorem 3.1. The eigenvalues {λj } of −Δ on the compact Riemannian manifold M have the behavior (3.6)

λ → +∞,

N (λ) ∼ C(M )λn/2 ,

with (3.7)

vol M a0 = . n + 1) Γ( 2 + 1)(4π)n/2

C(M ) =

Γ( n2

That (3.6) follows from (3.4) is a special case of a result known as Karamata’s Tauberian theorem. The following neat proof follows one in [Si3]. Let μ be a positive

(locally finite) Borel measure on [0, ∞); in the example above, μ [0, λ] = N (λ). Proposition 3.2. If μ is a positive measure on [0, ∞), α ∈ (0, ∞), then 



(3.8) 0

e−tλ dμ(λ) ∼ at−α ,

t  0,

implies 

x

d μ(λ) ∼ bxα ,

(3.9) 0

x  ∞,

with b=

(3.10)

a . Γ(α + 1)

Proof. Let d μt be the measure given by μt (A) = tα μ(t−1 A), and let d ν(λ) = αλα−1 d λ; then νt = ν. The hypothesis (3.8) becomes  (3.11)

lim

t→0

−λ

e

 d μt (λ) = b

e−λ d ν(λ),

with b given by (3.10), and the desired conclusion becomes 

 (3.12)

lim

t→0

χ(λ) d μt (λ) = b

χ(λ) d ν(λ)

when χ is the characteristic function of [0, 1]. It would suffice to show that (3.12) holds for all continuous χ(λ) with compact support in [0, ∞).

3. Heat asymptotics and eigenvalue asymptotics

121

From (3.11) we deduce that the measures e−λ d μt are uniformly bounded, for t ∈ (0, 1]. Thus (3.12) follows if we can establish  (3.13)

lim

t→0

g(λ)e−λ d μt (λ) = b



g(λ)e−λ d ν(λ),

for g in a dense subspace of C0 (R+ ), the space of continuous functions on [0, ∞) that vanish at infinity. Indeed, the hypothesis implies that (3.13) holds for all g in A, the space of finite, linear combinations of functions of λ ∈ [0, ∞) of the form ϕs (λ) = e−sλ , s ∈ (0, ∞), as can be seen by dilating the variables in (3.11). By the Stone-Weierstrass theorem, A is dense in Co (R+ ), so the proof is complete. We next want to establish similar results on N (λ) for the Laplace operator Δ on a compact manifold Ω with boundary, with Dirichlet boundary condition. At the end of §13 in Chap. 7 we sketched a construction of a parametrix for etΔ in this case which, when carried out, would yield an expansion (3.14)



Tr etΔ ∼ t−n/2 a0 + a1/2 t1/2 + a1 t + · · · ,

t  0,

extending (3.2). However, we will be able to verify the hypothesis of Proposition 3.2 with less effort than it would take to carry out the details of this construction, and for a much larger class of domains. For simplicity, we will restrict attention to bounded domains in Rn and to the flat Laplacian, though more general cases can be handled similarly. Now, let Ω be an arbitrary bounded, open subset of Rn , with closure Ω. The Laplace operator on Ω, with Dirichlet boundary condition, was studied in §5 of Chap. 5 Lemma 3.3. For any bounded, open Ω ⊂ Rn , Δ with Dirichlet boundary condition, etΔ is trace class for all t > 0. Proof. Let Ω ⊂ B, a large open ball. Then the variational characterization of eigenvalues shows that the eigenvalues λj (Ω) of −Δ on Ω and λj (B) of L = −Δ on B, both arranged in increasing order, have the relation (3.15)

λj (Ω) ≥ λj (B).

But we know that e−tL has integral kernel in C ∞ (B × B) for each t > 0, hence is trace class. Since e−tλj (Ω) ≤ e−tλj (B) , this implies that the positive self-adjoint operator etΔ is also trace class. Limiting arguments, which we leave to the reader, allow one to show that, even in this generality, if H(t, x, y) ∈ C ∞ (Ω × Ω) is, for fixed t > 0, the integral kernel of etΔ on L2 (Ω), then  (3.16) Tr etΔ = H(t, x, x) dx. Ω

122 8. Spectral Theory

See Exercises 1–5 at the end of this section. Proposition 3.4. If Ω is a bounded, open subset of Rn and Δ has the Dirichlet boundary condition, then (3.17)

Tr etΔ ∼ (4πt)−n/2 vol Ω,

t  0. 2

Proof. We will compare H(t, x, y) with H0 (t, x, y) = (4πt)−n/2 e|x−y| /4t , the free-space heat kernel. Let E(t, x, y) = H0 (t, x, y) − H(t, x, y). Then, for fixed y ∈ Ω, (3.18)

∂E − Δx E = 0 on R+ × Ω, ∂t

E(0, x, y) = 0,

and (3.19)

E(t, x, y) = H0 (t, x, y), for x ∈ ∂Ω.

To make simple sense out of (3.19), one might assume that every point of ∂Ω is a regular boundary point, though a further limiting argument can be made to lift such a restriction. The maximum principle for solutions to the heat equation implies (3.20) 0 ≤ E(t, x, y) ≤

sup

0≤s≤t,z∈Ω

H0 (s, z, y) ≤ sup (4πs)−n/2 e−δ(y) 0≤s≤t

2

/4s

,

where δ(y) = dist(y, ∂Ω). Now the function ψδ (s) = (4πs)−n/2 e−δ

2

/4s

on (0, ∞) vanishes at 0 and ∞ and has a unique maximum at s = δ 2 /2n; we have ψδ (δ 2 /2n) = Cn δ −n . Thus (3.21)

  2 0 ≤ E(t, x, y) ≤ max (4πt)−n/2 e−δ(y) /4t , Cn δ(y)−n .

Of course, E(t, x, y) ≤ H0 (t, x, y) also. Now, let O ⊂⊂ Ω be such that vol(Ω \ O) < ε. For t small enough, namely for t ≤ δ12 /2n where δ1 = dist(O, ∂Ω), we have (3.22)

0 ≤ E(t, x, x) ≤ (4πt)−n/2 e−δ(x)

2

/4t

,

x ∈ O,

while of course 0 ≤ E(t, x, x) ≤ (4πt)−n/2 , for x ∈ Ω \ O. Therefore,  (3.23) lim sup (4πt)n/2 E(t, x, x) dx ≤ ε, t→0

Ω

3. Heat asymptotics and eigenvalue asymptotics

so

123

 vol Ω − ε ≤ lim inf (4πt)

n/2

H(t, x, x) dx

t→0

Ω



(3.24)

≤ lim sup (4πt)n/2 t→0

H(t, x, x) dx ≤ vol Ω. Ω

As ε can be taken arbitrarily small, we have a proof of (3.17). Corollary 3.5. If Ω is a bounded, open subset of Rn , N (λ) the counting function of the eigenvalues of −Δ, with Dirichlet boundary condition, then (3.6) holds. Note that if Oε is the set of points in Ω of distance ≥ ε from ∂Ω and we define v(ε) = vol(Ω \ Oε ), then the estimate (3.24) can be given the more precise reformulation (3.25)

√ 0 ≤ vol Ω − (4πt)n/2 Tr etΔ ≤ ω( 2nt),

where  (3.26)

ω(ε) = v(ε) +



e−ns

2

/2ε2

d v(s).

ε

The fact that such a crude argument works, and works so generally, is a special property of the Dirichlet problem. If one uses the Neumann boundary condition, then for bounded Ω ⊂ Rn with nasty boundary, Δ need not even have compact resolvent. However, Theorem 3.1 does extend to the Neumann boundary condition provided ∂Ω is smooth. One can do this via the sort of parametrix for boundary problems sketched in §13 of Chap. 7. We now look at the heat kernel H(t, x, y) on the complement of a smooth, bounded region K ⊂ Rn . We impose the Dirichlet boundary condition on ∂K. As before, 0 ≤ H(t, x, y) ≤ H0 (t, x, y), where H0 (t, x, y) is the free-space heat kernel. We can extend H(t, x, y) to be Lipschitz continuous on (0, ∞) × Rn × Rn by setting H(t, x, y) = 0 when either x ∈ K or y ∈ K. We now estimate E(t, x, y) = H0 (t, x, y) − H(t, x, y). Suppose K is contained in the open ball of radius R centered at the origin. Lemma 3.6. For |x − y| ≤ |y| − R, we have (3.27)

E(t, x, y) ≤ Ct−1/2 e−(|y|−R)

2

/4t

.

Proof. With y ∈ Ω = Rn \ K, write (3.28)

−1/2

H(t, x, y) = (4πt)





−∞

e−s

2

/4t

cos sΛ ds,

124 8. Spectral Theory

√ where Λ = −Δ and Δ is the Laplace operator on Ω, with the Dirichlet boundary √ condition. We have a similar formula for H0 (t, x, y), using instead Λ0 = −Δ0 , with Δ0 the free-space Laplacian. Now, by finite propagation speed, cos sΛ δy (x) = cos sΛ0 δy (x), provided |s| ≤ d = dist(y, ∂K), and |x − y| ≤ d Thus, as long as |x − y| ≤ d, we have    2 −1/2 e−s /4t cos sΛ0 δy (x) − cos sΛ δy (x) ds. (3.29) E(t, x, y) = (4πt) |s|≥d

Then the estimate (3.27) follows easily, along the same lines as estimates on heat kernels discussed in Chap. 6, §2. When we combine (3.27) with the obvious inequality (3.30)

0 ≤ E(t, x, y) ≤ H0 (t, x, y) = (4πt)−n/2 e−|x−y|

2

/4t

,

we see that, for each t > 0, E(t, x, y) is rapidly decreasing as |x| + |y| → ∞. Using this and appropriate estimates on derivatives, we can show that E(t, x, y) is the integral kernel of a trace class operator on L2 (Rn ). We can write (3.31)

Tr etΔ0 − etΔ P =

 E(t, x, x) dx , Rn

where P is the projection of L2 (Rn ) onto L2 (Ω) defined by restriction to Ω. Now, as t  0, (4πt)n/2 E(t, x, x) approaches 1 on K and 0 on Rn \ K. Together with the estimates (3.27) and (3.30), this implies  (3.32)

(4πt)n/2

E(t, x, x) dx −→ vol K,

Rn

as t  0. This establishes the following: Proposition 3.7. If K is a closed, bounded set in Rn , Δ is the Laplacian on L2 (Rn \ K), with Dirichlet boundary condition, and Δ0 is the Laplacian on L2 (Rn ), then etΔ0 − etΔ P is trace class for each t > 0 and (3.33)

Tr etΔ0 − etΔ P ∼ (4πt)−n/2 vol K,

as t  0. This result will be of use in the study of scattering by an obstacle K, in Chap. 9. It is also valid for the Neumann boundary condition if ∂K is smooth.

4. The Laplace operator on S n

125

Exercises In Exercises 1–4, let Ω ⊂ Rn be a bounded, open set and let Oj be open with smooth boundary such that O1 ⊂⊂ O2 ⊂⊂ · · · ⊂⊂ Oj ⊂⊂ · · ·  Ω. Let Lj be −Δ on Lj , with Dirichlet boundary condition; the corresponding operator on Ω is simply denoted −Δ. 1. Using material developed in §5 of Chap. 5, show that, for any t > 0, f ∈ L2 (Ω), e−tLj Pj f −→ etΔ f strongly in L2 (Ω), as j → ∞, where Pj is multiplication by the characteristic function of Oj . Don’t peek at Lemma 3.4 in Chap. 11! 2. If λν (Oj ) are the eigenvalues of Lj , arranged in increasing order for each j, show that, for each ν, λν (Oj )  λν (Ω), as j → ∞. 3. Show that, for each t > 0,

Tr e−tLj  Tr etΔ .

4. Let Hj (t, x, y) be the heat kernel on R+ × Oj × Oj . Extend Hj to R+ × Ω × Ω so as to vanish if x or y belongs to Ω \ Oj . Show that, for each x ∈ Ω, y ∈ Ω, t > 0, Hj (t, x, y)  H(t, x, y), as j → ∞. Deduce that, for each t > 0,   Hj (t, x, x) dx  H(t, x, x) dx Oj

Ω

5. Using Exercises 1–4, give a detailed proof of (3.16) for general bounded Ω ⊂ Rn . 6. Give an example of a bounded, open, connected set Ω ⊂ R2 (with rough boundary) such that Δ, with Neumann boundary condition, does not have compact resolvent.

4. The Laplace operator on S n A key tool in the analysis of the Laplace operator ΔS on S n is the formula for the Laplace operator on Rn+1 in polar coordinates: (4.1)

Δ=

1 ∂2 n ∂ + ΔS . + ∂r2 r ∂r r2

In fact, this formula is simultaneously the main source of interest in ΔS and the best source of information about it. The fact that we have already developed so many tools from functional analysis, elliptic operator theory, and spectral theory allows us to give a relatively compact account of basic results of the spectral theory of ΔS . For an account that makes use only of basic results of advanced calculus, see Chapter 7 of [T6].

126 8. Spectral Theory

To begin, we consider the Dirichlet problem for the unit ball in Euclidean space, B = {x ∈ Rn+1 : |x| < 1}: Δu = 0 in B,

(4.2)

u = f on S n = ∂B,

given f ∈ D (S n ). In Chap. 5 we obtained the Poisson integral formula for the solution:  1 − |x|2 f (y) (4.3) u(x) = dS (y), An |x − y|n+1 Sn

where An is the volume of S n . Equivalently, if we set x = rω with r = |x|, ω ∈ S n,  1 − r2 f (ω  ) (4.4) u(rω) = dS (ω  ). An (1 − 2rω · ω  + r2 )(n+1)/2 Sn

Now we can derive an alternative formula for the solution of (4.2) if we use (4.1) and regard Δu = 0 as an operator-valued ODE in r; it is an Euler equation, with solution u(rω) = rA−(n−1)/2 f (ω),

(4.5)

r ≤ 1,

where A is an operator on D (S n ), defined by  (n − 1)2 1/2 . A = −ΔS + 4

(4.6)

If we set r = e−t and compare (4.5) and (4.4), we obtain a formula for the semigroup e−tA as follows. Let θ(ω, ω  ) denote the geodesic distance on S n from ω to ω  , so cos θ(ω, ω  ) = ω · ω  . We can rewrite (4.4) as u(rω) = (4.7)

2 sinh(log r−1 ) r−(n−1)/2 An  f (ω  ) · dS(ω  ).   −1 ) − 2 cos θ(ω, ω  ) −(n+1)/2 2 cosh(log r Sn

In other words, by (4.5), (4.8)

e−tA f (ω) =

2 sinh t An



Sn

f (ω  ) 

(n+1)/2 dS(ω ).  2 cosh t − 2 cos θ(ω, ω )

Identifying an operator on D (S n ) with its Schwartz kernel in D (S n × S n ), we write

4. The Laplace operator on S n

sinh t 2 , An (2 cosh t − 2 cos θ)(n+1)/2

e−tA =

(4.9)

127

t > 0.

Note that the integration of (4.9) from t to ∞ produces the formula (4.10)

A−1 e−tA = 2Cn (2 cosh t − 2 cos θ)−(n−1)/2 ,

t > 0,

provided n ≥ 2, where n − 1 1 1 = π −(n+1)/2 Γ (n − 1)An 4 2

Cn =

With the exact formula (4.9) for the semigroup e−tA , we can proceed to give formulas for fundamental solutions to various important PDE, particularly (4.11)

∂2u − Lu = 0 ∂t2

(wave equation)

∂u − Lu = 0 ∂t

(heat equation),

and (4.12) where L = ΔS −

(4.13)

(n − 1)2 = −A2 . 4

If we prescribe Cauchy data u(0) = f, ut (0) = g for (4.11), the solution is u(t) = (cos tA)f + A−1 (sin tA)g.

(4.14)

Assume n ≥ 2. We obtain formulas for these terms by analytic continuation of the formulas (4.9) and (4.10) to Re t > 0 and then passing to the limit t ∈ iR. This is parallel to the derivation of the fundamental solution to the wave equation on Euclidean space in §5 of Chap. 3. We have (4.15)

 −(n−1)/2 A−1 e(it−ε)A = −2Cn 2 cosh(it − ε) − 2 cos θ ,  −(n+1)/2 2 sinh(it − ε) 2 cosh(it − ε) − 2 cos θ . e(it−ε)A = An

Letting ε  0, we have A−1 sin tA = (4.16)

lim −2Cn Im (2 cosh ε cos t − 2i sinh ε sin t − 2 cos θ)−(n−1)/2

ε0

128 8. Spectral Theory

and

(4.17)

cos tA = −2 lim Im(sin t)(2 cosh ε cos t − 2i sinh ε sin t − 2 cos θ)−(n+1)/2 . ε0 An

For example, on S 2 we have, for 0 ≤ t ≤ π, (4.18)

A−1 sin tA = 2C2 (2 cos θ − 2 cos t)−1/2 , 0,

θ < |t|, θ > |t|,

with an analogous expression for general t, determined by the identity (4.19)

A−1 sin(t + 2π)A = −A−1 sin tA

on D (S 2k ),

plus the fact that sin tA is odd in t. The last line on the right in (4.18) reflects the well-known finite propagation speed for solutions to the hyperbolic equation (4.11). To understand how the sign is determined in (4.19), note that, in (4.15), with ε > 0, for t = 0 we have a real kernel, produced by taking the −(n − 1)/2 = −k + 1/2 power of a positive quantity. As t runs from 0 to 2π, the quantity 2 cosh(it − ε) = 2 cosh ε cos t − 2i sinh ε sin t moves once clockwise around a circle of radius 2(cosh2 ε + sinh2 ε)1/2 , centered at 0, so 2 cosh ε cos t − 2i sinh ε sin t − 2 cos θ describes a curve winding once clockwise about the origin in C. Thus taking a half-integral power of this gives one the negative sign in (4.19). On the other hand, when n is odd, the exponents on the right side of (4.15)– (4.17) are integers. Thus (4.20)

A−1 sin(t + 2π)A = A−1 sin tA

on D (S 2k+1 ).

Also, in this case, the distributional kernel for A−1 sin tA must vanish for |t| = θ. In other words, the kernel is supported on the shell θ = |t|. This is the generalization to spheres of the strict Huygens principle. In case n = 2k + 1 is odd, we obtain from (4.16) and (4.17) that (4.21)

A−1 sin tA f (x) =

 1 ∂ k−1

1 sin2k−1 s f (x, s) s=t (2k − 1)!! sin s ∂s

and (4.22)

cos tA f (x) =

 1 ∂ k

1 sin2k−1 s f (x, s) s=t , sin s (2k − 1)!! sin s ∂s

where, as in (5.66) of Chap. 3, (2k − 1)!! = 3 · 5 · · · (2k − 1) and

4. The Laplace operator on S n

(4.23)

129

f (x, s) = mean value of f on Σs (x) = {y ∈ S n : θ(x, y) = |s|}.

We can examine general functions of the operator A by the functional calculus (4.24)

g(A) = (2π)−1/2





−∞

gˆ(t)eitA dt = (2π)−1/2





−∞

gˆ(t) cos tA dt,

where the last identity holds provided g is an even function. We can rewrite this, using the fact that cos tA has period 2π in t on D (S n ) for n odd, period 4π for n even. In concert with (4.22), we have the following formula for the Schwartz kernel of g(A) on D (S 2k+1 ), for g even: (4.25)

∞  1 1 ∂ k  gˆ(θ + 2kπ). g(A) = (2π)−1/2 − 2π sin θ ∂θ k=−∞

As an example, we compute the heat kernel on odd-dimensional spheres. Take 2 2 g(λ) = e−tλ . Then gˆ(s) = (2t)−1/2 e−s /4t and (4.26)

(2π)−1/2



gˆ(s + 2kπ) = (4πt)−1/2

k



e−(s+2kπ)

2

/4t

= ϑ(s, t),

k 2

where ϑ(s, t) is a “theta function.” Thus the kernel of e−tA on S 2k+1 is given by (4.27)

 1 1 ∂ k 2 ϑ(θ, t). e−tA = − 2π sin θ ∂θ

A similar analysis on S 2k gives an integral, with the theta function appearing in the integrand. The operator A has a compact resolvent on L2 (S n ), and hence a discrete set of eigenvalues, corresponding to an orthonormal basis of eigenfunctions. Indeed, the spectrum of A has the following description Proposition 4.1. The spectrum of the self-adjoint operator A on L2 (S n ) is (4.28)

SpecA =

1 2

 (n − 1) + k : k = 0, 1, 2, . . . .

Proof. Since 0 is the smallest eigenvalue of −ΔS , the definition (4.6) shows that (n − 1)/2 is the smallest eigenvalue of A. Also, (4.20) shows that all eigenvalues of A are integers if n is odd, while (4.19) implies that all eigenvalues of A are (nonintegral) half-integers if n is even. Thus Spec A is certainly contained in the right side of (4.28). Another way to see this containment is to note that since the function u(x) given by (4.5) must be smooth at x = 0, the exponent of r in that formula can take only integer values.

130 8. Spectral Theory

Let Vk denote the eigenspace of A with eigenvalue νk = (n − 1)/2 + k. We want to show that Vk = 0 for k = 0, 1, 2, . . . . Moreover, we want to identify Vk . Now if f ∈ Vk , it follows that u(x) = u(rω) = rA−(n−1)/2 f (ω) = rk f (ω) is a harmonic function defined on all of Rn+1 , which, being homogeneous and smooth at x = 0, must be a harmonic polynomial, homogeneous of degree k in x. If Hk denotes the space of harmonic polynomials, homogeneous of degree k, restriction to S n ⊂ Rn+1 produces an isomorphism: ≈

ρ : Hk − → Vk .

(4.29)

To show that each Vk = 0, it suffices to show that each Hk = 0. Indeed, for c = (c1 , . . . , cn+1 ) ∈ Cn+1 , consider pc (x) = (c1 x1 + · · · + cn+1 xn+1 )k . A computation gives Δpc (x) = k(k − 1) c, c (c1 x1 + · · · + ck xk )k−2 , c, c = c21 + · · · + c2k . Hence Δpc = 0 whenever c, c = 0, so the proposition is proved. We now want to specify the orthogonal projections Ek of L2 (S n ) on Vk . We can attack this via (4.10), which implies (4.30)

∞ 

νk−1 e−tνk Ek (x, y) = 2Cn (2 cosh t − 2 cos θ)−(n−1)/2 ,

k=0

where θ = θ(x, y) is the geodesic distance from x to y in S n . If we set r = e−t and use νk = (n − 1)/2 + k, we get the generating function identity ∞ 

(4.31)

rk νk−1 Ek (x, y) = 2Cn (1 − 2r cos θ + r2 )−(n−1)/2

k=0

=

∞ 

rk pk (cos θ);

k=0

in particular, (4.32)

Ek (x, y) = νk pk (cos θ).

These functions are polynomials in cos θ. To see this, set t = cos θ and write (4.33)

(1 − 2tr + r2 )−α =

∞  k=0

Ckα (t) rk ,

4. The Laplace operator on S n

131

thus defining coefficients Ckα (t). To compute these, use −α

=

(1 − z)

 ∞   j+α−1 j

j=0

zj ,

with z = r(2t − r), to write the left side of (4.33) as ∞    α j=0

j

  j  ∞   j+α−1 j

rj (2t − r)j =

j

j=0 =0 ∞ [k/2]  

=



 (−1)

k=0 =0

(−1) rj+ (2t)j−

k−+α−1 k−

  k− (2t)k−2 rk . 

Hence 

[k/2]

Ckα (t)

(4.34)

=



(−1)



=0

k−+α−1 k−

  k− (2t)k−2 . 

These are called Gegenbauer polynomials. Therefore, we have the following: Proposition 4.2. The orthogonal projection of L2 (S n ) onto Vk has kernel Ek (x, y) = 2Cn νk Ckα (cos θ),

(4.35)

α=

1 (n − 1), 2

with Cn as in (4.10). In the special case n = 2, we have C2 = 1/4π, and νk = k + 1/2; hence Ek (x, y) =

(4.36)

2k + 1 1/2 2k + 1 Ck (cos θ) = Pk (cos θ), 4π 4π

1/2

where Ck (t) = Pk (t) are the Legendre polynomials. The trace of Ek is easily obtained by integrating (4.35) over the diagonal, to yield (4.37)

(n−1)/2

Tr Ek = 2Cn An νk Ck

(1) =

2νk (n−1)/2 C (1). n−1 k

Setting t = 1 in (4.33), so (1 − 2r + r2 )−α = (1 − r)−2α , we obtain (4.38)

Ckα (1) =

  k + 2α − 1 , k

e.g., Pk (1) = 1.

Thus we have the dimensions of the eigenspaces Vk :

132 8. Spectral Theory

Corollary 4.3. The eigenspace Vk of −ΔS on S n , with eigenvalue 1 λk = νk2 − (n − 1)2 = k 2 + (n − 1)k, 4 satisfies (4.39) dim Vk =

      k+n−2 k+n−1 2k + n − 1 k + n − 2 = + . n−1 k k−1 k

In particular, on S 2 we have dim Vk = 2k + 1. Another natural approach to Ek is via the wave equation. We have  T 1 Ek = e−iνk t eitA dt 2T −T (4.40)  T 1 = cos t(A − νk ) dt, 2T −T where T = π or 2π depending on whether n is odd or even. (In either case, one can take T = 2π.) In the special case of S 2 , when (4.18) is used, comparison of (4.36) with the formula produced by this method produces the identity (4.41)

1 Pk (cos θ) = π



θ

−θ

cos(k + 12 )t dt, (2 cos t − 2 cos θ)1/2

for the Legendre polynomials, known as the Mehler-Dirichlet formula. Wave equation techniques also provide a valuable tool for the study of pointwise convergence of spherical harmonic expansions on S n , in [PT].

Exercises Exercises 1–5 deal with results that follow from symmetries of the sphere. The group SO(n + 1) acts as a group of isometries of S n ⊂ Rn+1 , hence as a group of unitary operators on L2 (S n ). Each eigenspace Vk of the Laplace operator is preserved by this action. Fix p = (0, . . . , 0, 1) ∈ S n , regarded as the “north pole.” The subgroup of SO(n + 1) fixing p is a copy of SO(n). 1. Show that each eigenspace Vk has an element u such that u(p) = 0. Conclude by forming  u(gx) dg SO(n)

that each eigenspace Vk of ΔS has an element zk = 0 such that zk (x) = zk (gx), for all g ∈ SO(n). Such a function is called a spherical function, or a zonal function. 2. Suppose Vk has a proper subspace W invariant under SO(n + 1). (Hence W ⊥ ⊂ Vk is also invariant.) Show that W must contain a nonzero spherical function.

Exercises

133

3. Suppose zk and yk are two nonzero spherical functions in Vk . Show that they must be multiples of each other. Hence the unique spherical functions (up to constant multiples) are given by (4.35), with y = p. (Hint: zk and yk are eigenfunctions of −ΔS , with eigenvalue λk = k2 + (n − 1)k. Pick a sequence of surfaces Σj = {x ∈ S n : θ(x, p) = εj } ⊂ S n , with εj → 0, on which zk = αj = 0. With βj = yk |Σj , it follows that βj zk − αj yk is an eigenfunction of −ΔS that vanishes on Σj . Show that, for j large, this forces βj zk − αj yk to be identically zero.) 4. Using Exercises 2 and 3, show that the action of SO(n + 1) on each eigenspace Vk is irreducible, that is, Vk has no proper invariant subspaces. 5. Show that each Vk is equal to the linear span of the set of polynomials of the form pc (x) = (c1 x1 + · · · + cn+1 xn+1 )k , with c, c = 0. (Hint: Show that this linear span is invariant under SO(n + 1).) 6. Using (4.9), show that (4.42)

Tr e−tA =

2 sinh t . (2 cosh t − 2)(n+1)/2

Find the asymptotic behavior as t  0. Use Karamata’s Tauberian theorem to determine the asymptotic behavior of the eigenvalues of A, hence of −ΔS . Compare this with the general results of §3 and also with the explicit results of Corollary 4.3. 7. Using (4.27), show that, for A on S n with n = 2k + 1,

(4.43)

 2 A2k+1  1 1 ∂ k −θ2 /4t  − Tr e−tA = √ e + O(t∞ )  2π sin θ ∂θ θ=0 4πt   = (4πt)−n/2 A2k+1 + O t−n/2+1 ,

as t  0. Compare the general results of §3. 8. Show that (4.44)

e−πi(A−(n−1)/2) f (ω) = f (−ω),

f ∈ L2 (S n ).

(Hint: Check it for f ∈ Vk , the restriction to S n of a homogeneous harmonic polynomial of degree k.) Exercises 9–13 deal with analysis on S n when n = 2. When doing them, look for generalizations to other values of n. 9. If Ξ(A) has integral kernel KΞ (x, y), show that when n = 2, (4.45)

KΞ (x, y) =

∞  1  1 P (cos θ), (2 + 1)Ξ  + 4π 2 =0

where cos θ = x · y and P (t) are the Legendre polynomials. 10. Demonstrate the Rodrigues formula for the Legendre polynomials: (4.46)

Pk (t) =

1 2k k!

(Hint: Use Cauchy’s formula to get

 d k  k t2 − 1 . dt

134 8. Spectral Theory Pk (t) =

1 2πi

 γ

(1 − 2zt + z 2 )−1/2 z −k−1 dz

from (4.33); then use the change of variable 1 − uz = (1 − 2tz + z 2 )1/2 . Then appeal to Cauchy’s formula again, to analyze the resulting integral.) ϕ P (x · y), for some y ∈ S 2 , 11. If f ∈ L2 (S 2 ) has the form f (x) = g(x · y) = show that    2 + 1 1 1 f (z)P (y · z) dS(z) =  + g(t)P (t) dt. (4.47) ϕ = 4π 2 −1 S2



(Hint: Use S 2 Ek (x, z)E (z, y) dS (z) = δk E (x, y).) Conclude that g(x · y) is the integral kernel of ψ(A − 1/2), where ψ() =

(4.48)

4π ϕ = 2π 2 + 1



1 −1

g(t)P (t) dt.

This result is known as the Funk-Hecke theorem. 12. Show that, for x, y ∈ S 2 , eikx·y =

(4.49)

∞  (2 + 1) i j (k) P (x · y), =0

where (4.50)

j (z) =

  π 1/2 1 1  z  1 J+1/2 (z) = (1 − t2 ) eizt dt. 2z 2 ! 2 −1

(Hint: Take g(t) = eikt in Exercise 11, apply the Rodrigues formula, and integrate by parts.) Thus eikx ·y is the integral kernel of the operator 4π e(1/2)πi(A−1/2) jA−1/2 (k) For another approach, see Exercises 10 and 11 in §9 of Chap. 9. 13. Demonstrate the identities d (4.51) (1 − t2 ) + t P (t) = P−1 (t) dt and (4.52)

d d (1 − t2 ) P (t) + ( + 1)P (t) = 0. dt dt

Relate (4.52) to the statement that, for fixed y ∈ S 2 , ϕ(x) = P (x · y) belongs to the ( + 1)-eigenspace of −ΔS . Exercises 14–19 deal with formulas for an orthogonal basis of Vk (for S 2 ). We will make use of the structure of irreducible unitary representations of SO(3), obtained in §9 of Appendix B, Manifolds, Vector Bundles, and Lie Groups. We recall some of the results established there. The Lie algebra skew(3) of SO(3) is spanned by X1 , X2 , X3 , satisfying [Xj , Xj+1 ] = Xj+2 , j ∈ Z/(3). If π is a unitary representation of SO(3), we set

Exercises Lj = dπ(Xj ),

135

L± = L2 ∓ iL1 .

It is shown in Appendix B that each irreducible unitary representation of SO(3) is equivalent to a representation Dk (k ∈ Z+ ) of SO(3) on C2k+1 ≈ P2k , the space of polynomials on Cn , homogeneous of degree 2k, having basis ϕkj = z12k−j z2j , 0 ≤ j ≤ 2k, on which L1 ϕkj = i(−k + j)ϕkj ,

L+ ϕkj = −(2k − j)ϕk,j+1 ,

L− ϕkj = jϕk,j−1 .

The element wk = ϕk,2k , satisfying L1 wk = ikwk , is (up to scaling) the highest weight vector. The relevance to the action of SO(3) on L2 (S 2 ) = ⊕Vk is that, by Exercise 4 above, the action on each eigenspace Vk is irreducible, and, by Corollary 4.3, dim Vk = 2k + 1. 14. Show that the representation of SO(3) on Vk is equivalent to the representation Dk , for each k = 0, 1, 2, . . . . 15. Show that if we use coordinates (θ, ψ) on S 2 , where θ is the geodesic distance from (1, 0, 0) and ψ is the angular coordinate about the x1 -axis in R3 , then (4.53)

L1 =

∂ , ∂ψ

∂ ∂ . L± = i e±iψ ± + i cot θ ∂θ ∂ψ

16. Set (4.54)

wk (x) = (x2 + ix3 )k = sink θ eikψ .

Show that wk ∈ Vk and that it is the highest-weight vector for the representation, so L1 wk = ik wk 17. Show that an orthogonal basis of Vk is given by wk , L− wk , . . . , L2k − wk 18. Show that the functions ζkj = Lk−j − wk , j ∈ {−k, −k + 1, . . . , k − 1, k}, listed in Exercise 17 coincide, up to nonzero constant factors, with zkj , given by zk0 = zk , the spherical function considered in Exercises 1–3, and, for 1 ≤ j ≤ k, zk,−j = Lj− zk ,

zkj = Lj+ zk

19. Show that the functions zkj coincide, up to nonzero constant factors, with (4.55)

eijψ Pkj (cos θ),

−k ≤ j ≤ k,

where Pkj (t), called associated Legendre functions, are defined by (4.56)

Pkj (t) = (−1)j (1 − t2 )|j|/2

 d |j| Pk (t). dt

20. For a formula for the projections alternative to (4.35), replace (4.30)–(4.31) by the following consequence of (4.4)–(4.5):

136 8. Spectral Theory ∞ 

rk Ek f (ω) =

k=0

to obtain Ek f (ω) =

1 An



1 − r2 An

 Sn

(n+1)/2

Ck

f (ω  ) dS(ω  ), (1 − 2rω · ω  + r2 )(n+1)/2

(n+1)/2

(ω · ω  ) − Ck−2

(ω · ω  ) f (ω  ) dS(ω  ).

Sn

By convention Ckα = 0 for k < 0. Sum over 0 ≤ k ≤ N to get  1 (n+1)/2 (n+1)/2 SN f (ω) = CN (ω · ω  ) + CN −1 (ω · ω  ) f (ω  ) dS(ω  ), An Sn

where SN = E0 + E1 + · · · + EN .

5. The Laplace operator on hyperbolic space The hyperbolic space Hn shares with the sphere S n the property of having constant sectional curvature, but for Hn it is −1. One way to describe Hn is as a set of vectors with square length 1 in Rn+1 , not for a Euclidean metric, but rather for a Lorentz metric 2 v, v = −v12 − · · · − vn2 + vn+1 ,

(5.1) namely,

Hn = {v ∈ Rn+1 : v, v = 1, vn+1 > 0},

(5.2)

with metric tensor induced from (5.1). The connected component G of the identity of the group O(n, 1) of linear transformations preserving the quadratic form (5.1) acts transitively on Hn , as a group of isometries. In fact, SO(n), acting on Rn ⊂ Rn+1 , leaves invariant p = (0, . . . , 0, 1) ∈ Hn and acts transitively on the unit sphere in Tp Hn . Also, if A(u1 , . . . , un , un+1 )t = (u1 , . . . , un+1 , un )t , then etA is a one-parameter subgroup of SO(n, 1) taking p to the curve γ = {(0, . . . , 0, xn , xn+1 ) : x2n+1 − x2n = 1, xn+1 > 0} Together these facts imply that Hn is a homogeneous space. There is a map of Hn onto the unit ball in Rn , defined in a fashion similar to the stereographic projection of S n . The map (5.3)

s : Hn −→ B n = {x ∈ Rn : |x| < 1}

is defined by (5.4)

s(x, xn+1 ) = (1 + xn+1 )−1 x.

5. The Laplace operator on hyperbolic space

137

The metric on Hn defined above then yields the following metric tensor on B n : (5.5)

n

−2  ds 2 = 4 1 − |x|2 dx 2j . j=1

Another useful representation of hyperbolic space is as the upper half space Rn+ = {x ∈ Rn : xn > 0}, with a metric we will specify shortly. In fact, with en = (0, . . . , 0, 1), (5.6)

1 τ (x) = |x + en |−2 (x + en ) − en 2

defines a map of the unit ball B n onto Rn+ , taking the metric (5.5) to ds2 = x−2 n

(5.7)

n 

dx2j .

j=1

The Laplace operator for the metric (5.7) has the form Δu = (5.8)

n 



xnn ∂j x2−n ∂j u n

j=1

= x2n

n 

∂j2 u + (2 − n)xn ∂n u.

j=1

which is convenient for a number of computations, such as (5.9) in the following: Proposition 5.1. If Δ is the Laplace operator on Hn , then Δ is essentially self-adjoint on C0∞ (Hn ), and its natural self-adjoint extension has the property (5.9)

spec(−Δ) ⊂

1 4

 (n − 1)2 , ∞ .

Proof. Since Hn is a complete Riemannian manifold, the essential self-adjointness on C0∞ (Hn ) follows from Proposition 2.4. To establish (5.9), it suffices to show that (n − 1)2 u 2L2 (Hn ) , (−Δu, u)L2 (Hn ) ≥ 4 for all u ∈ C0∞ (Hn ). Now the volume element on Hn , identified with the upper half-space with the metric (5.7), is x−n n dx1 · · · dxn , so for such u we have

138 8. Spectral Theory



(5.10)

  1 −Δ − (n − 1)2 u, u 2 L  4  (n − 1)u 2  (∂n u)2 − x2−n = dx1 · · · dxn n 2xn n−1  + dx1 · · · dxn . (∂j u)2 x2−n n j=1

Now, by an integration by parts, the first integral on the right is equal to  

2 (5.11) ∂n x−(n−1)/2 u xn dx1 · · · dxn . n Rn +

Thus the expression (5.10) is ≥ 0, and (5.9) is proved. We next describe how to obtain the fundamental solution to the wave equation on Hn . This will be obtained from the formula for S n , via an analytic continuation in the metric tensor. Let p be a fixed point (e.g., the north pole) in S n , taken to be the origin in geodesic normal coordinates. Consider the one-parameter family of metrics given by dilating the sphere, which has constant curvature K = 1. Spheres dilated to have radius > 1 have constant curvature K ∈ (0, 1). On such a space, the fundamental kernel A−1 sin tA δp (x), with (5.12)

1/2  K A = −Δ + (n − 1)2 , 4

can be obtained explicitly from that on the unit sphere by a change of scale. The explicit representation so obtained continues analytically to all real values of K and at K = −1 gives a formula for the wave kernel, (5.13)

A−1 sin tA δp (x) = R(t, p, x),

1/2  1 A = −Δ − (n − 1)2 . 4

We have (5.14)

 −(n−1)/2 , R(t, p, x) = lim −2Cn Im 2 cos(it − ε) − 2 cosh r ε0

where r = r(p, x) is the geodesic distance from p to x. Here, as in (4.10), Cn = 1/(n − 1)An . This exhibits several properties similar to those in the case of S n discussed in §4. Of course, for r > |t|, the limit vanishes, exhibiting the finite propagation speed phenomenon. Also, if n is odd, the exponent (n − 1)/2 is an integer, which implies that (5.14) is supported on the shell r = |t|. In case n = 3, a use of the Plemelj formula similar to that done in Chapter 3, §5 yields the formula

6. The harmonic oscillator

sin tA δ(t − r) δp (x) = , A 4π sinh r

(5.15)

139

on H3 .

In analogy with (4.25), we have the following formula for g(A)δp (x), for g ∈ S(R), when acting on L2 (Hn ), with n = 2k + 1:  1 1 ∂ k gˆ(r). g(A) = (2π)−1/2 − 2π sinh r ∂r

(5.16)

If n = 2k, we have

(5.17)

g(A) =  ∞

−1/2 1 1 ∂ k 1 − gˆ(s) cosh s − cosh r sinh s ds. 1/2 2π sinh s ∂s π r

R EMARK is also of interest to have formulas for the Schwartz kernel of √ . It √ (sin t −Δ)/ −Δ on hyperbolic space. How to obtain such formulas from (5.13)–(5.14) follows from a general result relating wave equations and shifted wave equations, treated in Appendix B to this chapter.

Exercises −1  1. If n = 2k + 1, show that the Schwartz kernel of −Δ − (n − 1)2 /4 − z 2 on Hn , for Im z > 0, is 1 ∂ k izr 1  1 − e , (5.18) Gz (x, y) = − 2iz 2π sinh r ∂r where r = r(x, y) is geodesic distance, and the integral kernel of et(Δ+(n−1) t > 0, is 1  1 1 ∂ k −r2 /4t − e Ht (x, y) = √ 2π sinh r ∂r 4πt

2

/4)

, for

R EMARK . The formula (5.18) leads to an analysis of the Lp -spectrum of Δ on Hn , showing a dependence on p. This analysis is pursued further in [DST], yielding results of the Lp -spectrum of Δ on a class of quotients of hyperbolic space.

6. The harmonic oscillator We consider the differential operator H = −Δ + |x|2 on L2 (Rn ). By Proposition 2.7, H is essentially self-adjoint on C0∞ (Rn ). Furthermore, as a special case of Proposition 2.8, we know that H has compact resolvent, so L2 (Rn ) has an orthonormal basis of eigenfunctions of H. To work out the spectrum, it suffices to work with the case n = 1, so we consider H = D2 + x2 , where D = −i d/dx.

140 8. Spectral Theory

The spectral analysis follows by some simple algebraic relations, involving the operators  1 d +x , i dx  1 d a+ = D + ix = −x . i dx a = D − ix =

(6.1)

Note that on D (R), H = aa+ − I = a+ a + I,

(6.2) and

[H, a] = −2a,

(6.3)

[H, a+ ] = 2a+ .

Suppose that uj ∈ C ∞ (R) is an eigenfunction of H, that is, uj ∈ D(H),

(6.4)

Huj = λj uj .

Now, by material developed in §2, D(H 1/2 ) = {u ∈ L2 (R) : Du ∈ L2 (R), xu ∈ L2 (R)}, (6.5)

D(H) = {u ∈ L2 (R) : D2 u + x2 u ∈ L2 (R)}.

Since certainly each uj belongs to D(H 1/2 ), it follows that au j and a+ uj belong to L2 (R). By (6.3), we have (6.6)

H(au j ) = (λj − 2)auj ,

H(a+ uj ) = (λj + 2)a+ uj .

It follows that auj and a+ uj belong to D(H) and are eigenfunctions. Hence, if (6.7)

Eigen(λ, H) = {u ∈ D(H) : Hu = λu},

we have, for all λ ∈ R, (6.8)

a+ : Eigen(λ, H) → Eigen(λ + 2, H), a : Eigen(λ + 2, H) → Eigen(λ, H).

From (6.2) it follows that (Hu, u) ≥ u 2L2 , for all u ∈ C0∞ (R); hence, in view of essential self-adjointness, (6.9)

spec H ⊂ [1, ∞),

for n = 1.

6. The harmonic oscillator

141

Now each space Eigen(λ, H) is a finite-dimensional subspace of C ∞ (R), and, by (6.2), we conclude that, in (6.8), a+ is an isomorphism of Eigen(λj , H) onto Eigen(λj + 2, H), for each λj ∈ spec H. Also, a is an isomorphism of Eigen(λj , H) onto Eigen(λj − 2, H), for all λj > 1. On the other hand, a must annihilate Eigen(λ0 , H) when λ0 is the smallest element of spec H, so u0 ∈ Eigen(λ0 , H) =⇒ u0 (x) = −xu0 (x)

(6.10)

2

=⇒ u0 (x) = K e−x

/2

.

Thus λ0 = 1,

(6.11)

2 Eigen(1, H) = span e−x /2 .

2

Since e−x /2 spans the null space of a, acting on C ∞ (R), and since each nonzero space Eigen(λj , H) is mapped by some power of a to this null space, it follows that, for n = 1, spec H = {2k + 1 : k = 0, 1, 2, . . . }

(6.12) and

 (6.13)

Eigen(2k + 1, H) = span

 k 2 ∂ − x e−x /2 . ∂x

One also writes (6.14)

k  ∂ 2 2 − x e−x /2 = Hk (x) e−x /2 , ∂x

where Hk (x) are the Hermite polynomials, given by 2

Hk (x) = (−1)k ex (6.15)

[k/2]

=

 j=0

(−1)j

 d k 2 e−x dx k! (2x)k−2j . j!(k − 2j)!

We define eigenfunctions of H: (6.16)

hk (x) = ck

 ∂ k 2 2 − x e−x /2 = ck Hk (x)e−x /2 , ∂x

where ck is the unique positive number such that hk L2 (R) = 1. To evaluate ck , note that (6.17)

a+ hk 2L2 = (aa+ hk , hk )L2 = 2(k + 1) hk 2L2 .

142 8. Spectral Theory

Thus, if hk L2 = 1, in order for hk+1 = γk a+ hk to have unit norm, we need γk = (2k + 2)−1/2 . Hence  −1/2 ck = π 1/2 2k (k!) .

(6.18)

Of course, given the analysis above of H on L2 (R), then for H = −Δ + |x|2 on L2 (Rn ), we have spec H = {2k + n : k = 0, 1, 2, . . . }.

(6.19)

In this case, an orthonormal basis of Eigen(2k + n, H) is given by (6.20)

ck1 · · · ckn Hk1 (x1 ) · · · Hkn (xn )e−|x|

2

/2

,

k1 + · · · + kn = k,

where kν ∈ {0, . . . , k}, the Hkν (xν ) are the Hermite polynomials, and the ckν are given by (6.18). The dimension of this eigenspace is the same as the dimension of the space of homogeneous polynomials of degree k in n variables. We now want to derive a formula for the semigroup e−tH , t > 0, called the Hermite semigroup. Again it suffices to treat the case n = 1. To some degree paralleling the analysis of the eigenfunctions above, we can produce this formula via some commutator identities, involving the operators (6.21)

X = D2 = −∂x2 ,

Y = x2 ,

Z = x∂x + ∂x x = 2x ∂x + 1.

Note that H = X + Y . The commutator identities are (6.22)

[X, Y ] = −2Z,

[X, Z] = 4X,

[Y, Z] = −4Y.

Thus, X, Y , and Z span a three-dimensional, real Lie algebra. This is isomorphic to sl(2, R), the Lie algebra consisting of 2 × 2 real matrices of trace zero, spanned by  (6.23)

n+ =

0 0

 1 , 0

 n− =

0 1

 0 , 0

 α=

1 0

 0 . −1

We have (6.24)

[n+ , n− ] = α,

[n+ , α] = −2n+ ,

[n− , α] = 2n− .

The isomorphism is implemented by (6.25)

X ↔ 2n+ ,

Now we will be able to write

Y ↔ 2n− ,

Z ↔ −2α.

6. The harmonic oscillator

(6.26)

143

e−t(2n+ +2n− ) = e−2σ1 (t)n+ e−2σ3 (t)α e−2σ2 (t)n− ,

as we will see shortly, and, once this is accomplished, we will be motivated to suspect that also e−tH = e−σ1 (t)X eσ3 (t)Z e−σ2 (t)Y .

(6.27)

To achieve (6.26), write −2σ1 n+

e

e−2σ3 α

(6.28)

e−2σ2 n−

    1 −2σ1 1 x = = , 0 1 0 1  −2σ    e 3 0 y 0 = = , 0 e2σ3 0 1/y     1 0 1 0 = = , 1 −2σ2 z 1

and (6.29)

e−2t(n+ +n− ) =



cosh 2t − sinh 2t

− sinh 2t cosh 2t

 =

 u v

Then (6.26) holds if and only if y=

(6.30)

1 1 = , u cosh 2t

x=z=

v = − tanh 2t, u

so the quantities σj (t) are given by σ1 (t) = σ2 (t) =

(6.31)

1 tanh 2t, 2

e2σ3 (t) = cosh 2t.

Now we can compute the right side of (6.27). Note that −σ1 X

e (6.32)

−1/2



u(x) = (4πσ1 )

e−(x−y)

2

/4σ1

2

e−σ2 Y u(x) = e−σ2 x u(x), eσ3 Z u(x) = eσ3 u(e2σ3 x).

Upon composing these operators we find that, for n = 1, (6.33)

e−tH u(x) =

 Kt (x, y)u(y) dy,

u(y) dy,

 v . u

144 8. Spectral Theory

with

(6.34)

Kt (x, y) =

exp

   − 12 (cosh 2t)(x2 + y 2 ) + xy sinh 2t .

1/2 2π sinh 2t

This is known as Mehler’s formula for the Hermite semigroup. Clearly, for general n, we have (6.35)

e−tH u(x) =

 Kn (t, x, y)u(y) dy,

with (6.36)

Kn (t, x, y) = Kt (x1 , y1 ) · · · Kt (xn , yn ).

The idea behind passing from (6.26) to (6.27) is that the Lie algebra homomorphism defined by (6.25) should give rise to a Lie group homomorphism from (perhaps a covering group G of) SL(2, R) into a group of operators. Since this involves an infinite-dimensional representation of G (not necessarily by bounded operators here, since e−tH is bounded only for t ≥ 0), there are analytical problems that must be overcome to justify this reasoning. Rather than take the space to develop such analysis here, we will instead just give a direct justification of (6.33)–(6.34). Indeed, let v(t, x) denote the right side of (6.33), with u ∈ L2 (R) given. The rapid decrease of Kt (x, y) as |x| + |y| → ∞, for t > 0, makes it easy to show that (6.37)



u ∈ L2 (R) =⇒ v ∈ C ∞ (0, ∞), S(R) .

Also, it is routine to verify that (6.38)

∂v = −Hv. ∂t

Simple estimates yielding uniqueness then imply that, for each s > 0, (6.39)

v(t + s, ·) = e−tH v(s, ·).

Indeed, if w(t, ·) denotes the difference between the two sides of (6.39), then we have w(0) = 0, w ∈ C(R+ , D(H)), ∂w/∂t ∈ C(R+ , L2 (R)), and d w(t) 2L2 = −2(Hw, w) ≤ 0, dt so w(t) = 0, for all t ≥ 0. Finally, as t  0, we see from (6.31) that each σj (t)  0. Since v(t, x) is also given by the right side of (6.27), we conclude that

6. The harmonic oscillator

145

v(t, ·) → u in L2 (R), as t  0.

(6.40)

Thus we can let s  0 in (6.39), obtaining a complete proof that e−tH u is given by (6.33) when n = 1. It is useful to write down the formula for e−tH using the Weyl calculus, introduced in §14 of Chap. 7. We recall that it associates to a(x, ξ) the operator a(X, D)u = (2π)−n (6.41) −n

 

= (2π)

a ˆ(q, p)ei(q·X+p·D) u(x) dq dp x + y  , ξ ei(x−y)·ξ u(y) dy d ξ. a 2

In other words, the operator a(X, D) has integral kernel Ka (x, y), for which  a(X, D)u(x) = given by Ka (x, y) = (2π)−n



Ka (x, y)u(y) dy, x + y  , ξ ei(x−y)·ξ d ξ a 2

Recovery of a(x, ξ) from Ka (x, y) is an exercise in Fourier analysis. When it is applied to the formulas (6.33)–(6.36), this exercise involves computing a Gaussian integral, and we obtain the formula (6.42)

e−tH = ht (X, D)

on L2 (Rn ), with (6.43)

ht (x, ξ) = (cosh t)−n e−(tanh t)(|x|

2

+|ξ|2 )

.

It is interesting that this formula, while equivalent to (6.33)–(6.36), has a simpler and more symmetrical appearance. In fact, the formula (6.43) was derived in §15 of Chap. 7, by a different method, which we briefly recall here. For reasons of symmetry, involving the identity (14.19), one can write (6.44)

ht (x, ξ) = g(t, Q),

Q(x, ξ) = |x|2 + |ξ|2 .

Note that (6.42) gives ∂t ht (X, D) = −Hht (X, D). Now the composition formula for the Weyl calculus implies that ht (x, ξ) satisfies the following evolution equation:

146 8. Spectral Theory

∂ ht (x, ξ) = −(Q ◦ ht )(x, ξ) ∂t

1 = −Q(x, ξ)ht (x, ξ) − {Q, ht }2 (x, ξ) 2

1  2 ∂xk + ∂ξ2k ht (x, ξ). = −(|x|2 + |ξ|2 )ht (x, ξ) + 4

(6.45)

k

Given (6.44), we have for g(t, Q) the equation ∂2g ∂g ∂g = −Qg + Q 2 + n . ∂t ∂Q ∂Q

(6.46)

It is easy to verify that (6.43) solves this evolution equation, with h0 (x, ξ) = 1. We can obtain a formula for e−tQ(X,D) = hQ t (X, D),

(6.47)

for a general positive-definite quadratic form Q(x, ξ). First, in the case Q(x, ξ) =

(6.48)

n 

μj (x2j + ξj2 ),

μj > 0,

j=1

it follows easily from (6.43) and multiplicativity, as in (6.36), that ⎧ ⎫ n n ⎨  ⎬  

−1 cosh tμj (6.49) hQ · exp − (tanh tμj ) x2j + ξj2 . t (x, ξ) = ⎩ ⎭ j=1

j=1

Now any positive quadratic form Q(x, ξ) can be put in the form (6.48) via a linear symplectic transformation, so to get the general formula we need only rewrite (6.49) in a symplectically invariant fashion. This is accomplished using the “Hamilton map” FQ , a skew-symmetric transformation on R2n defined by Q(u, v) = σ(u, FQ v),

(6.50)

u, v ∈ R2n ,

where Q(u, v) is the bilinear form polarizing Q, and σ is the symplectic form on   R2n ; σ(u, v) = x · ξ  − x · ξ if u =  (x, ξ), v =  (x , ξ ). When Q has the form 0 μj (6.48), FQ is a sum of 2 × 2 blocks , and we have −μj 0 (6.51)

n −1/2 

−1  cosh tμj = det cosh itFQ . j=1

Passing from FQ to

Exercises

147

1/2 AQ = −FQ2 ,

(6.52)

the unique positive-definite square root, means passing to blocks 

μj 0

 0 , μj

and when Q has the form (6.48), then n 

(tanh tμj )(x2j + ξj2 ) = tQ ϑ(tAQ )ζ, ζ ,

(6.53)

j=1

where ζ = (x, ξ) and ϑ(t) =

(6.54)

tanh t . t

Thus the general formula for (6.47) is −1/2  hQ (x, ξ) = cosh tA e−tQ(ϑ(tAQ )ζ,ζ) . Q t

(6.55)

Exercises 1. Define an unbounded operator A on L2 (R) by D(A) = {u ∈ L2 (R) : Du ∈ L2 (R), xu ∈ L2 (R)},

Au = Du − ixu.

Show that A is closed and that the self-adjoint operator H satisfies H = A∗ A + I = AA∗ − I (Hint: Note Exercises 5–7 of §2.) 2. If Hk (x) are the Hermite polynomials, show that there is the generating function identity ∞  2 1 Hk (x)sk = e2xs−s k! k=0

(Hint: Use the first identity in (6.15).) 3. Show that Mehler’s formula (6.34) is equivalent to the identity ∞ 

π

−1/2

2 −1/2

(1 − s )

j=0

hj (x)hj (y)sj =

  2 2 exp (1 − s2 )−1 2xys − (x2 + y 2 )s2 · e−(x +y )/2 ,

for 0 ≤ s < 1. Deduce that

148 8. Spectral Theory ∞ 

Hj (x)2

j=0

2 sj = (1 − s2 )−1/2 e2sx /(1+s) , j 2 j!

4. Using H −s =

1 Γ(s)





e−tH ts−1 dt,

|s| < 1.

Re s > 0,

0

find the integral kernel As (x, y) such that  H −s u(x) = As (x, y)u(y) dy. Writing Tr H −s =



As (x, x) dx, Re s > 1, n = 1, show that ζ(s) =

1 Γ(s)



∞ 0

y s−1 dy ey − 1

See [Ing], pp. 41–44, for a derivation of the functional equation for the Riemann zeta function, using this formula. 5. Let Hω = −d2 /dx2 + ω 2 x2 . Show that e−tHω has integral kernel Ktω (x, y) = (4πt)−1/2 γ(2ωt)1/2 e−γ(2ωt)[(cosh 2ωt)(x where γ(z) =

2

+y 2 )−2xy]/4t

,

z . sinh z

6. Consider the operator 2  ∂ 2  ∂ − iωx2 − + iωx1 ∂x1 ∂x2  ∂ ∂  2 2 . = −Δ + ω |x| + 2iω x2 − x1 ∂x1 ∂x2

Q(X, D) = −

Note that Q(x, ξ) is nonnegative, but not definite. Study the integral kernel KtQ (x, y) of e−tQ(X,D) . Show that 2

KtQ (x, 0) = (4πt)−1 γ(2ωt) e−τ (2ωt)|x|

/4t

,

where τ (z) = z coth z. 7. Let (ωjk ) be an invertible, n × n, skew-symmetric matrix of real numbers (so n must be even). Suppose  2 n   ∂ −i ωjk xk . L=− ∂xj j=1 k

KtL (x, y), +

particularly at y = 0. Evaluate the integral kernel 8. In terms of the operators a, a given by (6.1) and the basis of L2 (R) given by (6.16)– (6.18), show that √ √ a+ hk = 2k + 2 hk+1 , ahk = 2k hk−1 .

7. The quantum Coulomb problem

149

7. The quantum Coulomb problem In this section we examine the operator Hu = −Δu − K|x|−1 u,

(7.1)

acting on functions on R3 . Here, K is a positive constant. This provides a quantum mechanical description of the Coulomb force between two charged particles. It is the first step toward a quantum mechanical description of the hydrogen atom, and it provides a decent approximation to the observed behavior of such an atom, though it leaves out a number of features. The most important omitted feature is the spin of the electron (and of the nucleus). Giving rise to further small corrections are the nonzero size of the proton, and relativistic effects, which confront one with great subtleties since relativity forces one to treat the electromagnetic field quantum mechanically. We refer to texts on quantum physics, such as [Mes], [Ser], [BLP], and [IZ], for work on these more sophisticated models of the hydrogen atom. We want to define a self-adjoint operator via the Friedrichs method. Thus we want to work with a Hilbert space (7.2)

H=

" !  |x|−1 |u(x)|2 dx < ∞ , u ∈ L2 (R3 ) : ∇u ∈ L2 (R3 ),

with inner product  (7.3)

(u, v)H = (∇u, ∇v)L2 + A(u, v)L2 − K

|x|−1 u(x)v(x) dx,

where A is a sufficiently large, positive constant. We must first show that A can be picked to make this inner product positive-definite. In fact, we have the following: Lemma 7.1. For all ε ∈ (0, 1], there exists C(ε) < ∞ such that  (7.4)

|x|−1 |u(x)|2 dx ≤ ε ∇u 2L2 + C(ε) u 2L2 ,

for all u ∈ H 1 (R3 ). Proof. Here and below we will use the inclusion (7.5)

H s (Rn ) ⊂ Lp (Rn ),

 ∀ p ∈ 2,

2n  , n − 2s

0≤s
3/2; take q  ∈ (3/2, 3). Then (7.6) holds for some  σ < 1, for which L2q (R3 ) ⊃ H σ (R3 ). From this, (7.4) follows immediately. Thus the Hilbert space H in (7.2) is simply H 1 (R3 ), and we see that indeed, for some A > 0, (7.3) defines an inner product equivalent to the standard one on H 1 (R3 ). The Friedrichs method then defines a positive, self-adjoint operator H + AI, for which

D (H + AI)1/2 = H 1 (R3 ).

(7.7) Then (7.8)

D(H) = {u ∈ H 1 (R3 ) : −Δu − K|x|−1 u ∈ L2 (R3 )},

where −Δu − K|x|−1 u is a priori regarded as an element of H −1 (R3 ) if u ∈ H 1 (R3 ). Since H 2 (R3 ) ⊂ L∞ (R3 ), we have (7.9)

u ∈ H 2 (R3 ) =⇒ |x|−1 u ∈ L2 (R3 ),

so (7.10)

D(H) ⊃ H 2 (R3 ).

Indeed, we have: Proposition 7.2. For the self-adjoint extension H of −Δ−K|x|−1 defined above, (7.11)

D(H) = H 2 (R3 ).

Proof. Pick λ in the resolvent set of H; for instance, λ ∈ C \ R. If u ∈ D(H) and (H − λ)u = f ∈ L2 (R3 ), we have (7.12)

u − KRλ V u = Rλ f = gλ ,

where V (x) = |x|−1 and Rλ = (−Δ − λ)−1 . Now the operator of multiplication by V (x) = |x|−1 has the property (7.13)

MV : H 1 (R3 ) −→ L2−ε (R3 ),

7. The quantum Coulomb problem

151

for all ε > 0, since H 1 (R3 ) ⊂ L6 (R3 )∩L2 (R3 ) and V ∈ L3−ε on |x| < 1. Hence MV : H 1 (R3 ) −→ H −ε (R3 ), for all ε > 0. Let us apply this to (7.12). We know that u ∈ D(H) ⊂ D(H 1/2 ) = H 1 (R3 ), so KRλ V u ∈ H 2−ε (R3 ). Thus u ∈ H 2−ε (R3 ), for all ε > 0. But, for ε > 0 small enough, (7.14)

MV : H 2−ε (R3 ) −→ L2 (R3 ),

so then u = KRλ (V u) + Rλ f ∈ H 2 (R3 ). This proves that D(H) ⊂ H 2 (R3 ) and gives (7.11). Since H is self-adjoint, its spectrum is a subset of the real axis, (−∞, ∞). We next show that there is only point spectrum in (−∞, 0) Proposition 7.3. The part of spec H lying in C \ [0, ∞) is a bounded, discrete subset of (−∞, 0), consisting of eigenvalues of finite multiplicity and having at most {0} as an accumulation point. Proof. Consider the equation (H − λ)u = f ∈ L2 (R3 ), that is, (7.15)

(−Δ − λ)u − KV u = f,

with V (x) = |x|−1 as before. Applying Rλ = (−Δ − λ)−1 to both sides, we again obtain (7.12): (7.16)

(I − KRλ MV )u = gλ = Rλ f.

Note that Rλ is a holomorphic function of λ ∈ C \ [0, ∞), with values in L(L2 (R3 ), H 2 (R3 )). A key result in the analysis of (7.16) is the following: Lemma 7.4. For λ ∈ C \ [0, ∞), (7.17)

Rλ MV ∈ K(L2 (R3 )),

where K is the space of compact operators. We will establish this via the following basic tool. For λ ∈ C \ [0, ∞), ϕ ∈ C0 (R3 ), the space of continuous functions vanishing at infinity, we have (7.18)

Mϕ Rλ ∈ K(L2 ) and Rλ Mϕ ∈ K(L2 ).

To see this, note that, for ϕ ∈ C0∞ (R3 ), the first inclusion in (7.18) follows from Rellich’s theorem. Then this inclusion holds for uniform limits of such ϕ, hence for ϕ ∈ C0 (R3 ). Taking adjoints yields the rest of (7.18). Now, to establish (7.17), write

152 8. Spectral Theory

V = V1 + V 2 ,

(7.19)

where V1 = ψV, ψ ∈ C0∞ (R3 ), ψ(x) = 1 for |x| ≤ 1. Then V2 ∈ C0 (R3 ), so Rλ MV2 ∈ K. We have V1 ∈ Lq (R3 ), for all q ∈ [1, 3), so, taking q = 2, we have (7.20)

MV1 : L2 (R3 ) −→ L1 (R3 ) ⊂ H −3/2−ε (R3 ),

for all ε > 0, hence (7.21)

Rλ MV1 : L2 (R3 ) −→ H 1/2−ε (R3 ) ⊂ L2 (R3 ).

Given V1 supported on a ball BR , the operator norm in (7.21) is bounded by a constant times V1 L2 . You can approximate V1 in L2 -norm by a sequence wj ∈ C0∞ (R3 ). It follows that Rλ MV1 is a norm limit of a sequence of compact operators on L2 (R3 ), so it is also compact, and (7.17) is established. The proof of Proposition 7.4 is finished by the following result, which can be found as Proposition 7.4 in Chap. 9 Proposition 7.5. Let O be a connected, open set in C. Suppose C(λ) is a compact-operator-valued holomorphic function of λ ∈ O. If I − C(λ) is invertible at one point p ∈ O, then it is invertible except at most on a discrete set in O, and (I − C(λ))−1 is meromorphic on O. This applies to our situation, with C(λ) = KRλ MV ; we know that I − C(λ) is invertible for all λ ∈ C \ R in this case. One approach to analyzing the negative eigenvalues of H is to use polar coordinates. If −K|x|−1 is replaced by any radial potential V(|x|), the eigenvalue equation Hu = −Eu becomes (7.22)

∂ 2 u 2 ∂u 1 + + 2 ΔS u − V(r)u = Eu. ∂r r ∂r r

We can use separation of variables, writing u(rθ) = v(r)ϕ(θ), where ϕ is an eigenfunction of ΔS , the Laplace operator on S 2 , (7.23)

1 2 1 λ= k+ − = k 2 + k. 2 4

ΔS ϕ = −λϕ,

Then we obtain for v(r) the ODE (7.24)

2 v  (r) + v  (r) + f (r)v(r) = 0, r

f (r) = −E −

One can eliminate the term involving v  by setting (7.25) Then

w(r) = rv(r).

λ − V(r). r2

7. The quantum Coulomb problem

(7.26)

153

w (r) + f (r)w(r) = 0.

For the Coulomb problem, this becomes  λ K − 2 w(r) = 0. w (r) + −E + r r √ If we set W (r) = w(βr), β = 1/2 E, we get a form of Whittaker’s ODE:

(7.27)

(7.28)

 1 κ W  (z) + − + + 4 z

1 4

− μ2  W (z) = 0, z2

with (7.29)

K κ= √ , 2 E

μ2 = λ +

1 = 4

 k+

1 2

2 .

This in turn can be converted to the confluent hypergeometric equation (7.30)

zψ  (z) + (b − z)ψ  (z) − aψ(z) = 0

upon setting (7.31)

W (z) = z μ+1/2 e−z/2 ψ(z),

with

(7.32)

a=μ−κ+

K 1 =k+1− √ , 2 2 E

b = 2μ + 1 = 2k + 2. Note that ψ and v are related by (7.33)

√ √ √ v(r) = (2 E)k+1 rk e−2 Er ψ(2 Er).

Looking at (7.28), we see that there are two independent solutions, one behaving roughly like e−z/2 and the other like ez/2 , as z → +∞. Equivalently, (7.30) has two linearly independent solutions, a “good” one growing more slowly than exponentially and a “bad” one growing like ez , as z → +∞. Of course, for a solution to give rise to an eigenfunction, we need v ∈ L2 (R+ , r2 dr), that is, w ∈ L2 (R+ , dr). We need to have simultaneously w(z) ∼ ce−z/2 (roughly) as z → +∞ and w square integrable near z = 0. In view of (7.8), we also need v  ∈ L2 (R+ , r2 dr). To examine the behavior near z = 0, note that the Euler equation associated with (7.28) is

154 8. Spectral Theory

(7.34)

z 2 W  (z) +

1 4

 − μ2 W (z) = 0,

with solutions z 1/2+μ and z 1/2−μ , i.e., z k+1 and z −k , k = 0, 1, 2, . . . . If k = 0, both are square integrable near 0, but for k ≥ 1 only one is. Going to the confluent hypergeometric equation (7.30), we see that two linearly independent solutions behave respectively like z 0 and z −2μ = z −2k−1 as z → 0. As a further comment on the case k = 0, note that a solution W behaving like z 0 at z = 0 gives rise to v(r) ∼ C/r as r → 0, with c = 0, hence v  (r) ∼ −C/r2 . This is not square integrable near r = 0, with respect to r2 dr, so also this case does not produce an eigenfunction of H. If b ∈ / {0, −1, −2, . . . }, which certainly holds here, the solution to (7.30) that is “good” near z = 0 is given by the confluent hypergeometric function (7.35)

∞  (a)n z n , 1 F1 (a; b; z) = (b)n n! n=0

an entire function of z. Here, (a)n = a(a + 1) · · · (a + n − 1); (a)0 = 1. If also a∈ / {0, −1, −2, . . . }, it can be shown that (7.36)

1 F1 (a; b; z)



Γ(b) z −(b−a) e z , Γ(a)

z → +∞.

See the exercises below for a proof of this. Thus the “good” solution near z = 0 is “bad” as z → +∞, unless a is a nonpositive integer, say a = −j. In that case, as is clear from (7.35), 1 F1 (−j; b; z) is a polynomial in z, thus “good” as z → +∞. Thus the negative eigenvalues of H are given by −E, with (7.37)

K √ = j + k + 1 = n, 2 E

that is, by (7.38)

E=

K2 , 4n2

n = 1, 2, 3 . . . .

Note that, for each value of n, one can write n = j + k + 1 using n choices of k ∈ {0, 1, 2, . . . , n − 1}. For each such k, the (k 2 + k)-eigenspace of ΔS has dimension 2k + 1, as established in Corollary 4.3. Thus the eigenvalue −E = −K 2 /4n2 of H has multiplicity (7.39)

n−1  k=0

(2k + 1) = n2 .

7. The quantum Coulomb problem

155

Let us denote by Vn the n2 -dimensional eigenspace of H, associated to the eigenvalue λn = −K 2 /4n2 . The rotation group SO(3) acts on each Vn , via ρ(g)f (x) = f (g −1 x),

g ∈ SO(3), x ∈ R3

By the analysis leading to (7.39), this action on Vn is not irreducible, but rather has n irreducible components. This suggests that there is an extra symmetry, and indeed, as W. Pauli discovered early in the history of quantum mechanics, there is one, arising via the Lenz vector (briefly introduced in §16 of Chap. 1), which we proceed to define. The angular momentum vector L = x × p, with p replaced by the vector operator (∂/∂x1 , ∂/∂x2 , ∂/∂x3 ), commutes with H as a consequence of the rotational invariance of H. The components of L are L = xj

(7.40)

∂ ∂ − xk , ∂xk ∂xj

where (j, k, ) is a cyclic permutation of (1, 2, 3). Then the Lenz vector is defined by  x 1 L×p−p×L − , (7.41) B= K r with components Bj , 1 ≤ j ≤ 3, each of which is a second-order differential operator, given explicitly by (7.42)

Bj =

1 xj (Lk ∂ + ∂ Lk − L ∂k − ∂k L ) − , K r

where (j, k, ) is a cyclic permutation of (1, 2, 3). A calculation gives (7.43)

[H, Bj ] = 0,

in the sense that these operators commute on C ∞ (R3 \ 0). It follows that if u ∈ Vn , then Bj u is annihilated by H − λn , on R3 \ 0. Now, we have just gone through an argument designed to glean from all functions that are so annihilated, those that are actually eigenfunctions of H. In view of that, it is important to establish the next lemma Lemma 7.6. We have (7.44)

Bj : Vn −→ Vn .

Proof. Let u ∈ Vn . We know that u ∈ D(H) = H 2 (R3 ). Also, from the analysis of the ODE (7.28), we know that u(x) decays as |x| → ∞, roughly like 1/2 e−|λn | |x| . It follows from (7.42) that Bj u ∈ L2 (R3 ). It will be useful to obtain a bit more regularity, using Vn ⊂ D(H 2 ) together with the following.

156 8. Spectral Theory

Proposition 7.7. If u ∈ D(H 2 ), then, for all ε > 0, u ∈ H 5/2−ε (R3 ).

(7.45) Furthermore,

g ∈ S(R3 ), g(0) = 0 =⇒ gu ∈ H 7/2−ε (R3 ).

(7.46)

Proof. We proceed along the lines of the proof of Proposition 7.2, using (7.12), i.e., u = KRλ V u + Rλ f,

(7.47)

where f = (H − λ)u, with λ chosen in C \ R. We know that f = (H − λ)u belongs to D(H), so Rλ f ∈ H 4 (R3 ). We know that u ∈ H 2 (R3 ). Parallel to (7.13), we can show that, for all ε > 0, MV : H 2 (R3 ) −→ H 1/2−ε (R3 ),

(7.48)

so KRλ V u ∈ H 5/2−ε (R3 ). This gives (7.45). Now, multiply (7.47) by g and write gu = KRλ gV u + K[Mg , Rλ ]V u + gRλ f.

(7.49) This time we have

MgV : H 2 (R3 ) −→ H 3/2−ε (R3 ), so Rλ gV u ∈ H 7/2−ε (R3 ). Furthermore, (7.50)

[Mg , Rλ ] = Rλ [Δ, Mg ] Rλ : H s (R3 ) −→ H s+3 (R3 ),

so [Mg , Rλ ]V u ∈ H 7/2−ε (R3 ). This establishes (7.46). We can now finish the proof of Lemma 7.6. Note that the second-order derivatives in Bj have a coefficient vanishing at 0. Keep in mind the known exponential decay of u ∈ Vn . Also note that Mxj /r : H 2 (R3 ) → H 3/2−ε (R3 ). Therefore, u ∈ Vn =⇒ Bj u ∈ H 3/2−ε (R3 ).

(7.51) Consequently, (7.52)

Δ(Bj u) ∈ H −1/2−ε (R3 ), and V (Bj u) ∈ L1 (R3 ) + L2 (R3 ).

7. The quantum Coulomb problem

157

Thus (H −λn )(Bj u), which we know vanishes on R3 \0, must vanish completely, since (7.52) does not allow for a nonzero quantity supported on {0}. Using (7.8), we conclude that Bj u ∈ D(H), and the lemma is proved. With Lemma 7.6 established, we can proceed to study the action of Bj and Lj on Vn . When (j, k, ) is a cyclic permutation of (1, 2, 3), we have [Lj , Lk ] = L ,

(7.53) and, after a computation, (7.54)

[Lj , Bk ] = B ,

[Bj , Bk ] = −

4 HL . K

Of course, (7.52) is the statement that Lj span the Lie algebra so(3) of SO(3). The identities (7.54), when Lj and Bj act on Vn , can be rewritten as (7.55)

[Lj , Ak ] = A ,

[Aj , Ak ] = A ,

K Aj = √ Bj . 2 −λn

If we set (7.56)

M=

1 (L + A), 2

N=

1 (L − A), 2

we get, for cyclic permutations (j, k, ) of (1, 2, 3), (7.57)

[Mj , Mk ] = M ,

[Nj , Nk ] = N ,

[Mj , Nj  ] = 0,

which is clearly the set of commutation relations for the Lie algebra so(3)⊕so(3). We next aim to show that this produces an irreducible representation of SO(4) on Vn , and to identify this representation. A priori, of course, one certainly has a representation of SU(2) × SU(2) on Vn . We now examine the behavior on Vn of the Casimir operators M 2 = M12 + M22 + M32 and N 2 . A calculation using the definitions gives B · L = 0, hence A · L = 0, so, on Vn , 1 2 (A + L2 ) 4 1 2 K 2 2 = L − B . 4 4λn

M2 = N2 = (7.58)

We also have the following key identity: (7.59)

K 2 (B 2 − I) = 4H(L2 + I),

which follows from the definitions by a straightforward computation. If we compare (7.58) and (7.59) on Vn , where H = λn , we get

158 8. Spectral Theory

(7.60)

 K2  I 4M 2 = 4N 2 = − 1 + 4λn

on Vn .

Now the representation σn we get of SU(2) × SU(2) on Vn is a direct sum (possibly with only one summand) of representations Dj/2 ⊗ Dj/2 , where Dj/2 is the standard irreducible representation of SU(2) on Cj+1 , defined in §9 of Appendix B. The computation (7.60) implies that all the copies in this sum are isomorphic, that is, for some j = j(n), (7.61)

σn =

μ #

Dj(n)/2 ⊗ Dj(n)/2 .

=1



2 A dimension count gives μ j(n) + 1 = n2 . Note that on Dj/2 ⊗ Dj/2 , we have M 2 = N 2 = (j/2)(j/2 + 1). Thus (7.60) implies j(j + 2) = −1 + K 2 /4λn , or (7.62)

λn = −

K2 , 4(j + 1)2

j = j(n).

Comparing (7.38), we have (j + 1)2 = n2 , that is, (7.63)

j(n) = n − 1.

Since we know that dim Vn = n2 , this implies that there is just one summand in (7.61), so (7.64)

σn = D(n−1)/2 ⊗ D(n−1)/2 .

This is an irreducible representation of SU(2) × SU(2), which is a double cover of SO(4), κ : SU(2) × SU(2) −→ SO(4). It is clear that σn is the identity operator on both elements in ker κ, and so σn actually produces an irreducible representation of SO(4). Let ρn denote the restriction to Vn of the representation ρ of SO(3) on L3 (R3 ), described above. If we regard this as a representation of SU(2), it is clear that ρn is the composition of σn with the diagonal map SU(2) → SU(2)×SU(2). Results established in §9 of Appendix B imply that such a tensor-product representation of SU(2) has the decomposition into irreducible representations: (7.65)

ρn ≈

n−1 #

Dk .

k=0

This is also precisely the description of ρn given by the analysis leading to (7.39).

Exercises

159

There are a number of other group-theoretic perspectives on the quantum Coulomb problem, which can be found in [Eng] and [GS2]. See also [Ad] and [Cor], Vol. 2.

Exercises 1. For H = −Δ − K|x|−1 with domain given by (7.8), show that (7.66)

D(H) = {u ∈ L2 (R3 ) : −Δu − K|x|−1 u ∈ L2 (R3 )},

where a priori, if u ∈ L2 (R3 ), then Δu ∈ H −2 (R3 ) and |x|−1 u ∈ L1 (R3 ) + L2 (R3 ) ⊂ H −2 (R3 ). (Hint: Parallel the proof of Proposition 7.2. If u belongs to the right side of (7.66), and if you pick λ ∈ C \ R, then, as in (7.12), u − KRλ V u = Rλ f ∈ H 2 (R3 ).)

(7.67) Complement (7.13) with

MV : L2 (R3 ) −→ 

(7.68) MV :



H −3/2−ε (R3 ),

ε>0

H

1/2−ε

(R3 ) −→

ε>0



H −3/4−δ (R3 ).

δ>0

(Indeed, sharper results can be obtained.) Then deduce from (7.67) first that u ∈ H 1/2−ε (R3 ) and then that u ∈ H 5/4−δ (R3 ) ⊂ H 1 (R3 ).) 2. As a variant of (7.4), show that, for u ∈ H 1 (R3 ),   (7.69) |x|−2 |u(x)|2 dx ≤ 4 |∇u(x)|2 dx. Show that 4 is the best possible constant on the right. (Hint: Use the Mellin transform to show that the spectrum of r d/dr − 1/2 on L2 (R+ , r−1 dr) (which coincides with the spectrum of r d/dr on L2 (R+ , dr)) is {is − 1/2 : s ∈ R}, hence  ∞  ∞ (7.70) |u(r)|2 dr ≤ 4 |u (r)|2 r2 dr. 0

0

This is sometimes called an “uncertainty principle” estimate. Why might that be? (Cf. [RS], Vol. 2, p. 169.) 3. Show that H = −Δ − K/|x| has no non-negative eigenvalues, i.e., only continuous spectrum in [0, ∞). (Hint: Study the behavior as r → +∞ of solutions to the ODE (7.28), when −E is replaced by +E ∈ [0, ∞). Consult [Olv] for techniques. See also [RS], Vol. 4, for general results.) 4. Generalize the propositions of this section, with modifications as needed, to other classes of potentials V (x), such as V ∈ L2 + εL∞ , the set of functions V such that, for each ε > 0, one can write V = V1 + V2 , V1 ∈ L2 , V2 L∞ ≤ ε. Consult [RS], Vols. 2–4, for further generalizations.

160 8. Spectral Theory

Exercises on the confluent hypergeometric function 1. Taking (7.35) as the definition of 1 F1 (a; b; z), show that  1 Γ(b) ezt ta−1 (1 − t)b−a−1 dt, 1 F1 (a; b; z) = Γ(a)Γ(b − a) 0 (7.71) Re b > Re a > 0. (Hint: Use the beta function identity, (A.23)–(A.24) of Chap. 3.) Show that (7.71) implies the asymptotic behavior (7.36), provided Re b > Re a > 0, but that this is insufficient for making the deduction (7.37). Exercises 2–5 deal with the analytic continuation of (7.71) in a and b, and a complete justification of (7.36). To begin, write Γ(b) Γ(b) Aψ (a, −z) + Aϕ (b − a, z)ez , Γ(b − a) Γ(a)   where, for Re c > 0, ψ ∈ C ∞ [0, 1/2] , we set

(7.72)

1 F1 (a; b; z)

(7.73)

=

1 Γ(c)

Aψ (c, z) =



1/2

e−zt ψ(t)tc−1 dt,

0

and, in (7.72), ψ(t) = (1 − t)b−a−1 ,

ϕ(t) = (1 − t)a−1 .

2. Given Re c > 0, show that Aψ (c, z) ∼ ψ(0)z −c ,

(7.74)

z → +∞,

and (7.75)

Aψ (c, −z) ∼

ψ( 12 ) −1 z/2 z e , Γ(c)

z → +∞.

3. For j = 0, 1, 2, . . . , set (7.76)

Aj (c, t) =

1 Γ(c)



1/2

e−zt tj tc−1 dt,

0

so Aj (c, z) = Aψ (c, z), with ψ(t) = tj . Show that  ∞ Γ(c + j) −c−j 1 − e−zt tc+j−1 dt, z Aj (c, z) = Γ(c) Γ(c) 1/2 for Re z > 0. Deduce that Aj (c, t) is an entire function of c, for Re z > 0, and that Aj (c, z) ∼ if c ∈ / {0, −1, −2, . . . }. 4. Given k = 1, 2, 3, . . . , write

Γ(c + j) −c−j , z Γ(c)

z → +∞,

Exercises ψ(t) = a0 + a1 t + · · · + ak−1 tk−1 + ψk (t)tk ,

ψk ∈ C ∞

161

 1  0, 2

Thus (7.77)

Aψ (c, z) =

k−1 

aj Aj (c, z) +

j=0

1 Γ(c)



1/2

e−zt ψk (t)tk+c−1 dt.

0

Deduce that Aψ (c, z) can be analytically continued to Re c > −k when Re z > 0 and that (7.74) continues to hold if c ∈ / {0, −1, −2, . . . }, a0 = 0. 5. Using tc−1 = c−1 (d/dt)tc and integrating by parts, show that (7.78)

A0 (c, z) = zA0 (c + 1, z) −

1 e−z/2 , 2c Γ(c + 1)

for Re c > 0, all z ∈ C. Show that this provides an entire analytic continuation of A0 (c, z) and that (7.74)–(7.75) hold, for ψ(t) = 1. Using Γ(c + j) A0 (c + j, z) Γ(c)   and (7.77), verify (7.75) for all ψ ∈ C ∞ [0, 1/2] . (Also again verify (7.74)). Hence, verify the asymptotic expansion (7.36). The approach given above to (7.36) is one the author learned from conversations with A. N. Varchenko. In Exercises 6–15 below, we introduce another solution to the confluent hypergeometric equation and follow a path to the expansion (7.36) similar to one described in [Leb] and in [Olv]. 6. Show that a solution to the ODE (7.30) is also given by Aj (c, z) =

z 1−b 1 F1 (1 + a − b; 2 − b; z), in addition to 1 F1 (a; b; z), defined by (7.35). Assume b = 0, −1, −2, . . . . Set Ψ(a; b; z) = (7.79)

Γ(1 − b) 1 F1 (a; b; z) Γ(1 + a − b) Γ(b − 1) 1−b + z 1 F1 (1 + a − b; 2 − b; z). Γ(a)

Show that the Wronskian is given by   Γ(b) −b z W 1 F1 (a; b; z), Ψ(a; b; z) = − z e . Γ(a) 7. Show that (7.80)

1 F1 (a; b; z)

= ez 1 F1 (b − a; b; −z),

b∈ / {0, −1, −2, . . . }

(Hint: Use the integral in Exercise 1, and set s = 1−t, for the case Re b > Re a > 0.) 8. Show that  ∞ 1 e−zt ta−1 (1 + t)b−a−1 dt, Re a > 0, Re z > 0. (7.81) Ψ(a; b; z) = Γ(a) 0 (Hint: First show that the right side solves (7.30). Then check the behavior as z → 0.) 9. Show that

162 8. Spectral Theory (7.82)

Ψ(a; b; z) = zΨ(a + 1; b + 1; z) + (1 − a − b)Ψ(a + 1; b; z).

(Hint: To get this when Re a > 0, use the integral expression (7.81) for Ψ(a + 1; b + 1; z), write ze−zt = −(d/dt)e−zt , and integrate by parts.) 10. Show that Γ(b) e±πai Ψ(a; b; z) 1 F1 (a; b; z) = Γ(b − a) Γ(b) ±π(a−b)i z (7.83) e Ψ(b − a; b; −z), e + Γ(a) where −z = e∓πi z, b = 0, −1, −2, . . . . (Hint: Make use of (7.80) as well as (7.79).) 11. Using the integral representation (7.81), show that under the hypotheses δ > 0, b ∈ / {0, −1, −2, . . . }, and Re a > 0, we have (7.84)

Ψ(a; b; z) ∼ z −α ,

|z| → ∞,

in the sector |Arg z| ≤

(7.85)

π − δ. 2

12. Extend (7.84) to the sector |Arg z| ≤ π − δ. (Hint: Replace (7.81) by an integral along the ray γ = {eiα s : 0 ≤ s < ∞}, given |α| < π/2.) 13. Further extend (7.84) to the case where no restriction is placed on Re a. (Hint: Use (7.82).) 14. Extend (7.84) still further, to be valid for |Arg z| ≤

(7.86)

3π − δ. 2

(Hint: See Theorem 2.2 on p. 235 of [Olv], and its application to this problem on p. 256 of [Olv].) 15. Use (7.83)–(7.86) to prove (7.36), that is, (7.87)

1 F1 (a; b; z)



Γ(b) z −(b−a) , e z Γ(a)

z → +∞,

provided a, b ∈ / {0, −1, −2, . . . }.

Remarks: For the analysis of Ψ(b − a; b; −z) as z → +∞, the result of Exercise 14 suffices, but the result of Exercise 13 does not. This point appears to have been neglected in the discussion of (7.87) on p. 271 of [Leb].

8. Potential well–quantum model of a deuteron In this section, we consider a quantum model for the deuteron, and compute its ground state. We see that, in this model, the nucleons have a greater probability of lying outside the potential well than in it, as noted in nuclear physics texts, such as [BM], [F], and [S].

8. Potential well–quantum model of a deuteron

163

We take a simple potential well. In more detail, given a, V0 ∈ (0, ∞), x ∈ R3 , we set V (x) = −V0 , |x| < a,

(8.1)

0,

|x| > a.

We consider whether −Δ + V has negative eigenvalues, and if so, how its ground state behaves. If −Δ + V has negative eigenvalues, denote by −E the one with largest absolute value. We must have E ∈ (0, V0 ), and the ground state will be given by a function ψ ∈ C 1 (R3 ), rapidly decreasing at infinity, positive and radially symmetric, satisfying (8.2)

on R3 .

Δψ = [V (x) + E]ψ

In particular, with r = |x|, ψ(x) =

(8.3)

u(r) , r

where u ∈ C 1 ((0, ∞)) satisfies (8.4)

u (r) = [V (r) + E]u(r).

The properties of E and ψ detailed above demand that, for some A, B ∈ (0, ∞), (8.5)

u(r) = A sin kr, −γr

Be

r ≤ a,

, r ≥ a,

with (8.6)

k=

$ V0 − E,

γ=

√ E.

The fact that u ∈ C 1 ((0, ∞)) yields the relations (8.7)

A sin ka = Be−γa ,

kA cos ka = −Bγe−γa ,

hence (8.8)

B = Aeγa sin ka,

k cot ka = −γ.

Also (8.9)

ψ > 0 =⇒ 0 < ka < π =⇒ (V0 − E)a2 < π 2 .

164 8. Spectral Theory

Note that A is a positive multiple of B; hence the second part of (8.7) yields cos ka < 0,

(8.10)

so ka >

π . 2

Comparison with (8.9) gives (8.11)

π < ka < π, 2

π2 < (V0 − E)a2 < π 2 . 4

hence

In particular: Proposition 8.1. If −Δ + V has a negative eigenvalue, then V0 a2 >

(8.12)

π2 . 4

Given that there is a negative eigenvalue −E with largest absolute value, we next strive for a formula: E = E(V0 , a). To get this, it is convenient to set E = δ 2 V0 , 0 < δ < 1, π π ka = + ε, 0 < ε < , 2 2

(8.13)

and get formulas relating these quantities. Note that cot ka = cot

(8.14)

π 2

 + ε = − tan ε,

and bringing in (8.8) we have (8.15)

γ tan ε = = k

%

E = V0 − E

%

δ2 , 1 − δ2

δ ∈ (0, 1).

Equivalently, δ = sin ε,

(8.16)

0 π 2 /4, we have 2 1 π + ε , cos2 ε 2 =⇒ E = V0 sin2 ε,

V0 a2 =

(8.18)

0 a} is much larger than the integral over {|x| < a}, for ε small enough. As for how small ε is, we note that (8.18) plus the identity E = γ 2 yield (8.23)

π 2

 + ε tan ε = γa.

Information on a and on E (hence on γ) would allow one to solve for ε, and then for V0 = E/ sin2 ε. We next see how this plays out for the deuteron, for which (8.2) arises as a model for the ground state. Actually, (8.2) is the nondimensionalized form. The physical form is (8.24)

Δψ =

2m & & [V (x) + E]ψ, 2

where, with m = mp mn /(mp + mn ) ≈ mp /2 and c ≈ 3 × 108 m/sec, (8.25)

2m ≈ mass of a proton ≈ 938 MeV/c2 ,  = Planck’s constant ≈ 6.6 × 10−22 MeV-sec,

8. Potential well–quantum model of a deuteron

167

& are measured in MeV. This leads to (8.2) with and V& (x) and E (8.26)

V (x) =

2m & V (x), 2

E=

2m & E, 2

where V& (x) = −V&0 on |x| < a, and V0 = (2m/2 )V&0 . Experiments shooting gamma rays at deuterium show that & ≈ 2.225 MeV. E

(8.27) This corresponds via γ =



E=

$

& to 2mE/

γ −1 ≈ 4.32 fm,

(8.28)

where 1 fm=10−15 m. The meson model of nuclear forces suggests a ≈ 2.8 fm.

(8.29) Cf. [S], p. 449. This gives

γa ≈ 0.648,

(8.30) and solving (8.23) then gives

ε ≈ 0.329.

(8.31) Hence

δ ≈ 0.323,

(8.32) so

& ≈ 21.34 MeV. V&0 = δ −2 E

(8.33)

Referring to (8.20)–(8.21), we see that in this case  (8.34) |x|a

4πA2 |ψ(x)|2 dx ≈ (1.387) √ . V0

168 8. Spectral Theory

The figure (8.31) for ε is not terribly consistent with the hypothesis that & The figures ε > E. & for γ and V0 given in (8.28) and (8.33) agree with those given in [S] (p. 449). We note however that the integral (8.35) is only a little larger than (8.34). This contrasts with the statement in [S] that it is “about twice as large.” In more detail, the ratio of (8.35) to (8.34) is (8.36)

R≈

1.387 ≈ 1.191, 1.165

which is not close to 2. On the other hand, if we take ε as in (8.31) and plug it into the “small ε approximation” (8.22), we get the “approximation” (8.37)

R≈

2 ≈ 1.935, πε

in close agreement with the calculation in [S]. However, (8.36) seems to be a more accurate calculation for the model at hand. Exercises 1. Formulate conditions on V0 and a in (8.1) that imply that L = −Δ + V has exactly one negative eigenvalue. 2. Give conditions on V0 and a under which L must have more than one negative eigenvalue.

9. The Laplace operator on cones Generally, if N is any compact Riemannian manifold of dimension m, possibly with boundary, the cone over N, C(N ), is the space R+ × N together with the Riemannian metric (9.1)

dr2 + r2 g,

where g is the metric tensor on N . In particular, a cone with vertex at the origin in Rm+1 can be described as the cone over a subdomain Ω of the unit sphere S m in Rm+1 . Our purpose is to understand the behavior of the Laplace operator Δ, a negative, self-adjoint operator, on C(N ). If ∂N = ∅, we impose Dirichlet boundary conditions on ∂C(N ), though many other boundary conditions could be equally easily treated. The analysis here follows [CT]. The initial step is to use the method of separation of variables, writing Δ on C(N ) in the form

9. The Laplace operator on cones

Δ=

(9.2)

169

1 ∂2 m ∂ + 2 ΔN , + 2 ∂r r ∂r r

where ΔN is the Laplace operator on the base N . Let μj , ϕj (x) denote the eigenvalues and eigenfunctions of −ΔN (with Dirichlet boundary condition on ∂N if ∂N = ∅), and set νj = (μj + α2 )1/2 ,

(9.3) If

g(r, x) =



α=−

m−1 . 2

gj (r)ϕj (x),

j

with gj (r) well behaved, and if we define the second-order operator Lμ by  2  ∂ μ m ∂ − + (9.4) Lμ g(r) = g(r), ∂r2 r ∂r r2 then we have Δg(r, x) =

(9.5)



Lμj gj (r)ϕj (x).

j

In particular, Δ(gj ϕj ) = −λ2 gj ϕj

(9.6) provided

gj (r) = r−(m−1)/2 Jνj (λr).

(9.7)

Here Jν (z) is the Bessel function, introduced in §6 of Chap. 3; there in (6.6) it is defined to be  1 (z/2)ν (9.8) Jν (z) = (1 − t2 )ν−1/2 eizt dt, Γ( 12 )Γ(ν + 12 ) −1 for Re ν > −1/2; in (6.11) we establish Bessel’s equation ' (9.9)

 ( d2 ν2 1 d + 1− 2 + Jν (z) = 0, dz 2 z dz z

which justifies (9.6); and in (6.19) we produced the formula (9.10)

Jν (z) =

∞  z ν 

2

k=0

 z 2k (−1)k . k!Γ(k + ν + 1) 2

170 8. Spectral Theory

We also recall, from (6.56) of Chap. 3, the asymptotic behavior (9.11)

Jν (r) ∼

 2 1/2  π πν − cos r − + O(r−3/2 ), πr 2 4

r → +∞.

This suggests making use of the Hankel transform, defined for ν ∈ R+ by  ∞ (9.12) Hν (g)(λ) = g(r)Jν (λr)r dr. 0



Clearly, Hν : C0∞ (0, ∞) → L∞ (R+ ). We will establish the following:

Proposition 9.1. For ν ≥ 0, Hν extends uniquely from C0∞ (0, ∞) to (9.13)

Hν : L2 (R+ , r dr) −→ L2 (R+ , λ dλ), unitary.

Furthermore, for each g ∈ L2 (R+ , r dr), Hν ◦ Hν g = g.

(9.14)

To prove this, it is convenient to consider first  ∞ Jν (λr) 2ν+1 & ν f (λ) = f (r) r dr, (9.15) H (λr)ν 0 since, by (9.10), (λr)−ν Jν (λr) is a smooth function of λr. Set (9.16)

S(R+ ) = {f |R+ : f ∈ S(R) is even}.

Lemma 9.2. If ν ≥ −1/2, then (9.17)

& ν : S(R+ ) −→ S(R+ ). H

Proof. By (9.10), Jν (λr)/(λr)ν is a smooth function of λr. The formula (9.8) yields ) J (λr) ) ) ) ν (9.18) ) ≤ Cν < ∞, ) (λr)ν for λr ∈ [0, ∞), ν > −1/2, a result that, by the identity (9.19)

J−1/2 (z) =

 2 1/2 cos z, πz

established in (6.35) of Chap. 3, also holds for ν = −1/2. This readily yields (9.20)

& ν : S(R+ ) −→ L∞ (R+ ), H

9. The Laplace operator on cones

171

& ν , given by whenever ν ≥ −1/2. Now consider the differential operator L   & ν f (r) = −r−2ν−1 ∂ r2ν+1 ∂f L ∂r ∂r

(9.21)

=−

∂2f 2ν + 1 ∂f . − 2 ∂r r ∂r

Using Bessel’s equation (9.9), we have   & ν Jν (λr) = λ2 Jν (λr) , L (λr)ν (λr)ν

(9.22) and, for f ∈ S(R+ ),

& ν f (λ), & ν (L & ν f )(λ) = λ2 H H & ν (r2 f )(λ) = L & ν f (λ). &ν H H

(9.23)

Since f ∈ L∞ (R+ ) belongs to S(R+ ) if and only if arbitrary iterated applications & ν and multiplication by r2 to f yield elements of L∞ (R+ ), the result (9.17) of L follows. We also have that this map is continuous with respect to the natural Frechet space structure on S(R+ ). Lemma 9.3. Consider the elements Eb ∈ S(R+ ), given for b > 0 by 2

Eb (r) = e−br .

(9.24) We have

& ν E1/2 (λ) = E1/2 (λ), H

(9.25) and more generally (9.26)

& ν Eb (λ) = (2b)−ν−1 E1/4b (λ). H

Proof. To establish (9.25), plug the power series (9.10) for Jν (z) into (9.15) and integrate term by term, to get (9.27)

& ν E1/2 (λ) = H

 ∞  (−1)k 2−ν−2k 2k ∞ 2k+2ν+1 −r2 /2 λ r e dr. k!Γ(k + ν + 1) 0

k=0

This last integral is seen to equal 2k+ν Γ(k + ν + 1), so we have

172 8. Spectral Theory

& ν E1/2 (λ) = H

(9.28)

∞  2 1  λ 2 k = e−λ /2 = E1/2 (λ). − k! 2

k=0

Having (9.25), we get (9.26) by an easy change of√variable argument. √ In more detail, set r2 /2 = bs2 , or s = r/ 2b. Then set μ = 2bλ, so λr = μs. Then (9.28), which we can write as 



(9.29)

e−r

0

2

/2

2

Jν (λr)rν+1 dr = λν e−λ

/2

,

translates to  ∞ 2 2 (9.30) e−bs Jν (μs)(2b)(ν+1)/2 sν+1 (2b)1/2 ds = (2b)−ν/2 μν e−μ /4b , 0

or, changing notation back, 



(9.31) 0

2

2

e−bs Jν (λs)sν+1 ds = (2b)−ν−1 λν e−λ

/4b

,

which gives (9.26). From (9.26) we have, for each b > 0, (9.32)

& ν Eb = (2b)−ν−1 H & ν E1/4b = Eb , &ν H H

which verifies our stated Hankel inversion formula for f = Eb , b > 0. To get the inversion formula for general f ∈ S(R+ ), it suffices to establish the following. Lemma 9.4. The space V = Span {Eb : b > 0}

(9.33) is dense in S(R+ ).

Proof. Let V denote the closure of V in S(R+ ). From (9.34)

2

2 1 −br2 e − e−(b+ε)r → r2 e−br , ε 2

we deduce that r2 e−br ∈ V, and inductively, we get (9.35)

2

r2j e−br ∈ V,

∀ j ∈ Z+ .

From here, one has (9.36)

2

(cos ξr)e−r ∈ V,

∀ ξ ∈ R.

9. The Laplace operator on cones

173

Now each even ω ∈ S  (R) annihilating (9.36) for all ξ ∈ R has the property that 2 e−r ω has Fourier transform zero, which implies ω = 0. The assertion (9.33) then follows by the Hahn-Banach theorem. Putting the results of Lemmas 9.2–9.4 together, we have Proposition 9.5. Given ν ≥ −1/2, we have &ν H & ν f = f, H

(9.37) for all f ∈ S(R+ ). We promote this to

& ν from S(R+ ) to Proposition 9.6. If ν ≥ −1/2, we have a unique extension of H (9.38)

& ν : L2 (R+ , r2ν+1 dr) −→ L2 (R+ , λ2ν+1 dλ), H

as a unitary operator, and (9.37) holds for all f ∈ L2 (R+ , r2ν+1 dr). Proof. Take f, g ∈ S(R+ ), and use the inner product  (9.39)

(f, g) =



f (r)g(r)r2ν+1 dr.

0

Using Fubini’s theorem and the fact that Jν (λr)/(λr)ν is real valued and symmetric in (λ, r), we get the first identity in (9.40)

& ν f, g) = (f, g), & ν f, H & ν g) = (H &ν H (H

the second identity following by Proposition 9.5. From here, given that the linear space S(R+ ) ⊂ L2 (R+ , r2ν+1 dr) is dense, the assertions of Proposition 9.6 are apparent. We return to the Hankel transform (9.12). Note that (9.41)

& ν f (λ), Hν (rν f )(λ) = λν H

and that Mν f (r) = rν f (r) has the property that (9.42)

Mν : L2 (R+ , r2ν+1 dr) −→ L2 (R+ , r dr) is unitary.

Thus Proposition 9.6 yields Proposition 9.1. Another proof is sketched in the exercises. An elaboration of Hankel’s original proof is given on pp. 456–464 of [Wat]. In view of (9.23) and (9.41), we have

174 8. Spectral Theory

Hν (r (9.43)

−α

 Lμ g) =



Lμ (rα Jν (λr))g rm dr  ∞ 2 grα Jν (λr)rm dr = −λ 0

0

= −λ2 Hν (r−α g). Now from (9.5)–(9.13), it follows that the map H given by   Hg = Hν0 (r−α g0 ), Hν1 (r−α g1 ), . . .

(9.44)

provides an isometry of L2 (C(N )) onto L2 (R+ , λ dλ, 2 ), such that Δ is carried into multiplication by −λ2 . Thus (9.44) provides a spectral representation of Δ. Consequently, for well-behaved functions f , we have

(9.45)

f (−Δ)g(r, x)   ∞ = rα f (λ2 )Jνj (λr)λ j

0

∞ 0

s1−α Jνj (λs)gj (s) ds dλ ϕj (x).

Now we can interpret (9.45) in the following fashion. Define the operator ν on N by (9.46)

1/2 . ν = −ΔN + α2

Thus νϕj = νj ϕj . Identifying operators with their distributional kernels, we can describe the kernel of f (−Δ) as a function on R+ × R+ taking values in operators on N , by the formula  ∞ α f (−Δ) = (r1 r2 ) f (λ2 )Jν (λr1 )Jν (λr2 )λ dλ (9.47) 0 = K(r1 , r2 , ν), since the volume element on C(N ) is rm dr dS(x) if the m-dimensional area element of N is dS(x). At this point it is convenient to have in hand some calculations of Hankel transforms, including some examples of the form (9.47). We establish some here; many more can be found in [Wat]. Generalizing (9.31), we can compute  ∞ −br 2 e Jν (λr)rμ+1 dr in a similar fashion, replacing the integral in (9.27) by 0  ∞  μ ν 2 1 + +k+1 . r2k+μ+ν+1 e−br dr = b−k−μ/2−ν/2−1 Γ (9.48) 2 2 2 0 We get

9. The Laplace operator on cones

 (9.49)



0

175

2

e−br Jν (λr)rμ+1 dr = λν 2−ν−1 b−μ/2−ν/2−1

∞  Γ( μ2 + ν2 + k + 1)  λ2 k − . k!Γ(k + ν + 1) 4b

k=0

We can express the infinite series in terms of the confluent hypergeometric function, introduced in §7. A formula equivalent to (7.35) is ∞

Γ(b)  Γ(a + k) z k , 1 F1 (a; b; z) = Γ(a) Γ(b + k) k!

(9.50)

k=0

since (a)k = a(a + 1) · · · (a + k − 1) = Γ(a + k)/Γ(a). We obtain, for Re b > 0, Re(μ + ν) > −2, (9.51)  ∞ 2 e−br Jν (λr)rμ+1 dr 0

= λν 2−ν−1 b−μ/2−ν/2−1

μ ν Γ( μ2 + ν2 + 1) λ2  + + 1; ν + 1; − . 1 F1 Γ(ν + 1) 2 2 4b 2

We can apply a similar attack when e−br is replaced by e−br , obtaining 



0

(9.52)

e−br Jν (λr)rμ−1 dr =

 λ ν 2

b−μ−ν

∞  Γ(μ + ν + 2k)  λ2 k − 2 , k!Γ(ν + k + 1) 2b

k=0

at least provided Re b > |λ|, ν ≥ 0, and μ + ν > 0; here we use  (9.53)



e−br r2k+μ+ν−1 dr = b−2k−μ−ν Γ(μ + ν + 2k).

0

The duplication formula for the gamma function (see (A.22) of Chap. 3) implies (9.54) Γ(2k + μ + ν) = π −1/2 22k+μ+ν−1 Γ

μ 2

+

 μ ν ν 1 +k Γ + +k+ , 2 2 2 2

so the right side of (9.52) can be rewritten as (9.55)

π −1/2 λν 2μ−1 b−μ−ν

∞  Γ( μ2 + k=0

ν 2

+ k)Γ( μ2 + ν2 + k!Γ(ν + 1 + k)

1 2

+ k)  λ2 k − 2 . b

This infinite series can be expressed in terms of the hypergeometric function, defined by

176 8. Spectral Theory 2 F1 (a1 , a2 ; b; z) =

∞  (a1 )k (a2 )k z k (b)k k!

k=0

(9.56)



=

 Γ(a1 + k)Γ(a2 + k) z k Γ(b) , Γ(a1 )Γ(a2 ) Γ(b + k) k! k=0

for a1 , a2 ∈ / {0, −1, −2, . . . }, |z| < 1. If we put the sum in (9.55) into this form, and use the duplication formula, to write μ ν  μ ν 1 + Γ + + = π 1/2 2−μ−ν+1 Γ(μ + ν), Γ(a1 )Γ(a2 ) = Γ 2 2 2 2 2 we obtain  (9.57)

0

=



e−br Jν (λr)rμ−1 dr

 λ ν 2

b

−μ−ν

Γ(μ + ν) · 2 F1 Γ(ν + 1)



μ ν μ ν 1 λ2 + , + + ; ν + 1; − 2 2 2 2 2 2 b

 .

This identity, established so far for |λ| < Re b (and ν ≥ 0, μ + ν > 0), continues analytically to λ in a complex neighborhood of (0, ∞). 2 To evaluate the integral (9.47) with f (λ2 ) = e−tλ , we can use the power series (9.10) for Jν (λr1 ) and for Jν (λr2 ) and integrate the resulting double series term by term using (9.48). We get (9.58)  ∞ 2 e−tλ Jν (r1 λ)Jν (r2 λ)λ dλ 0

=

 1  r12 j  r22 k Γ(ν + j + k + 1) 1  r1 r2 ν − − × , 2t 4t Γ(ν + j + 1)Γ(ν + k + 1) j!k! 4t 4t j,k≥0

for any t, r1 , r2 > 0, ν ≥ 0. This can be written in terms of the modified Bessel function Iν (z), given by Iν (z) =

(9.59)

∞  z ν 

2

k=0

 z 2k 1 . k!Γ(ν + k + 1) 2

One obtains the following, known as the Weber identity. Proposition 9.7. For t, r1 , r2 > 0,  (9.60) 0



2

e−tλ Jν (r1 λ)Jν (r2 λ)λ dλ =

1 −(r12 +r22 )/4t  r1 r2  e . Iν 2t 2t

Proof. The left side of (9.60) is given by (9.58). Meanwhile, by (9.59), the right side of (9.60) is equal to (1/2t)(r1 r2 /4t)ν times

9. The Laplace operator on cones



(9.61)

,m≥0

177

 r r 2n 1  r12   r22 m  1 1 2 . − − !m! 4t 4t n!Γ(ν + n + 1) 4t n=0 ∞

If we set yj = −rj2 /4t, we see that the asserted identity (9.60) is equivalent to the identity  j,k≥0

(9.62)

Γ(ν + j + k + 1) 1 j k y y Γ(ν + j + 1)Γ(ν + k + 1) j!k! 1 2 =

 ,m,n≥0

1 1 y +n y2m+n . !m! n!Γ(ν + n + 1) 1

We compare coefficients of y1j y2k in (9.62). Since both sides of (9.62) are symmetric in (y1 , y2 ), it suffices to treat the case j ≤ k,

(9.63)

which we assume henceforth. Then we take  + n = j, m + n = k and sum over n ∈ {0, . . . , j}, to see that (9.62) is equivalent to the validity of (9.64) j 

Γ(ν + j + k + 1) 1 1 = , (j − n)!(k − n)!n!Γ(ν + n + 1) Γ(ν + j + 1)Γ(ν + k + 1) j!k! n=0 whenever 0 ≤ j ≤ k. Using the identity Γ(ν + j + 1) = (ν + j) · · · (ν + n + 1)Γ(ν + n + 1) and its analogues for the other Γ-factors in (9.64), we see that (9.64) is equivalent to the validity of (9.65)

j 

j!k! (ν+j) · · · (ν+n+1) = (ν+j+k) · · · (ν+k+1), (j − n)!(k − n)!n! n=0

for 0 ≤ j ≤ k. Note that the right side of (9.65) is a polynomial of degree j in ν, and the general term on the left side of (9.65) is a polynomial of degree j − n in ν. In order to establish (9.65), it is convenient to set (9.66)

μ=ν+j

and consider the associated polynomial identity in μ. With

178 8. Spectral Theory

(9.67)

p0 (μ) = 1, p1 (μ) = μ, p2 (μ) = μ(μ − 1), . . . pj (μ) = μ(μ − 1) · · · (μ − j + 1),

we see that {p0 , p1 , . . . , pj } is a basis of the space Pj of polynomials of degree j in μ, and our task is to write (9.68)

pj (μ + k) = (μ + k)(μ + k − 1) · · · (μ + k − j + 1)

as a linear combination of p0 , . . . , pj . To this end, define T : Pj −→ Pj ,

(9.69)

T p(μ) = p(μ + 1).

By explicit calculation, (9.70)

p1 (μ + 1) = p1 (μ) + p0 (μ), p2 (μ + 1) = (μ + 1)μ = μ(μ − 1) + 2μ = p2 (μ) + 2p1 (μ),

and an inductive argument gives T pi = pi + ipi−1 .

(9.71)

By convention we set pi = 0 for i < 0. Our goal is to compute T k pj . Note that T = I + N,

(9.72)

N pi = ipi−1 ,

and Tk =

(9.73)

j    k N n, n n=0

if j ≤ k. By (9.72), (9.74)

N n pi = i(i − 1) · · · (i − n + 1)pi−n ,

so we have T k pj =

j    k n=0

(9.75) =

n

j(j − 1) · · · (j − n + 1)pj−n

j 

k! j! pj−n . (k − n)!n! (j − n)! n=0

This verifies (9.65) and completes the proof of (9.60).

9. The Laplace operator on cones

179

Similarly we can evaluate (9.47) with f (λ2 ) = e−tλ /λ, as an infinite series, using (9.53) to integrate each term of the double series. We get (9.76)  0

=



e−tλ Jν (r1 λ)Jν (r2 λ) dλ

Γ(2ν + 2j + 2k + 1) 1  r12 j  r22 k 1  r1 r2 ν  − 2 , − 2 t t2 Γ(ν + j + 1)Γ(ν + k + 1) j!k! 4t 4t j,k≥0

provided t > rj > 0. It is possible to express this integral in terms of the Legendre function Qν−1/2 (z). Proposition 9.8. One has, for all y, r1 , r2 > 0, ν ≥ 0, (9.77)  2   ∞ r1 + r22 + y 2 1 e−yλ Jν (r1 λ)Jν (r2 λ) dλ = (r1 r2 )−1/2 Qν−1/2 . π 2r1 r2 0 The Legendre functions Pν−1/2 (z) and Qν−1/2 (z) are solutions to (9.78)

  d d 1 (1 − z 2 ) u(z) + ν 2 − u(z) = 0; dz dz 4

Compare with (4.52). Extending (4.41), we can set (9.79)

Pν−1/2 (cos θ) =



2 π

θ

2 cos s − 2 cos θ

0

−1/2

cos νs ds,

and Qν−1/2 (z) can be defined by the integral formula  (9.80)



Qν−1/2 (cosh η) =

2 cosh s − 2 cosh η

−1/2

e−sν ds.

η

The identity (9.77) is known as the Lipschitz-Hankel integral formula. Proof of Proposition 9.8. We derive (9.77) from the Weber identity (9.60). Recall (9.81)

Iν (y) = e−πiν/2 Jν (iy),

y > 0.

To work with (9.60), we use the subordination identity (9.82)

λ e−yλ = √ π





e−y

2

/4t −tλ2 −1/2

e

t

dt;

0

cf. Chap. 3, (5.31) for a proof. Plugging this into the left side of (9.77), and using (9.60), we see that the left side of (9.77) is equal to

180 8. Spectral Theory



1 √ 2 π

(9.83)



2

2

e−(r1 +r2 +y

2

)/4t

0



r r  1 2 t−3/2 dt. 2t

The change of variable s = r1 r2 /2t gives % (9.84)

1 (r1 r2 )−1/2 2π





2

2

e−s(r1 +r2 +y

2

)/2r1 r2

0

Iν (s)s−1/2 ds.

Thus the asserted identity (9.77) follows from the identity 



% −sz

e

(9.85) 0

−1/2

Iν (s)s

ds =

2 Qν−1/2 (z), π

z > 0.

As for the validity of (9.85), we mention two identities. Recall from (9.57) that (9.86) 



−sz

e 0

Jν (λs)s

μ−1

 ν λ Γ(μ + ν) ds = z −μ−ν 2 Γ(ν + 1)   μ ν 1 μ ν λ2 + + , + ; ν + 1; − 2 . · 2 F1 2 2 2 2 2 z

Next, there is the classical representation of the Legendre function Qν−1/2 (z) as a hypergeometric function: (9.87) Qν−1/2 (z) =

Γ

1   1 ν 3 ν 1 1 2 Γ ν + 2 (2z)−ν−1/2 2 F1 + , + ; ν + 1; 2 ; Γ (ν + 1) 2 4 2 4 z

cf. [Leb], (7.3.7) If we apply (9.86) with λ = i, μ = 1/2 (keeping (9.81) in mind), then (9.85) follows. Remark: Formulas (9.77) and (9.60) are proven in the opposite order in [W]. By analytic continuation, we can treat f (λ2 ) = e−ελ λ−1 sin λt for any ε > 0. We apply this to (9.47). Letting ε  0, we get for the fundamental solution to the wave equation: (−Δ)−1/2 sin t(−Δ)1/2 = − lim (r1 r2 ) Im



α

(9.88)

ε0

0



e−(ε+it)λ Jν (λr1 )Jν (λr2 ) dλ

 r2 + r2 + (ε + it)2  1 2 . = − (r1 r2 )α−1/2 lim Im Qν−1/2 1 ε0 π 2r1 r2

Using the integral formula (9.80), where the path of integration is a suitable path from η to +∞ in the complex plane, one obtains the following alternative integral

9. The Laplace operator on cones

181

representation of (−Δ)−1/2 sin t(−Δ)1/2 . The Schwartz kernel is equal to (9.89) (9.90)

1 (r1 r2 )α π

if t < |r1 − r2 |,

0,



β1 

−1/2 t2 − (r12 + r22 − 2r1 r2 cos s) cos νs ds,

0

if |r1 − r2 | < t < r1 + r2 , and (9.91)

1 (r1 r2 )α cos πν π



∞

r12 + r22 + 2r1 r2 cosh s − t2

β2

−1/2

e−sν ds,

if t > r1 + r2 , where (9.92)

β1 = cos−1

 r 2 + r 2 − t2  1

2

2r1 r2

,

β2 = cosh−1

 t2 − r 2 − r 2  1

2r1 r2

2

.

Recall that α = −(m − 1)/2, where m = dim N . We next show how formulas (9.89)–(9.91) lead to an analysis of the classical problem of diffraction of waves by a slit along the positive x-axis in the plane R2 . In fact, if waves propagate in R2 with this ray removed, on which Dirichlet boundary conditions are placed, we can regard the space as the cone over an interval of length 2π, with Dirichlet boundary conditions at the endpoints. By the method of images, it suffices to analyze the case of the cone over a circle of circumference 4π (twice the circumference of the standard unit circle). Thus C(N ) is a double cover of R2 \ 0 in this case. We divide up the spacetime into regions I, II, and III, respectively, as described by (9.89), (9.90), and (9.91). Region I contains only points on C(N ) too far away from the source point to be influenced by time t; that the fundamental solution is 0 here is consistent with finite propagation speed. Since the circle has dimension m = 1, we see that (9.93)

 d2 1/2 ν = (−ΔN )1/2 = − 2 dθ

in this case if θ ∈ R/(4πZ) is the parameter on the circle of circumference 4π. On the line, we have (9.94)

cos sν δθ1 (θ2 ) =

 1 δ(θ1 − θ2 + s) + δ(θ1 − θ2 − s) . 2

To get cos sν on R/(4πZ), we simply make (9.94) periodic by the method of images. Consequently, from (9.90), the wave kernel (−Δ)−1/2 sin t(−Δ)1/2 is equal to (9.95)

 −1/2 (2π)−1 t2 − r12 − r22 + 2r1 r2 cos(θ1 − θ2 ) 0

if |θ1 − θ2 | ≤ π, if |θ1 − θ2 | > π,

182 8. Spectral Theory

in region II. Of course, for |θ1 − θ2 | < π this coincides with the free space fundamental solution, so (9.95) also follows by finite propagation speed. We turn now to an analysis of region III. In order to make this analysis, it is convenient to make simultaneous use both of (9.91) and of another formula for the wave kernel in this region, obtained by choosing another path from η to ∞ in the integral representation (9.80). The formula (9.91) is obtained by taking a horizontal line segment; see Fig. 9.1. If instead we take the path indicated in Fig. 9.2, we obtain the following formula for (−Δ)−1/2 sin t(−Δ)1/2 in region III: π

−1

−(m−1)/2

*

(r1 r2 )

(9.96)

 − sin πν

0

π

0 β2

2

−1/2 t2 − r12 − r22 + 2r1 r2 cos s cos sν ds

t −

r12



r22

+

−1/2 −sν − 2r1 r2 cosh s e ds .

The operator ν on R/(4πZ) given by (9.93) has spectrum consisting of (9.97)

 1  3 Spec ν = 0, , 1, , 2, . . . , 2 2

all the eigenvalues except for 0 occurring with multiplicity 2. The formula (9.91) shows the contribution coming from the half-integers in Spec ν vanishes, since cos 12 πn = 0 if n is an odd integer. Thus we can use formula (9.96) and compose with the projection onto the sum of the eigenspaces of ν with integer spectrum. This projection is given by (9.98)

P = cos2 πν

on R/(4πZ). Since sin πn = 0, in the case N = R/(4πZ) we can rewrite (9.96) as

F IGURE 9.1 Integration Contour

9. The Laplace operator on cones

183

F IGURE 9.2 Alternative Contour

(9.99) π −1 (r1 r2 )−(m−1)/2

 0

π

−1/2 t2 − r12 − r22 + 2r1 r2 cos s P cos sν ds.

In view of the formulas (9.94) and (9.96), we have P cos sν δθ1 (θ2 ) 1 (9.100) = δ(θ1 − θ2 + s) + δ(θ1 − θ2 − s) 4  +δ(θ1 − θ2 + 2π + s) + δ(θ1 − θ2 + 2π − s) mod 4π. Thus, in region III, we have for the wave kernel (−Δ)−1/2 sin t(−Δ)1/2 the formula (9.101)



−1/2 . (4π)−1 t2 − r12 − r22 + 2r1 r2 cos(θ1 − θ2 )

Thus, in region III, the value of the wave kernel at points (r1 , θ1 ), (r2 , θ2 ) of the double cover of R2 \ 0 is given by half the value of the wave kernel on R2 at the image points. The jump in behavior from (9.95) to (9.101) gives rise to a diffracted wave. We depict the singularities of the fundamental solution to the wave equation for R2 minus a slit in Figs. 9.3 and 9.4. In Fig. 9.3 we have the situation |t| < r1 , where no diffraction has occurred, and region III is empty. In Fig. 9.4 we have a typical situation for |t| > r1 , with the diffracted wave labeled by a “D.” This diffraction problem was first treated by Sommerfeld [Som] and was the first diffraction problem to be rigorously analyzed. For other approaches to the diffraction problem presented above, see [BSU] and[Stk]. Generally, the solution (9.89)–(9.91) contains a diffracted wave on the boundary between regions II and III. In Fig. 9.5 we illustrate the diffraction of a wave by a wedge; here N is an interval of length  < 2π. We now want to provide, for general N , a description of the behavior of the distribution v = (−Δ)−1/2 sin t(−Δ)1/2 δ(r2 ,x2 ) near this diffracted wave, that is, a study of the limiting behavior as r1  t − r2 and as r1  t − r2 . We begin with region II. From (9.90), we have v equal to

184 8. Spectral Theory

F IGURE 9.3 Reflected Wave Front

F IGURE 9.4 Reflected and Diffracted Wave Fronts

F IGURE 9.5 Diffraction by a Wedge

(9.102)

1 (r1 r2 )α−1/2 Pν−1/2 (cos β1 ) δx2 2

in region II,

where Pν−1/2 is the Legendre function defined by (9.79) and β1 is given by (9.92). Note that as r1  t − r2 , β1  π. To analyze (9.102), replace s by π − s in (9.79), and, with δ1 = π − β1 , write

9. The Laplace operator on cones

(9.103)



π Pν−1/2 (cos β1 ) = cos πν 2

π

δ1



+ sin πν

185

−1/2 2 cos δ1 − 2 cos s cos sν ds

π

−1/2 2 cos δ1 − 2 cos s sin sν ds.

δ1

As δ1  0, the second term on the right tends in the limit to 

π

sin πν

(9.104)

0

sin sν ds. sin 12 s

Write the first term on the right side of (9.103) as  π cos πν (2 cos δ1 − 2 cos s)−1/2 (cos sν − 1) ds δ1 (9.105)  π (2 cos δ1 − 2 cos s)−1/2 ds. + cos πν δ1

As δ1  0, the first term here tends in the limit to  π cos sν − 1 (9.106) cos πν ds. sin 12 s 0 The second integral in (9.105) is a scalar, independent of ν, and it is easily seen to have a logarithmic singularity. More precisely,  π 1 (2 cos δ1 − 2 cos s)− 2 ds (9.107)

δ1

∞ ∞   2  ∼ log Aj δ1j + Bj δ1j , δ1 j=0 j=1

A0 = 1.

Consequently, one derives the following. Proposition 9.9. Fix (r2 , x2 ) and t. Then, as r1  t − r2 ,

(9.108)

(−Δ)−1/2 sin t(−Δ)1/2 δ(r2 ,x2 ) ! 1 2 = (r1 r2 )α−1/2 log cos πν δx2 π δ1 "  π cos sν − cos πν + + R δ , ds δ x 1 x 2 2 2 cos 12 s 0

where, for s > (m + 1)/2, (9.109)

R1 δx2 D−s−1 ≤ Cδ1 log

1 , as δ1  0. δ1

186 8. Spectral Theory

The following result analyzes the second term on the right in (9.108). Proposition 9.10. We have  π 0

(9.110)

1 −1 2 cos s (cos sν − cos πν) ds 2 ⎧ ⎫ K ⎨ ⎬ π  aj ν −2j + sin πν + SK (ν), = cos πν − log ν + ⎩ ⎭ 2 j=0

where SK (ν) : Ds → Ds+2K , for all s. The spaces Ds are spaces of generalized functions on N , introduced in Chap. 5, Appendix A. We turn to the analysis of v in region III. Using (9.91), we can write v as (9.111)

1 (r1 r2 )α−1/2 cos πν Qν−1/2 (cosh β2 ) δx2 , π

in region III,

where Qν−1/2 is the Legendre function given by (9.80) and β2 is given by (9.92). It is more convenient to use (9.96) instead; this yields for v the formula

(9.112)

1 (r1 r2 )α−1/2 π

! 0

π

(2 cosh β2 + 2 cos s)−1/2 cos sν ds 

− sin πν

0

β2

" (2 cosh β2 − 2 cosh s)−1/2 e−sν ds .

Note that as r1  t − r2 , β2  0. The first integral in (9.112) has an analysis similar to that arising in (9.103); first replace s by π − s to rewrite the integral as 

π

(2 cosh β2 − 2 cos s)−1/2 cos sν ds  π + sin πν (2 cosh β2 − 2 cos s)−1/2 sin sν ds.

cos πν

0

(9.113)

0

As β2  0, the second term in (9.113) tends to the limit (9.104), and the first term also has an analysis similar to (9.105)–(9.107), with (9.107) replaced by  (9.114)

0

π

(2 cosh β2 − 2 cos s)−1/2 ds

 2   j   j ∼ log Aj β2 + Bj β2 , β2 j≥0

j≥1

A0 = 1.

9. The Laplace operator on cones

187

It is the second term in (9.112) that leads to the jump across r1 = t − r2 , hence to the diffracted wave. We have  (9.115) 0

β2

(2 cosh β2 − 2 cosh s)−1/2 e−sν ds ∼

 0

β2

$

ds β22



s2

=

π . 2

Thus we have the following: Proposition 9.11. For r1  t − r2 ,

(9.116)

(−Δ)−1/2 sin t(−Δ)1/2 δ(r2 ,x2 ) ! 1 2 = (r1 r2 )α−1/2 log cos πν δx2 π β2 "  π cos sν − cos πν π & sin πν δ + − + R δ ds δ x2 x2 1 x2 , 2 2 cos 12 s 0

where, for s > (m + 1)/2, (9.117)

R1 δx2 D−s−1 ≤ Cβ2 log

1 , as β2  0. β2

Note that (9.116) differs from (9.108) by the term π −1 (r1 r2 )α−1/2 times (9.118)



π sin πν δx2 . 2

This contribution represents a jump in the fundamental solution across the diffracted wave D. There is also the logarithmic singularity, (r1 r2 )α−1/2 times (9.119)

2 1 log cos πν δx2 , π δ

where δ = δ1 in (9.108) and δ = β2 in (9.116). In the special case where N is an interval [0, L], so dim C(N ) = 2, cos πν δx2 is a sum of two delta functions. Thus its manifestation in such a case is subtle. We also remark that if N is a subdomain of the unit sphere S 2k (of even dimension), then cos πν δx2 vanishes on the set N \ N0 , where (9.120)

N0 = {x1 ∈ N : for some y ∈ ∂N, dist(x2 , y) + dist(y, x1 ) ≤ π}.

Thus the log blow-up disappears on N \ N0 . This follows from the fact that cos πν0 = 0, where ν0 is the operator (9.46) on S 2k , together with a finite propagation speed argument. While Propositions 9.9–9.11 contain substantial information about the nature of the diffracted wave, this information can be sharpened in a number of respects. A much more detailed analysis is given in [CT].

188 8. Spectral Theory

There has been work on the propagation of singularities on manifolds with conic singularities, pursued in [MW], involving metric tensors of the form dr2 + r2 g, where g has a more general form, for example possibly depending smoothly on r. The base N is assumed to be a compact manifold without boundary, but within that category the metrics considered are more general than those treated in this section. The analytical tools are quite different from those used here. Further work, on propagation of singularities and diffraction effects, involving manifolds with “edge type” and “corner type” singularities, is given in [MVW, MVW2].

Exercises 1. Using (7.36) and (7.80), work out the asymptotic behavior of 1 F1 (a; b; −z) as z → +∞, given b, b − a ∈ / {0, −1, −2, . . . }. Deduce from (9.51) that whenever ν ≥ 0, s ∈ R, 1   ∞ −br 2 −is −is Γ 2 (ν + 1 − is)  . e Jν (r)r dr = 2 (9.121) lim b0 0 Γ 12 (ν + 1 + is) 2. Define operators Mr f (r) = rf (r),

(9.122)

J f (r) = f (r−1 ).

Show that (9.123)

Mr : L2 (R+ , r dr) −→ L2 (R+ , r−1 dr), J : L2 (R+ , r−1 dr) −→ L2 (R+ , r−1 dr)

are unitary. Show that Hν# = J Mr Hν Mr−1

(9.124) is given by (9.125)

Hν# f (λ) = (f  ν )(λ),

where  denotes the natural convolution on R+ , with Haar measure r−1 dr:  ∞ (9.126) (f  g)(λ) = f (r)g(r−1 λ)r−1 dr, 0

and ν (r) = r−1 Jν (r−1 ).

(9.127) 3. Consider the Mellin transform: (9.128)

M# f (s) =





f (r)ris−1 dr.

0

As shown in (A.17)–(A.20) of Chap. 3, we have

Exercises

189

(2π)−1/2 M# : L2 (R+ , r−1 dr) −→ L2 (R, ds), unitary.

(9.129) Show that

M# (f  g)(s) = M# f (s) · M# g(s),

(9.130) and deduce that

M# Hν# f (s) = Ψ(s)M# f (s),

(9.131) where (9.132)





Jν (r

Ψ(s) =

−1

)r

is−2

 dr =

0





Jν (r)r

−is

dr =

0

Γ 1 (ν 2−is  21 Γ 2 (ν

 + 1 − is) . + 1 + is)

4. From (9.126)–(9.132), give another proof of the unitarity (9.13) of Hν . Using symmetry, deduce that spec Hν = {−1, 1}, and hence deduce again the inversion formula (9.14). 5. Verify the asymptotic expansion (9.107). (Hint: Write 2 cos δ − 2 cos s = (s2 − δ 2 ) F (s, δ) with F smooth and positive, F (0, 0) = 1. Then, with G(s, δ) = F (s, δ)−1/2 ,  π  π ds −1/2 (9.133) (2 cos δ − 2 cos s) ds = G(s, δ) √ . 2 − δ2 s δ δ Write G(s, δ) = g(s) + δH(s, δ), g(0) = 1, and verify that (9.133) is equal to A1 + A2 , where  π  ds 1 1 , A1 = G(s, δ) = g(0) log + O δ log s δ δ δ  π 1 1 ds + O(δ) = B2 + O(δ). A2 = g(s) √ − 2 2 s s −δ δ Show that  B2 = g(0)

π/δ 1



1 1 dt + O(δ) = C2 + O(δ), − t −1

t2 

with

1 1 dt − t t2 − 1 1 Use the substitution t = cosh u to do this integral and get C2 = log 2.) Next, verify the expansion (9.114). C2 =





Exercises on the hypergeometric function 1. Show that 2 F1 (a1 , a2 ; b; z), defined by (9.56), satisfies (9.134)

2 F1 (a1 , a2 ; b; z) =

Γ(b) Γ(a2 )Γ(b − a2 )



1 0

ta2 −1 (1 − t)b−a2 −1 (1 − tz)−a1 dt,

190 8. Spectral Theory for Re b > Re a2 > 0, |z| < 1. (Hint: Use the beta function identity, (A.23)–(A.24) of Chap. 3, to write  1 Γ(b) (a2 )k = ta2 −1+k (1 − t)b−a2 −1 dt, k = 0, 1, 2, . . . , (b)k Γ(a2 )Γ(b − a2 ) 0 and substitute this into (9.39). Then use ∞  (a1 )k (zt)k = (1 − tz)−a1 , k!

0 ≤ t ≤ 1,

|z| < 1.)

k=0

2. Show that, given Re b > Re a2 > 0, (9.134) analytically continues in z to z ∈ C \ [1, ∞). 3. Show that the function (9.134) satisfies the ODE z(1 − z)

 du d2 u  + b − (a1 + a2 + 1)z − a 1 a2 u = 0 2 dz dz

Note that u(0) = 1, u (0) = a1 a2 /b, but zero is a singular point for this ODE. Show that another solution is u(z) = z 1−b 2 F1 (a1 − b + 1, a2 − b + 1; 2 − b; z). 4. Show that 2 F1 (a1 , a2 ; b; z)

  = (1 − z)−a1 2 F1 a1 , b − a2 ; b; (z − 1)−1 z .

(Hint: Make a change of variable s = 1 − t in (9.134).) For many other important transformation formulas, see [Leb] or [WW]. 5. Show that −1 z). 1 F1 (a; b; z) = lim 2 F1 (a, c; b; c c∞

We mention the generalized hypergeometric function, defined by p Fq (a; b; z)

=

∞  (a)k z k , (b)k k! k=0

where p ≤ q + 1, a = (a1 , . . . , ap ), b = (b1 , . . . , bq ), bj ∈ C \ {0, −1, −2, . . . }, |z| < 1, and (a)k = (a1 )k · · · (ap )k ,

(b)k = (b1 )k · · · (bq )k ,

and where, as before, for c ∈ C, (c)k = c(c + 1) · · · (c + k − 1). For more on this class of functions, see [Bai]. 6. The Legendre function Qν−1/2 (z) satisfies the identity (9.87), for ν ≥ 0, |z| > 1, and |Arg z| < π; cf. (7.3.7) of [Leb]. Take z = (r12 + r22 + t2 )/2r1 r2 , and compare the resulting power series for the right side of (9.77) with the power series in (9.76).

10. Quantum adiabatic limit and parallel transport

191

10. Quantum adiabatic limit and parallel transport Let H(t), t ∈ I = [0, 1], be a smooth family of self adjoint operators on a Hilbert space H, with a smoothly varying family of eigenspaces E(t), of constant dimension k, with eigenvalues λ(t). Assume the spectrum of H(t) on E(t)⊥ is bounded away from λ(t). Making a trivial adjustment, we will assume λ(t) = 0. For simplicity, assume all the operators H(t) have the same domain. Consider the solution operator S(t, s) to the “Schr¨odinger equation” (10.1)

∂u = iH(t)u, ∂t

taking u(s) to u(t). Now slow down the rate of change of H, and consider the solution operators Sn (t, s) to (10.2)

t ∂u = iH u. ∂t n

The claim is that, if u(0) = u0 ∈ E(0), then Sn (nt, 0)u0 → w(t) ∈ E(t) as n → ∞, and there is a simple geometrical description of w(t). This limit is called the quantum adiabatic limit of the Schr¨odinger equation (10.1). This result was established by T. Kato [Kat], and rediscovered by M. Berry [Ber1], the geometrical content brought out by B. Simon [Si4]. Berry worked with the case dim E(t) = 1, but that restriction is not necessary. The more general case was already dealt with by Kato; that the argument can be so extended was rediscovered by F. Wilczek and A. Zee [WZ]. A collection of subsequent literature can be found in [SW]. The geometrical structure is the following. The family E(t) gives a vector bundle E → I, a subbundle of the product bundle I × H. If P (t) denotes the orthogonal projection of H on E(t), then, as seen in §1 of Appendix C, Connections and Curvature, we have a covariant derivative on sections of E defined by (10.3)

∇T w(t) = P (t)DT w(t),

where DT is the standard componentwise derivative of H-valued functions. Parallel transport is defined by ∇T w = 0. One can see, using (10.4)

P P + P P  = P ,

that parallel transport is also characterized by (10.5)

dw = P  (t)w, if w(0) ∈ E(0). dt

192 8. Spectral Theory

See Exercise 3 in §1 of Appendix C, or Exercise 1 below. The claim is that the adiabatic limit w(t) mentioned above exists and is equal to the solution to (10.5), with w(0) = u0 ∈ E(0). To prove this, we will rescale the t-variable in the equation (10.2). Thus we compare the solutions u and w to ∂u = inH(t)u ∂t ∂w = P  (t)w, ∂t

(10.6)

given u(0) = w(0) = u0 ∈ E(0). Then we know w(t) ∈ E(t) for each t, so H(t)w(t) = 0. Also, by (10.4), P  (t)w = (I − P )P  w = (I − P )w . Let v(t) = u(t) − w(t). Then v(0) = 0 and ∂v − inH(t)v = −P ⊥ (t)w = f (t). ∂t

(10.7)

Here we have used (10.4) and set P ⊥ = I − P. Thus f (t) ⊥ E(t) for each t. Now let Sn (t, s) denote the solution operator to ∂u/∂t = inH(t)u. Then the solution v(t) to (10.7) is given by  v(t) =

(10.8)

0

t

Sn (t, s)f (s) ds.

We know that ∂ Sn (t, s) = −inSn (t, s)H(s). ∂s

(10.9)

Now the spectral hypothesis on H(t) implies we can set f (t) = H(t)g(t),

(10.10)

with g(t) a smooth family of elements of H. Hence  0

(10.11)

Hence

t

Sn (t, s)H(s)g(s) ds  t ∂ 1 Sn (t, s) g(s) ds =− in 0 ∂s  t  1 1 g(t) − Sn (t, 0)g(0) + Sn (t, s)g  (s) ds. =− in in 0

v(t) =

10. Quantum adiabatic limit and parallel transport

(10.12)

v(t) H ≤

193

K . n

This proves the following quantum adiabatic theorem. Proposition 10.1. Take H(t) and E(t), t ∈ [0, 1], as described in the first paragraph of this section, and let Sn (t, s) denote the solution operator to the first equation in (10.6). Then (10.13)

Sn (t, 0)u0 −→ w(t) as n → ∞ if u0 ∈ E(0),

where w(t) is obtained from u0 by parallel translation, as in the second equation in (10.6). Note that if H(1) = H(0), so E(1) = E(0), then w(1) will typically differ from u0 by the application of a unitary operator on E0 , since the connection (10.3) on E is typically not flat. In the case Berry considered, where dimC E(0) = 1, this could only be multiplication by eiθ , θ being called Berry’s phase. If we assume that H(t) has purely distinct spectrum λ1 (t) < λ2 (t) < · · · , of constant multiplicity, and no crossings, then we can analyze the behavior of solutions to ∂u/∂t = inH(t)u via superpositions. Let Pj (t) denote the , orthogonal uj (0), with projection of H onto the λj (t)-eigenspace of H(t). Write u(0) = uj (0) ∈ R(Pj (0)) = Ej (0). Let Tj (t) denote parallel translation in the vector bundle R(Pj (t)) = Ej (t), i.e., the solution operator to (10.14)

dwj = Pj (t)wj , dt

wj (0) ∈ Ej (0).

Then we can compare wj (t) = Tj (t)uj (0) to the solution uj (t) to (10.15)

∂uj = inH(t)uj , ∂t

uj (0) = Pj (0)u(0),

i.e., to Sn (t, 0)Pj (0)u(0). By (10.11), we have

(10.16)

  1 uj (t) = einΛj (t) wj (t) − gj (t) − Sjn (t, 0)gj (0) in  t  1 Sjn (t, s)gj (s) ds , − in 0

where Sjn (t, s) denotes the solution operator for

∂ − in H(t) − λj (t) , ∂t and gj (t) is obtained in a fashion similar to g(t) in (10.11). Also, we have set

194 8. Spectral Theory

 Λj (t) =

0

t

λj (s) ds.

If all the spectral gaps are bounded below: λj+1 (t) − λj (t) ≥ C > 0,

(10.17)

then we can decompose any u(0) ∈ H and sum over j, obtaining Sn (t, 0)u(0) =  einΛj (t) Tj (t)u(0) (10.18)

j

+

1  inΛj (t)  gj (t) − Sjn (t, 0)gj (0) + e in j

 0

t

 Sjn (t, s)gj (s) ds .

Similar approximations, for Sjn (t, 0) and Sjn (t, s), can be made on the right, and this process iterated, to obtain higher order asymptotic expansions. Of course, the hypothesis (10.17) is rather restrictive. If one weakens it to (10.19)

|λj+1 (t) − λj (t)| ≥ C λj (t) −K > 0,

then one can iterate (10.18), at least a finite number of times, provided u(0) belongs to the domain of some power of H(0). Exercise 1. Let w(t) solve (10.5) (so P (0)w(0) = w(0)). Verify that P (t)w(t) = w(t) for all t ∈ [0, 1]. Then verify that P (t)w (t) ≡ 0. Hint. Set ξ(t) = (I − P (t))w(t), and use (10.4) to show that ξ  (t) = −P  (t)ξ(t),

ξ(t) = 0,

hence ξ(t) ≡ 0.

11. A quantum ergodic theorem Let M√ be a compact Riemannian manifold, with Laplace operator Δ, and set Λ = −Δ. Let {uk : k ∈ N} be an orthonormal basis of L2 (M ) consisting of eigenfunctions, (11.1)

Λuk = λk uk ,

λk  +∞.

11. A quantum ergodic theorem

195

Let X = S ∗ M ⊂ T ∗ M denote the unit cosphere bundle. The symplectic form on T ∗ M induces a volume form dS on X, which we normalize to have unit volume. The Hamiltonian vector field associated to the principal symbol σ(Λ) generates a smooth flow Gt on X, preserving the volume form dS. Let P be the orthogonal projection of L2 (X, dS) onto V = {b ∈ L2 (X, dS) : b ◦ Gt ≡ b}.

(11.2)

Our first goal in this section is to prove the following quantum ergodic theorem. Theorem 11.1. There is a set N ⊂ N, of density zero, with the following property. Let A ∈ OP S 0 (M ) and assume a = σ(A)|X satisfies  Pa = a =

(11.3)

a dS. X

Then lim

(11.4)

k∈N / ,k→∞

(Auk , uk ) = a.

If the flow {Gt } is ergodic on X, then V consists of constants, and (11.3) holds for all a. The ergodic case has been studied in [Shn], [CdV], [HMR], [Zel], [Don], and other works. Theorem 11.1 applies to cases where such ergodicity does not hold. Such a formulation was mentioned in [ST], and pursued in [T4] and [T5]. Our proof of Theorem 11.1 is adapted from [CdV]. We start with the following Weyl law. Proposition 11.2. Let A ∈ OP S 0 (M ), with principal symbol σ(A) C ∞ (T ∗ M \ 0), homogenous of degree 0. Then (11.5)

lim

N →∞

 ∞ 1  (Auk , uk ) = σ(A) dS. N k=1

X

Proof. We have  (11.6)

e−tλk (Auk , uk ) =

k



(Auk , e−tΛ uk )

k

= Tr Ae−tΛ . A parametrix construction for e−tΛ yields



196 8. Spectral Theory −tΛ

Tr Ae





 σ(A) dS Tr e−tΛ

X

(11.7) =



σ(A) dS



e−tλk .

k

X

The result (11.5) follows from Karamata’s Tauberian theorem. To proceed, we recall from Chapter 7 the notion of a “quantization,” which is a continuous linear map op : C ∞ (X) −→ L(L2 (M )),

(11.8)

with the property that, given a ∈ C ∞ (X), A = op(a) is a pseudodifferential operator with principal symbol a. We also impose the condition op(1) = I, the identity map. The existence of quantizations follows via local coordinate charts and partitions of unity from pseudodifferential operator calculus on Euclidean space. Each one gives rise to a sequence of elements μk ∈ D (X), defined by a, μk = (op(a)uk , uk ).

(11.9)

p is Since (Kuk , uk ) → 0 whenever K is compact on L2 (M ), it follows that if oanother quantization, yielding μ ˜k ∈ D (X), then for each a ∈ C ∞ (X), a, μk − a, μ ˜k → 0 as k → ∞. Basic examples are “Kohn-Nirenberg” quantizations: (11.10)

0 (M ). opKN : C ∞ (X) −→ OP S 0 (M ) ⊂ OP S1,0

For our analysis, it is useful to bring in the existence of a “Friedrichs quantization,” opF : C ∞ (X) −→ L(L2 (M )),

(11.11) having the property (11.12)

a ≥ 0 =⇒ opF (a) ≥ 0,

opF (1) = I.

This is constructed on the Euclidean space level from opKN (a) via “Friedrichs symmetrization,” in Chapter 7. Results given there imply (11.13)

−δ (M ), a ∈ C ∞ (X) =⇒ opF (a) − opKN (a) ∈ OP Sρ,1−ρ

for some δ > 0, ρ ∈ (1/2, 1). From (11.12) it follows that (11.14)

opF (a) L(L2 ) ≤ sup |a|, X

11. A quantum ergodic theorem

197

and hence (11.11) has a unique continuous linear extension to opF : C(X) −→ L(L2 (M )),

(11.15)

with (11.12) holding for all a ∈ C(X). From here on we take a Friedrichs quantization, and set A = opF (a). In such a case, the distributions μk ∈ D (X) defined by (11.9) satisfy a ≥ 0 =⇒ a, μk ≥ 0.

(11.16)

Also 1, μk = (uk , uk ) = 1. Consequently, for each k, μk is a probability measure on X.

(11.17) We write

 (Auk , uk ) =

(11.18)

a dμk . X

The next step is to establish the following consequence of Egorov’s theorem. Proposition 11.3. Given a ∈ C ∞ (X), we have  (a − a ◦ Gt ) dμk −→ 0,

(11.19)

as k → ∞,

X

locally uniformly in t. Proof. Set A = opF (a) and At = e−itΛ AeitΛ .

(11.20)

By Egorov’s theorem (and (11.13)) At − opF (a ◦ Gt ) ∈ L(L2 (M ), H δ (M )),

(11.21)

and this holds locally uniformly in t, so  a ◦ Gt dμk − (AeitΛ uk , eitΛ uk ) −→ 0,

(11.22) X

locally uniformly in t. But (11.23)

(AeitΛ uk , eitΛ uk ) = (Auk , uk ),

as k → ∞,

198 8. Spectral Theory

so (11.22) leads to (11.19). Proof of Theorem 11.1. We will actually establish a more general result, in which we take A = opF (a),

(11.24)

a ∈ C(X),

and make the hypothesis (11.3), i.e., P a = a, where P is the orthogonal projection of L2 (X, dS) onto the space V , given by (11.2). Note that Proposition 11.3 readily extends to a ∈ C(X). To proceed, given a ∈ C(X), set (11.25)

1 aT = T

 0



T

a ◦ Gt dt,

a=

a dS. X

Then von Neumann’s mean ergodic theorem (established in an appendix to this chapter) implies that, as T → ∞, aT −→ P a in L2 -norm.

(11.26)

Under the hypothesis P a = a, we then have  |aT − a| dS −→ 0 as T → ∞.

(11.27) X

Thus, for ε ∈ (0, 1], there exists Tε < ∞ such that  T ≥ Tε =⇒

(11.28)

|aT − a| dS ≤ ε. X

Now, Proposition 11.3 gives, for all a ∈ C(X), T < ∞,  (aT − a) dμk −→ 0,

(11.29) X

as k → ∞, hence 

 (aT − a) dμk −

(11.30) X

(a − a) dμk −→ 0, X

as k → ∞. Furthermore, Proposition 11.2 implies

11. A quantum ergodic theorem

lim

(11.31)

N →∞

199

 N  1  b dμk = b dS, N k=1 X

X

for each b ∈ C ∞ (X), and then (11.17) gives this result for all b ∈ C(X). Taking b = |aT − a| gives (11.32)

 N  1  |aT − a| dμk = |aT − a| dS, N →∞ N lim

k=1 X

X

for each T < ∞. Comparison with (11.28) gives (11.33)

N  1  T ≥ Tε =⇒ lim |aT − a| dμk ≤ ε, N →∞ N k=1 X

if a satisfies (11.3). It follows from (11.33) that, if a ∈ C(X) satisfies (11.3), there exists a set Nε (a) ⊂ N, of density zero, such that  (11.34)

T = Tε =⇒

lim sup k∈N / ε (a),k→∞

|aT − a| dμk ≤ 2ε. X

Hence, by (11.30), for all ε > 0, lim sup

(11.35)

k∈N / ε (a),k→∞

 ) ) ) ) )a − a dμk ) ≤ 2ε. X

Now we can produce (11.36)

N (a) ⊂ N,

of density zero,

such that, for all  ∈ N, (11.37)

N2− (a) \ N (a) is finite.

Then (11.35) gives, for all ε ∈ (0, 1], (11.38)

lim sup k∈N / (a),k→∞

so, if a ∈ C(X) satisfies (11.3),

 ) ) ) ) )a − a dμk ) ≤ 2ε, X

200 8. Spectral Theory

(11.39)

lim

k∈N / (a),k→∞

 ) ) ) ) )a − a dμk ) = 0. X

To proceed, let (11.40)

I = {a ∈ C(X) : P a = a},

which is a closed, linear subspace of C(X) (equal to C(X) if {Gt } is ergodic). We can take a countable set {aν }, dense in I, and produce (11.41)

N ⊂ N,

of density zero,

such that, for all ν, N (aν ) \ N is finite.

(11.42) Then (11.43)

lim

k∈N / ,k→∞

 ) ) ) ) )a − a dμk ) = 0, X

whenever a = aν , and hence, by a limiting argumant, using (11.17), for all a ∈ I. This proves Theorem 11.1. In fact, it proves the following more general result. Proposition 11.4. There is a set N ⊂ N, of density zero, with the following property. Take a ∈ C(X) satisfying P a = a, and set A = opF (a). Then lim

(11.44)

k∈N / ,k→∞

(Auk , uk ) = a.

Here is a further generalization. Replace (11.3) by a, P a ∈ C(X).

(11.45) Then (11.46)

b = a − P a =⇒ b ∈ C(X), P b = 0,

and Proposition 11.4 yields (11.47)

lim

k∈N / ,k→∞

(opF (b), uk , uk ) = 0.

If we set (11.48)

A = opF (a),

AP = opF (P a),

11. A quantum ergodic theorem

201

then opF (b) = A − AP , and we have the following quantum ergodic theorem. Proposition 11.5. Assume a, P a ∈ C(X), and define A, AP as in (11.48). Then (11.49)

lim

k∈N / ,k→∞

  (Auk , uk ) − (AP uk , uk ) = 0.

As mentioned, work on quantum ergodic theorems started with the case that {Gt } acts ergodically on X, or equivalently that I = C(X), where I is defined by (11.40). The conclusion that (11.4) then holds for all A ∈ OP S 0 (M ) can be called a microlocal quantum ergodic theorem. The version stated in [Shn] was the more restricted local version, that (11.4) holds for all multiplication operators, Au(x) = a(x)u(x),

(11.50)

with a ∈ C(M ). A consequence of Proposition 11.4 is that for this result to hold for all such A, we need not require that {Gt } act on X ergodically. It is enough to require that (11.51)

C ⊂ I,

C = {a ∈ C(X) : a(x, ξ) = a(x)}.

An example for which (11.51) holds is M = Tn = Rn /Zn , a flat torus. In such a case, {Gt } is clearly not ergodic on X = Tn × S n−1 . See [T4] for further examples illustrating Theorem 11.1 and Proposition 11.5, including cases when / C 1 (X). a ∈ C ∞ (X) and P a ∈ C(X), but P a ∈ It is of interest to record the following local quantum ergodic theorem. Proposition 11.6. Assume {Gt } acts ergodically on X, or more generally that (11.51) holds. Then if a ∈ R(M ),

(11.52)

i.e., a is bounded and Riemann integrable on M , we have  (11.53)

a(x)|uk (x)|2 dV (x) = a.

lim

k∈N / ,k→∞ M

In particular, given O ⊂ M open,  (11.54)

lim

k∈N / ,k→∞

|uk (x)|2 dV (x) =

O

provided χO ∈ R(M ), i.e., m∗ (∂O) = 0. Proof. Given (11.52), we can find

V (O) , V (M )

202 8. Spectral Theory

 (11.55)

fν , gν ∈ C(M ),

fν ≤ a ≤ gν ,

(gν − fν ) dV = δν → 0, M

and apply Proposition 11.4 to multiplication by fν and gν . Taking ν → ∞ yields (11.53). It is also of interest to see if (11.53) extends to more general a ∈ L∞ (M ). Some results on this are discussed in [T5].

A. Von Neumann’s mean ergodic theorem Let {U (t) : t ∈ R} be a strongly continuous unitary group on a Hilbert space H. A fundamental object of ergodic theory is the study of the “time average” 1 AT f = T

(A.1)



T

U (s)f ds, 0

with emphasis on the limit as T → ∞. To get this, we can set U (t) = eitA , where iA generates U (t), with A self-adjoint (possibly unbounded) and use the spectral representation in Theorem 1.7 (or Proposition 1.3 plus a cyclic space decomposition of H) to write 1 T

(A.2)



T

U (s)f ds = G(T A)f, 0

where G(λ) = λ−1 (eitλ − 1),

(A.3)

1,

λ = 0, λ = 0.

In other words, using a unitary map W : L2 (Ω, μ) → H as in (1.29), (A.4)

W −1 AT W g(x) = G(T a(x))g(x),

g ∈ L2 (Ω, μ).

Now the dominated convergence theorem implies that, for each g ∈ L2 (Ω, μ), (A.5)

G(T a)g −→ χg,

in L2 -norm,

as T → ∞, where (A.6)

χ(x) = 1,

if a(x) = 0,

0,

if a(x) = 0.

A. Von Neumann’s mean ergodic theorem

203

We therefore have the following abstract mean ergodic theorem. Theorem A.1. If U (t) = eitA is a strongly continuous unitary group on H, and AT is given by (A.1), then, for all f ∈ H, AT f −→ P f in H-norm,

(A.7)

as T → ∞, where P is the orthogonal projection of H onto (A.8)

V = Ker A = {f ∈ H : U (t)f = f, ∀ t ∈ R}.

This theorem also has a discrete analogue, where we take a unitary operator U on H and consider N −1 1  k U f. AN f = N

(A.9)

k=0

We leave the formulation of such a result to the reader. In ergodic theory, such unitary groups arise as follows. Let μ be a probability measure on a σ-algebra F of subsets of X. One studies properties of groups of measure-preserving maps ϕt : X → X. That is, we assume (A.10)

−1 S ∈ F =⇒ ϕ−1 t (S) ∈ F and μ(ϕt (S)) = μ(S),

for all t ∈ R. We assume (A.11)

ϕ0 (x) ≡ x,

ϕs+t = ϕs ◦ ϕt ,

∀ s, t ∈ R.

The family {ϕt } defines a family of linear maps on functions: U (t)f (x) = f (ϕt (x)).

(A.12) From (A.10) we obtain



 f (ϕt (x)) dμ =

(A.13) X

f (x) dμ, X

for all f ∈ L1 (X, μ). Hence (A.14)

U (t) : Lp (X, μ) −→ Lp (X, μ)

is an isometry for all t ∈ R, p ∈ [1, ∞), and we have (A.15)

U (0) = I,

U (s + t) = U (s)U (t),

∀ s, t ∈ R.

204 8. Spectral Theory

We also impose a continuity requirement on {ϕt }, namely that {U (t)} be a strongly continuous group on Lp (X, μ), for each p ∈ [1, ∞). For a basic example of when this holds, let X be a compact metric space, F = B the σ-algebra of Borel sets, μ a probability measure on B, and let {ϕt } be a group of μ-preserving homeomorphisms of X, with continuous dependence on t. Theorem A.1 immediately applies to such situations, with H = L2 (X, μ). We next establish a more general result. Proposition A.2. In the setting described above, with U (t) given by (A.12), let P denote the orthogonal projection of L2 (X, μ) onto (A.16)

V = {f ∈ L2 (X, μ) : U (t)f = f, ∀ t ∈ R}.

Then, for p ∈ [1, 2], P extends to a continuous projection on Lp (X, μ), and (A.17)

f ∈ Lp (X, μ) =⇒ AT f → P f,

in Lp -norm, as T → ∞. Proof. Note that the Lp -operator norm AT L(Lp ) ≤ 1 for each T , and since g Lp ≤ g L2 for p ∈ [1, 2], we have (A.17) in Lp -norm for each f in the dense subspace L2 (X, μ) of Lp (X, μ). Now, given f ∈ Lp (X, μ), ε > 0, pick g ∈ L2 (X, μ) such that f − g Lp < ε. Then (A.18) AT f −AS f Lp ≤ AT g−AS g L2 + AT (f −g) Lp + AS (f −g) Lp . Hence (A.19)

lim sup AT f − AS f Lp ≤ 2ε, S,T →∞

∀ ε > 0.

This implies that (AT f ) is Cauchy in Lp (X, μ), for each f ∈ Lp (X, μ). Hence it has a limit; call it Qf . Clearly Qf is linear in f , Qf Lp ≤ f Lp , and Qf = P f for f ∈ L2 (X, μ). Hence Q is the unique continuous linear extension of P from L2 (X, μ) to Lp (X, μ) (so we change the name to P ). Note that P 2 = P on Lp (X, μ), since it holds on the dense linear subspace L2 (X, μ). Proposition A.2 is proved. In the setting of Proposition A.2, note that P = P ∗ , and hence (A.20)

P : Lp (X, μ) −→ Lp (X, μ),

∀ p ∈ [1, ∞].

Also, as indicated above, (A.21)

AT L(Lp ) ≤ 1,

∀ T ∈ R+ , p ∈ [1, ∞].

B. Wave equations and shifted wave equations

205

We aim to extend Proposition A.2 to p ∈ (2, ∞). The following yields a key ingredient. Lemma A.3. In the setting of Proposition A.2, given p < ∞, (A.22)

f ∈ L∞ (X, μ) =⇒ AT f → P f, in Lp -norm,

as T → ∞. Proof. If (A.22) fails, there exists ε > 0 and Tn → ∞ such that gn = ATn f satisfies gn − P f Lp ≥ ε,

(A.23)

∀ n.

We know that gn → P f in L2 -norm, so, passing to a subsequence, we have (A.24)

gn (x) −→ P f (x),

for μ-a.e. x ∈ X.

We also know gn L∞ ≤ f L∞ , so the convergence (A.24) occurs boundedly. The dominated convergence theorem then gives (A.25)

gn − P f Lp −→ 0,

contradicting (A.23). Now for our extension. Proposition A.4. In the setting of Proposition A.2, we have (A.17), with convergence in Lp -norm, for each p ∈ [1, ∞). Proof. This follows from Lemma A.3, the uniform operator bound (A.21), and the denseness of L∞ (X, μ) in Lp (X, μ), by arguments similar to those arising in the proof of Proposition A.2. R EMARK . The mean ergodic theorem has an extension to non-invertible measurepreserving transformations, which yield non-unitary operators on L2 (X, μ), and require a diffferent technique. This can be found in various texts, including Chapter 4 of [Haw] and Chapter 14 of [T3], which also treat the Birkhoff ergodic theorem.

B. Wave equations and shifted wave equations In Sections 4 and 5 we studied the Laplace operator on spheres S n and hyperbolic space Hn , respectively. Key identities included neat formulas for the solution to shifted wave equations,

206 8. Spectral Theory

∂2u − Lu = 0, ∂t2

(B.1) with (B.2)

L = Δ − a2n on S n ,

L = Δ + a2n on Hn ,

an =

n−1 . 2

Here we show how such formulas lead to formulas for the solutions to the regular wave equation ∂t2 u − Δu = 0. Following [MT], we establish an abstract result: Proposition B.1. Let L0 be a negative self-adjoint operator, and, for a ∈ R, set L1 = L0 + a2 ,

(B.3)

L2 = L0 − a2 .

Then, for t ∈ R, (B.4)

 $ $ cos t −L1 = cos t −L0 + at

t

 $ $ cos t −L2 = cos t −L0 − at

t

0

√ $ I1 (a t2 − s2 ) √ cos s −L0 ds, t2 − s2

and (B.5)

0

√ $ J1 (a t2 − s2 ) √ cos s −L0 ds. 2 2 t −s

Here J1 is the Bessel function J1 (λ) =

(B.6)

∞  k=0

(−1)k  λ 2k+1 , k!(k + 1)! 2

and (B.7)

∞ 

I1 (λ) = −iJ1 (iλ) =

k=0

 λ 2k+1 1 . k!(k + 1)! 2

To prove (B.4), we borrow a trick from §5 of Chapter 3. It starts with the identity (B.8)

1 etL1 f = √ 2π





−∞

$ ˆ t (s) cos s −L1 f ds, h

2

with ht (x) = e−tx , hence % (B.9)

ˆ t (s) = h

1 −s2 /4t e . 2t

B. Wave equations and shifted wave equations

207

√ If we set 4t = 1/λ, we see that w(t) = (cos t −L1 )f satisfies 



1 ds = 2

−λs2

w(s)e

(B.10) 0

%

π a2 /4λ (1/4λ)L0 e e f. λ

√ By comparison, w0 (t) = (cos t −L0 )f satisfies 



(B.11) 0

−λs2

w0 (s)e

1 ds = 2

%

π (1/4λ)L0 e f. λ

Consequently, 



(B.12)

2

2

w(s)e−λs ds = ea





/4λ

0

0

2

w0 (s)e−λs ds.

Now the change of variable σ = s2 makes (B.12) a relation between Laplace transforms of √ √ w( σ) w0 ( σ) . (B.13) ψ(σ) = √ , and ψ0 (σ) = √ σ σ 2

Hence a representation of ea

2

ea

(B.14)

/4λ

as a Laplace transform

/4λ

=





ϕ(σ)e−λσ dσ

0

will give rise to a convolution formula:  ψ(σ) =

(B.15)

0

σ

ϕ(τ )ψ0 (σ − τ ) dτ.

To identify the function ϕ(σ) in (B.14), we start with the following: 



(B.16) 0

2

Jν (as)sν+1 e−λs ds =

2 aν e−a /4λ . (2λ)ν+1

This is one of the most fundamental Hankel transforms and is used to prove the Hankel inversion formula in Proposition 9.1 (cf. (9.31)) of this chapter. We recall the method of proof was to replace Jν (as) in (B.16) by its power series expansion: (B.17)

Jν (as) =

∞  k=0

 as ν+2k (−1)k , k!Γ(k + ν + 1) 2

208 8. Spectral Theory

and integrate term by term. This works for ν > −1. We want to pass to the limit ν  −1. Of course, the integrand converges pointwise to 2

2

J−1 (as)e−λs = −J1 (as)e−λs .

(B.18)

The integral involving each term in (B.17) converges to the integral of the corre2 sponding term in the power series of J−1 (as) (times e−λs ) except for the term k = 0. Rather, due to the fact that, with ν = −1 + ε, B > 0, 1 Γ(ε)

(B.19)



B

sε−1 ds = 0

1 B ε → 1, as ε  0, εΓ(ε)

the result we arrive at is 



(B.20) 0

−λs2

J1 (as)e

2

1 − e−a ds = a

/4λ

.

Next we can analytically continue in a. We have (cf. (6.55) of Chapter 3), Iν (r) = e−πiν/2 Jν (ir),

(B.21)

r > 0,

and in particular I1 (r) = −iJ1 (ir), as in (B.7). Hence, for a, λ > 0,  (B.22) 0



2

2

I1 (as)e−λs ds =

ea

/4λ

−1

a

.

We have achieved (B.14), and hence (B.15), with √ a ϕ(σ) = √ I1 (a σ) + δ0 . 2 σ

(B.23)

Thus (B.15) yields a relation between ψ(σ) and ψ0 (σ) that, in view of (B.13), gives the desired identity (B.4). The proof of (B.5) follows along similar lines, as the reader can verify. Here is a specific conclusion for the wave equations on S n and Hn . Corollary B.2. For M = S n or Hn , let Δ be the Laplace operator, and define L and an as in (B.2). Then, on S n , (B.24)

√ √ cos t −Δ = cos t −L + an t



t

0

√ √ I1 (an t2 − s2 ) √ cos s −L ds, 2 2 t −s

and, on Hn , (B.25)

√ cos t −Δ = cos t −L − an t √

 0

t

√ √ J1 (an t2 − s2 ) √ cos s −L ds. 2 2 t −s

B. Wave equations and shifted wave equations

209

Proof. Apply (B.4), (B.5) to S n and Hn , respectively. In both cases, take L0 = L, a = an . For S n , we have L1 = Δ, and for Hn , we have L2 = Δ. For applications to wave equations, it$is useful to complement formulas (B.4)– $ (B.5) with formulas for (sin t −Lj )/ −Lj . For simplicity, we concentrate on the case j = 2. Computations for j = 1 are similar. To start, integrating (B.5) gives (B.26)

√ √ sin t −L0 sin t −L2 √ = √ + R(t), −L2 −L0

with √ $ J1 (a τ 2 − s2 ) R(t) = −a τ √ cos s −L0 ds dτ 2 2 τ −s 0 0 √  t  t $ J1 (a τ 2 − s2 )  = −a τ √ dτ cos s −L0 ds. 2 2 τ −s 0 s  t

(B.27)

τ

Using (B.28)

√ $ d sin s −L0 √ cos s −L0 = ds −L0

and integrating by parts gives

0

√ ∂ sin s −L0 A(s, t) √ ds, ∂s −L0



t

 (B.29)

R(t) = −a

t

with (B.30)

A(s, t) =

τ s

√ J1 (a τ 2 − s2 ) √ dτ. τ 2 − s2

If we write (B.31)

√ J1 (a z) √ ϕ(z) = , z

which is an entire function of z, then  (B.32)

A(s, t) = s

Using

t

τ ϕ(τ 2 − s2 ) dτ.

210 8. Spectral Theory

∂s ϕ(τ 2 − s2 ) = −2sϕ (τ 2 − s2 ),

(B.33)

∂τ ϕ(τ 2 − s2 ) = 2τ ϕ (τ 2 − s2 ),

we obtain ∂ A(s, t) = −sϕ(0) + ∂s (B.34)



t

τ ∂s ϕ(τ 2 − s2 ) dτ

s



= −sϕ(0) − s

t

∂τ ϕ(τ 2 − s2 ) dτ

s

= −sϕ(t2 − s2 ). Plugging this into (B.29) yields the following conclusion. √ √ Proposition B.3. In the setting of Proposition B.1, (sin t −L2 )/ −L2 is given by (B.26), with  R(t) = −a

(B.35)

0

t

√ √ J1 (a t2 − s2 ) sin s −L0 √ √ s ds. −L0 t2 − s2

Corollary B.4. In the setting of Corollary B.2, on Hn we have (B.36)

√ √ √ √  t J1 (an t2 − s2 ) sin s −L sin t −L sin t −Δ √ √ √ √ = − an ds. −Δ −L −L t2 − s2 0

Let us specialize to n = 3. It follows from (5.15) of §5 that, on H3 , (B.37)

√ sin s −L δ(s − r) √ , δp (x) = 4π sinh r −L

for s > 0, with r = dist(p, x), and (B.35) gives, for t > 0, (B.38)

√ J1 ( t2 − r2 ) r √ R(t)δp (x) = − χ{r≤t} , 4π sinh r t2 − r 2

as in (2.20) of [MT]. Such a formula was used in [MT] (and [MT2]) to obtain dispersive estimates on solutions to the wave equation on R × H3 , which then led to global existence of solutions to certain semilinear wave equations.

References [Ad] B. Adams, Algebraic Approach to Simple Quantum Systems, Springer, New York, 1994

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9 Scattering by Obstacles

Introduction In this chapter we study the phenomenon of scattering by a compact obstacle in Euclidean space R3 . We restrict attention to the three-dimensional case, though a similar analysis can be given for obstacles in Rn whenever n is odd. The Huygens principle plays an important role in part of the analysis, and for that part the situation for n even is a little more complicated, though a theory exists there also. The basic scattering problem is to solve the boundary problem (0.1)

(Δ + k 2 )v = 0 in Ω,

v = f on ∂K,

where Ω = R3 \ K is the complement of a compact set K. (We also assume Ω is connected.) We place on v the “radiation condition”  (0.2)

r

∂v − ikv ∂r

 −→ 0, as r −→ ∞,

in case k is real. We establish the existence and uniqueness of solutions to (0.1) and (0.2) in §1. Motivation for the condition (0.2) is also given there. Special choices of the boundary value f give rise to the construction of the Green function G(x, y, k) and of “eigenfunctions” u± (x, kω). In §2 we study analogues of the Fourier transform, arising from such eigenfunctions, providing Φ± , unitary operators from L2 (Ω) to L2 (R3 ) which intertwine the Laplace operator on Ω, with the Dirichlet boundary condition, and multiplication by |ξ|2 on L2 (R3 ). For any smooth f on ∂K, the solution to (0.1) and (0.2) has the following asymptotic behavior: (0.3)

v(rθ) = r−1 eikr α(f, θ, k) + o(r−1 ),

r −→ ∞,

known as the “far field expansion.” In case f (x) = −eikω·x on ∂K, the coefficient is denoted by a(ω, θ, k) and called the “scattering amplitude.” This is one of the c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  M. E. Taylor, Partial Differential Equations II, Applied Mathematical Sciences 116, https://doi.org/10.1007/978-3-031-33700-0 9

215

216 9. Scattering by Obstacles

fundamental objects of scattering theory; in §3 it is related to the unitary operator 2 3 S = Φ+ Φ−1 − on L (R ), the “scattering operator.” The term “scattering” refers to the scattering of waves. Connection with the wave equation is made in §§4 and 5, where the scattering operator is related to the long-time behavior of the solution operator for the wave equation, in counterpoint to the long-distance characterization of the scattering amplitude given in §1. In the study of the wave-equation approach to scattering theory, a useful tool is a semigroup Z(t), introduced by Lax and Phillips, which is described in §6. Section 7 considers the meromorphic continuation in k of the solution operator to (0.1) and (0.2). This operator has poles in the lower half-plane, called scattering poles. The analytical method used to effect this construction involves the classical use of integral equations. We also relate the scattering poles to the spectrum of the Lax–Phillips semigroup Z(t). In §8 we derive “trace formulas,” further relating the poles and Z(t). In §9 we illustrate material of earlier sections by explicit calculations for scattering by the unit sphere in R3 . In §§10 and 11, we discuss the “inverse” problem of determining an obstacle K, given scattering data. Section 10 focuses on uniqueness results, asserting that exact measurements of certain scattering data will uniquely determine K. In §11 we discuss some methods that have been used to determine K approximately, given approximate measurements of scattering data. This leads us to a discussion of “ill-posed” problems and how to regularize them. In §12 we present some material on scattering by a rough obstacle, pointing out similarities and differences with the smooth cases considered in the earlier sections. Appendix A at the end of this chapter is devoted to the proof of a trace identity used in §8. We have confined attention to the Dirichlet boundary condition. The scattering problem with the Neumann boundary condition, and for electromagnetic waves, with such boundary conditions as discussed in Chap. 5, are of equal interest. There are also studies of scattering for the equations of linear elasticity, with boundary conditions of the sort considered in Chap. 5. Many of the results in such cases can be obtained with only minor modifications of the techniques used here, while other results require further work. For further material on the theory of scattering by obstacles, consult [LP1], [Rm], [CK], and [Wil]. Another important setting for scattering theory is the Schr¨odinger operator −Δ + V ; see [RS], [New], and [Ho] for material on this. We include some exercises on some of the simplest problems in this quantum scattering theory. These exercises indicate that very similar techniques to those for scattering by a compact obstacle apply to scattering by a compactly supported potential. It would not take a much greater modification to handle potentials V (x) that decay very rapidly as |x| → ∞. Such potentials, with exponential fall-off, are used in crude models of two-body interactions involving nuclear forces. It takes more substantial modifications to treat long-range potentials, such as those that arise from the Coulomb force. The most interesting quantum scattering problems involve multiparticle interactions, and the analysis of these requires a much more elaborate set-up. See [Vas].

1. The scattering problem

217

Further directions include the scattering behavior of waves on noneuclidean spaces, with special structure at infinity. For major developments in this area, see [Me4, Bor], and [DZ].

1. The scattering problem In this section we establish the existence and uniqueness for the following boundary problem. Let K ⊂ R3 be a compact set with smooth boundary and connected complement Ω. Let f ∈ H s (∂K) be given, and let k > 0. We want to solve (1.1)

(Δ + k 2 )v = 0 on Ω,

(1.2)

v=f

on ∂K.

In addition, we impose a “radiation condition,” of the following form:  (1.3)

|rv(x)| ≤ C,

r

∂v − ikv ∂r

 −→ 0, as r −→ ∞,

where r = |x|. This condition will hold provided v satisfies the integral identity (1.4)

   ∂v ∂g v(x) = (x, y, k) − g(x, y, k) (y) dS (y), f (y) ∂νy ∂ν ∂K

for x ∈ Ω, where (1.5)

 −1 ik|x−y| g(x, y, k) = 4π|x − y| e .

Our existence proof will utilize the following fact. If k > 0 is replaced by k + iε, ε > 0, then −(k + iε)2 belongs to the resolvent set of the Laplace operator Δ on Ω, with Dirichlet boundary conditions on ∂K. Hence, for s ≥ 3/2, (1.1)– 2 To obtain (1.2) (with k replaced by k + iε) has a unique  solution vε2 ∈ #L (Ω). # 2 this, extend f to f ∈ H (Ω), and set ϕ = Δ + (k + iε) f ∈ L2 (Ω). Then −1  vε = f # − Δ + (k + iε)2 ϕ. Furthermore, in this case, the integral formula (1.4) does hold, as a consequence of Green’s theorem, with g(x, y, k) replaced by (1.6)

 −1 (ik−ε)|x−y| g(x, y, k + iε) = 4π|x − y| e ,

218 9. Scattering by Obstacles

which, as we saw in Chap. 3, is (the negative of) the resolvent kernel for (Δ+(k + iε)2 )−1 on free space R3 , a kernel that converges to (1.5) as ε  0. The strategy will be to show that, as ε  0, vε converges to the solution to (1.1)–(1.3). Before tackling the existence proof, we first establish the uniqueness of solutions to (1.1)–(1.3), as this uniqueness result will play an important role in the existence proof. Proposition 1.1. Given k > 0, if v satisfies (1.1)–(1.3) with f = 0, then v = 0. Proof. Let SR denote the sphere {|x| = R} in R3 ; for R large, SR ⊂ Ω, and, with vr = ∂v/∂r, we have     2    |vr | + k 2 |v|2 dS − ik vv r − vvr dS . |vr − ikv|2 dS = (1.7) SR

SR

SR

Now Green’s theorem applied to v and v implies     ∂v ∂v

v vv r − vvr dS = −v dS = 0, (1.8) ∂ν ∂ν SR

∂K

provided v|∂K = 0. Since the hypothesis (1.3) implies  vr − ikv 2 dS −→ 0, as R −→ ∞, (1.9) SR

we deduce from (1.7) that  (1.10) |v|2 dS −→ 0, as R −→ ∞. SR

The proof of Proposition 1.1 is completed by the following result. Lemma 1.2. If v satisfies (Δ + k 2 )v = 0 for |x| ≥ R0 and (1.10) holds, then v(x) = 0 for |x| ≥ R0 . Proof. It suffices to prove that, for r ≥ R0 ,  (1.11)

V (r) =

v(rω)ϕ(ω) dS (ω) S2

is identically zero, for each eigenfunction ϕ of the Laplace operator ΔS on the unit sphere S 2 : (1.12)

(ΔS + μ2 )ϕ = 0 (μ ≥ 0).

1. The scattering problem

219

In view of the formula for Δ on R3 in polar coordinates, Δ=

(1.13)

∂2 1 2 ∂ + ΔS , + ∂r2 r ∂r r2

it follows that V (r) satisfies the ODE (1.14)

V  (r) +

2  μ2

V (r) + k 2 − 2 V (r) = 0, r r

r ≥ R0 . (j)

This ODE has two linearly independent solutions of the form r−1/2 Hν (kr), (1) (2) j = 1, 2, where Hν (z) and Hν (z) are the Hankel functions discussed in Chap. 3, and ν 2 = μ2 + 1/4. In view of the integral formulas given there, it follows that the asymptotic behavior of these two solutions is of the form (1.15)

V± (r) = C± r−1 e±ikr + o(r−1 ),

r −→ ∞.

Clearly, no nontrivial linear combination of these two is o(r−1 ) as r → ∞. Since the hypothesis implies that V (r) = o(r−1 ), we deduce that V = 0. Applying Lemma 1.2, we see that under the hypotheses of Proposition 1.1, v = 0 for |x| ≥ R0 , given that K ⊂ {x : |x| ≤ R0 }. Since v satisfies the unique continuation property in Ω, this implies v = 0 in Ω, so Proposition 1.1 is proved. Remark: The uniqueness proof above really used (1.9), which is formally weaker than the radiation condition (1.3). Consequently, (1.9) is sometimes called the radiation condition. On the other hand, the existence theorem, to which we turn next, shows that the formally stronger condition (1.3) holds. The following result, which establishes the existence of solutions to (1.1)– (1.3), is known as the limiting absorption principle. Theorem 1.3. Let s ≥ 3/2, and suppose that as ε  0, fε −→ f

(1.16)

in H s (∂K).

Let vε be the unique element of L2 (Ω) satisfying (1.17)



 Δ + (k + iε)2 vε = 0

in Ω,

vε = fε

(1.18) Then, as ε  0, we have a unique limit (1.19)

vε −→ v = B(k)f,

on ∂K.

220 9. Scattering by Obstacles

satisfying (1.1)–(1.3). Convergence occurs in the norm topology of the space s+1/2 L2 (Ω, x −1−δ dx ) for any δ > 0, as well as in Hloc (Ω), and the limit v satisfies the identity (1.4). It is convenient to divide the proof into two parts. Fix R such that K ⊂ {|x| < R} and let OR = Ω ∩ {|x| < R}. Lemma 1.4. Assume vε OR is bounded in L2 (OR ) as ε  0. Then the conclusions of Theorem 1.3 hold. Proof. Fix S < R with K ⊂ {|x| < S}. The elliptic estimates of Chap. 5 imply that if vε L2 (OR ) is bounded, then (1.20)

vε H s+1/2 (OS ) ≤ Ck + Ck fε H s (∂K) .

Passing to a subsequence, which we continue to denote by vε , we have vε −→ v

(1.21)

weakly in H s+1/2 (OS ),

for some v ∈ H s+1/2 (OS ). The trace theorem implies weak convergence vε ∂K −→ v ∂K

(1.22)

in H s (∂K),

and ∂v ∂vε −→ ∂ν ∂ν

(1.23) Since each vε satisfies (1.24)

vε (x) =



∂K

fε (y)

in H s−1 (∂K).

∂vε (y)

∂gε − gε dS (y), ∂ν ∂ν

x ∈ Ω,

with gε = g(x, y, k + iε) given by (1.6), we deduce from (1.22) and (1.23) that the right side of (1.24) converges locally uniformly in x ∈ Ω, as ε  0, to a limit, call it v, that coincides with the limit (1.21) on OS . Furthermore, in view of the formula (1.6), we have the estimate (1.25)

|vε (x)| ≤ Ck x −1 ,

x ∈ Ω,

with Ck independent of ε. Thus the limit v satisfies this estimate, and we have vε → v in L2 (Ω, x −1−δ dx ) for any δ > 0. Furthermore, the limit v satisfies the identity (1.4), so the radiation condition (1.3) holds. So far we have convergence for subsequences, but in view of the uniqueness result of Proposition 1.1, this limit v is unique, so Lemma 1.4 is proved. The proof of Theorem 1.3 is completed by the following argument.

1. The scattering problem

221

Lemma 1.5. The hypotheses (1.16)–(1.18) of Theorem 1.3 imply that {vε } is bounded in L2 (Ω, x −1−δ dx ), for any δ > 0. Proof. Fix such a δ. Suppose Nε = vε L2 (Ω,x−1−δ dx ) → ∞ for a subsequence εn  0. Set wε = Nε−1 vε . Then Lemma 1.4 applies to wε , with wε ∂K = fε# = Nε−1 fε → 0 in H s (∂K). Thus the conclusion of Lemma 1.4 gives wε −→ w strongly in L2 (Ω, x −1−δ dx ). The limit w satisfies the scattering problem (1.1)–(1.3) with f = 0, so our uniqueness result implies w = 0. This contradicts the fact that each wε has norm 1 in L2 (Ω, x −1−δ dx ), so the proof is complete. Remark: Considering the dense subspace H s+1 (∂K) of H s (∂K), we can ims+1/2 prove weak convergence of vε → v in Hloc (Ω) to strong convergence in this space. We draw a couple of conclusions from Theorem 1.3. The first concerns the limiting behavior as ε  0 of the Green function G(x, y, k + iε), the kernel for −1  on Ω, which is of the form the resolvent Δ + (k + iε)2 (1.26)

G(x, y, k + iε) = g(x, y, k + iε) + h(x, y, k + iε),

where g(x, y, k + iε) is the free-space Green kernel (1.6) and h(x, y, k + iε) is, for each y ∈ Ω, the element of L2 (Ω) satisfying  (1.27)

 Δx + (k + iε)2 h = 0, h(x, y, k + iε) = −g(x, y, k + iε), for x ∈ ∂K.

Clearly, as ε  0, g(x, y, k + iε) → g(x, y, k), given by (1.5). On the other hand, for any y ∈ Ω, Theorem 1.3 applies to fε (x) = −g(x, y, k + iε), and we have (1.28)

h(x, y, k + iε) → h(x, y, k),

where h(x, y, k) solves the scattering problem (1.1)–(1.3), with h(x, y, k) = −g(x, y, k) for x ∈ ∂K. Consequently, as ε  0, (1.29)

G(x, y, k + iε) −→ G(x, y, k),

where (1.30)

G(x, y, k) = g(x, y, k) + h(x, y, k).

Another important family of functions defined by a scattering problem is the following. Note that we have (1.31)

(Δ + |ξ|2 )e−ix·ξ = 0

on R3 ,

222 9. Scattering by Obstacles

for any ξ ∈ R3 . We define the functions u(x, ξ) on Ω × R3 by (1.32)

u(x, ξ) = e−ix·ξ + v(x, ξ),

where v(x, ξ) satisfies the scattering problem (1.1)–(1.3), with k 2 = |ξ|2 and (1.33)

v(x, ξ) = −e−ix·ξ

on ∂K.

As we will see in the next section, u(x, ξ) plays a role on Ω of generalized eigenfunction of the Laplace operator on Ω, with Dirichlet boundary conditions, analogous to the role played by u0 (x, ξ) = e−ix·ξ on R3 . There is an interesting relation between the Green function G(x, y, k) and the “eigenfunctions” u(x, ξ), which we give here, which will play an important role in the analysis in the next section. It involves the behavior of G(x, y, k) as |y| → ∞. Proposition 1.6. For y = rω, ω ∈ S 2 , r → ∞, and any fixed k > 0, (1.34)

G(x, rω, k) = (4πr)−1 eikr u(x, kω) + O(r−2 ).

This is uniformly valid for (x, ω, k) in any bounded subset of Ω × S 2 × R+ . Proof. Write G(x, rω, k) = g(x, rω, k)+h(x, rω, k), as in (1.30). Thus hr (x) = h(x, rω, k) satisfies (1.35) (Δ + k 2 )hr (x) = 0, hr ∂K = −g(x, rω, k), together with the radiation condition as |x| → ∞. Now, in view of (1.5), as r → ∞, we have, for x ∈ ∂K, or indeed for x in any bounded subset of R3 , (1.36)

g(x, rω, k) = (4πr)−1 eikr e−ikω·x + O(r−2 ),

where the remainder is O(r−2 ) in C  (∂K) for any . Thus, in view of the estimates established in the proof of Theorem 1.3, we have (1.37)

hr = (4πr)−1 eikr v(x, kω) + O(r−2 ),

r −→ ∞,

with v(x, ξ) defined above. This gives the desired result (1.34). We remark that a similar argument gives (1.38)

∂ G(x, rω, k) = (4πr)−1 ik eikr u(x, kω) + O(r−2 ), ∂r

as r → ∞. Note that, for any f ∈ C ∞ (∂K), by (1.4) we have an asymptotic behavior of the form

1. The scattering problem

(1.39)

v(rθ) = r−1 eikr α(f, θ, k) + o(r−1 ),

223

r → ∞,

with θ ∈ S 2 , for the solution to the scattering problem (1.1)–(1.3), with a smooth coefficient α(f, ·, ·). Also, ik ∂ v(rθ) = eikr α(f, θ, k) + o(r−1 ). ∂r r

(1.40)

In particular, the function v(x, ξ) given by (1.33) has the asymptotic behavior (1.41)

v(rθ, kω) ∼ r−1 eikr a(−ω, θ, k),

r → ∞,

for fixed θ, ω ∈ S 2 , k ∈ R+ , and its r-derivative has an analogous behavior. The coefficient a(ω, θ, k) is called the scattering amplitude and is one of the fundamental objects of scattering theory. We will relate this to the scattering operator in §3. The radiation condition (1.3) is more specifically called the “outgoing radiation condition.” It has a counterpart, the “incoming radiation condition”: (1.42)

|rv(x)| ≤ C,

r

∂v ∂r

+ ikv −→ 0, as r −→ ∞.

Clearly there is a parallel treatment of the scattering problem (1.1), (1.2), (1.42). Indeed, if v(x) satisfies (1.1)–(1.3), then v(x) satisfies the incoming scattering problem, with f replaced by f , and conversely. In particular, we can define u− (x, ξ) = e−ix·ξ + v− (x, ξ),

(1.43)

where v− (x, ξ) satisfies the scattering problem (1.1), (1.2), (1.42), with v− (x, ξ) = −e−ix·ξ

(1.44)

on ∂K,

and we clearly have (1.45)

v− (x, ξ) = v(x, −ξ),

u− (x, ξ) = u(x, −ξ).

In analogy with (1.41), we have the asymptotic behavior (1.46)

v− (rθ, kω) ∼ r−1 e−ikr a− (ω, θ, k),

with (1.47)

a− (ω, θ, k) = a(ω, θ, k).

r → ∞,

224 9. Scattering by Obstacles

Sometimes, to emphasize the relation between these functions, we use the notation u+ (x, ξ), v+ (x, ξ) and a+ (ω, θ, k) for the functions defined by (1.32) and (1.33) and by (1.41). We note that while the discussion above has dealt with k > 0, the case k = 0 can also be included. In this case, the proof of Proposition 1.1 does not apply; for example, (1.7) no longer implies (1.10). However, the existence and uniqueness of a solution to (1.48)

Δv = 0 on Ω,

v = f on ∂K,

satisfying (1.49)

|rv(x)| ≤ C,

|r2 ∂r v| ≤ C, as r → ∞,

is easily established, as follows. We can assume that the origin 0 ∈ R3 is in the interior of K. Then the inversion ψ(x) = x/|x|2 interchanges 0 and the point at infinity, and the transformation (1.50)

v(x) = |x|−1 w(|x|−2 x)

preserves harmonicity. We let w be the unique harmonic function on the bounded domain ψ(Ω), with boundary value w(x) = |x|−1 f (ψ(x)) on ∂ψ(Ω) = ψ(∂K). It is easily verified that v(x) satisfies (1.49) in this case. Conversely, if v(x) satisfies (1.48) and w is defined by (1.50), then w is harmonic on ψ(Ω) \ 0 and equal to f ◦ ψ on ψ(∂K). If v also satisfies (1.49), then w is bounded near 0, and so is r ∂w/∂r. Now the boundedness of w near 0 implies that 0 is a removable singularity of w, since Δw ∈ D (ψ(Ω)) is a distribution supported at 0, hence a finite linear combination of derivatives of δ(x), which implies that w is the sum of a function harmonic on ψ(Ω) and a finite sum of derivatives of |x|−1 , and the latter cannot be bounded unless it is identically zero. Similarly, r ∂w/∂r is harmonic on ψ(Ω) \ 0, and if it is bounded near 0 then it extends to be harmonic on ψ(Ω), and this in turn implies that w extends to be harmonic on ψ(Ω). Therefore, either one of the two conditions in (1.49) gives uniqueness. Of course, if f ∈ C(∂K) the uniqueness of solutions to (1.48), satisfying the first condition in (1.49), follows from the maximum principle. With this result established, the limiting absorption principle, Theorem 1.3, also holds for k = 0. We also note that the proof of Theorem 1.3 continues to work if instead of using k + iε (ε  0) in (1.17), one replaces k + iε by any λ(ε) approaching k ∈ [0, ∞) from the upper half-plane. Furthermore, the limit v depends continuously on k. In particular, the functions u± (x, ξ) defined above are continuous in ξ ∈ R3 , and a± (ω, θ, k) is continuous on S 2 × S 2 × [0, ∞). There is a natural fashion in which u+ (x, ξ) and u− (x, ξ) fit together, which we describe. This will be useful in §4. Namely, for k ∈ R, ω ∈ S 2 , set

1. The scattering problem

(1.51)

225

U± (x, k, ω) = e−ikx·ω + V± (x, k, ω),

where V+ satisfies (1.1)–(1.3) and V− satisfies (1.1), (1.2), (1.42), with the boundary condition V± = −e−ikx·ω for x ∈ ∂K. In each case, k is not restricted to be positive; we take any k ∈ R (using (1.49) for k = 0). It is easy to see that, for any k > 0, V± (x, k, ω) = v± (x, kω), while, for k < 0, V± (x, k, ω) = v∓ (x, −|k|ω) = v∓ (x, kω). Consequently, (1.52)

k > 0 =⇒ U± (x, k, ω) = u± (x, kω), k < 0 =⇒ U± (x, k, ω) = u∓ (x, kω).

Similarly, we can define A± (ω, θ, k) for k ∈ R. Note that as r → +∞, e±ikr a± (∓ω, θ, k), r e∓ikr a∓ (±ω, θ, k). k < 0 =⇒ V± (rθ, k, ω) ∼ r

k > 0 =⇒ V± (rθ, k, ω) ∼ (1.53)

Exercises 1. Let v solve (1.1)–(1.3), with f ∈ H 1 (∂K), with k > 0. Show that  ∂v v dS Φ = π Im ∂ν ∂K



satisfies Φ = π Im

∂v v dS , ∂ν

|x|=R

for all R such that K ⊂ BR (0), and that    2  2  π  ∂v  2 2 Φ = lim dS = πk α(f, θ, k) dθ.   + k |v| R→∞ 2k ∂ν |x|=R

S2

The quantity Φ is called the flux of the solution v. Show that Φ = 0 implies v = 0. (Hint: Refer to the proof of Proposition 1.1.) 2. Investigate solutions of (1.1)–(1.3) for f ∈ H s (∂K) with s < 3/2. (Hint: When extending f to f # ∈ H s+1/2 (Ω), use a parametrix construction for the Dirichlet problem for Δ + k2 .) 3. If (Δ+k2 )v(x) = 0 for x ∈ O, open in Rn , note that w(x, y) = v(x)eky is harmonic on O ×R ⊂ Rn+1 . Deduce that v must be real analytic on O, as asserted in the unique continuation argument used to prove Proposition 1.1. 4. With a(ω, θ, k) defined for k ∈ R so that (1.53) holds, show that (1.54) Relate this to (1.47).

k > 0 =⇒ a(ω, θ, −k) = a(ω, θ, k).

226 9. Scattering by Obstacles 5. If the obstacle K2 is obtained from K1 by translation, K2 = K1 + η, show that the scattering amplitudes are related by aK2 (ω, θ, k) = eik(ω−θ)·η aK1 (ω, θ, k). The following exercises deal with the operator H = −Δ + V on R3 , assuming V (x) is a real-valued function in C0∞ (R3 ). We consider the following variant of (1.1)–(1.3), given f ∈ L2comp (R3 ): (1.55)

(Δ − V + k2 )v = f 

(1.56)

|rv(x)| ≤ C,

r

∂v − ikv ∂r

on R3 ,

 → 0, as r → ∞.

6. Show that if k > 0 and v satisfies (1.55)–(1.56) and f = 0, then v = 0. (Hint: Modify the proof of Proposition 1.1, to get v(x) = 0 on R3 \ BR , given V supported on BR . Then use the following unique continuation result: Theorem UCP. If L is a second-order, real, scalar, elliptic operator on a connected region Ω, Lv = 0 on Ω, and v = 0 on a nonempty open set O ⊂ Ω, then v = 0 on Ω. A proof of this theorem can be found in [Ho], or in Chap. 14 of [T3].) 7. Show that H has no positive eigenvalues. (Hint: Use similar reasoning, with an appropriate variant of Lemma 1.2.) Obtain an analogue of Proposition 7.3 of Chap. 8, regarding negative eigenvalues. 8. Modify the proof of Theorem −1 1.3 to obtain a (unique) solution of (1.55)–(1.56), as a f , as ε  0, given k > 0. Show that (parallel to (1.4)) limit of −H + (k + iε)2 the solution v satisfies    (1.57) v(x) = − V (y)v(y) + f (y) g(x, y, k) dy = R(k)(V v + f ). This is called the Lippman–Schwinger equation. 9. Let u(x, ξ) = e−ix·ξ + v(x, ξ), where v satisfies (Δ − V + k2 )v = V (x)e−ix·ξ ,

k2 = |ξ|2 ,

and (1.56). Establish an analogue of Proposition 1.6 and an analogue of (1.41), yielding a(−ω, θ, k). Note the following case of (1.57):  (1.58) v+ (x, ξ) = − V (y)u+ (x, ξ)g(x, y, k) dy. 10. Note that the argument involving (1.48)–(1.50) has no analogue for the k = 0 case of (1.55)–(1.56). Reconsider this fact after looking at Exercise 9 of §9.

2. Eigenfunction expansions

227

2. Eigenfunction expansions The Laplace operator on Ω with the Dirichlet boundary condition, that is, with domain D(Δ) = H01 (Ω) ∩ H 2 (Ω), is self-adjoint and negative, so by the spectral theorem there is a projection-valued measure dE(λ) such that 



ϕ(−Δ)v =

(2.1)

0

ϕ(λ) dE (λ)v,

for any bounded continuous function ϕ. Furthermore, this spectral measure is given in terms of the jump of the resolvent across the real axis: 1 ε→0 2πi

(2.2) ϕ(−Δ)v = lim





ϕ(λ) (Δ + λ − iε)−1 − (Δ + λ + iε)−1 v dλ.

 −1 Using the kernel G(x, y, k + iε) for Δ + (k + iε)2 , we can write this as   2 ∞ (2.3) ϕ(−Δ)v(x) = lim ϕ(k 2 ) Im G(x, y, k + iε)v(y) dy k dk . ε0 π 0 Ω

From the limiting behavior G(x, y, k + iε) −→ G(x, y, k) established in §1, we can draw the following conclusion. Proposition 2.1. The operator Δ on Ω has only absolutely continuous spectrum. For any continuous ϕ with compact support, we have   √ 2 ∞ (2.4) ϕ( −Δ)v(x) = Im G(x, y, k)v(y) dy ϕ(k) k dk . π 0 Ω

The meaning of the first statement of the proposition is that the spectral measure is absolutely continuous with respect to Lebesgue measure. The primary goal of this section is to give the spectral decomposition of the Laplace operator on Ω in terms of the “eigenfunctions” u(x, ξ) defined by (1.32)–(1.33). We use a modified version of an approach taken in [Rm]. In view of (2.4), the following result plays a key role in achieving the spectral decomposition. Proposition 2.2. We have the identity (2.5)

k Im G(x, y, k) = 16π 2

 u(x, kω) u(y, kω) dω. S2

228 9. Scattering by Obstacles

Proof. We obtain this identity from the asymptotic result of Proposition 1.6, as follows. Applying Green’s theorem to G(x, y, k) and G(x, y, k), and using the fact that they both vanish for x ∈ ∂K, we have Im G(x, y, k)    ∂ ∂ (2.6) = 1 G(x, y, k) G(z, y, k) dS (z), G(z, y, k) − G(x, y, k) 2i ∂|z| ∂|z| SR

for R large, where SR = {z ∈ R3 : |z| = R}. Letting R → ∞, and using (1.34) and (1.38), gives (2.5) in the limit. In view of (2.5), we can write the identity (2.4) as (2.7)

√ ϕ( −Δ)v(x) = (2π)−3

  u(x, ξ)u(y, ξ)v(y)ϕ(|ξ|) dy dξ. R3

Ω

Therefore, we are motivated to define the following analogues of the Fourier transform:    (2.8) Φv (ξ) = (2π)−3/2 v(y)u(y, ξ) dy Ω

and (2.9)

 ∗  Φ w (x) = (2π)−3/2

 u(x, ξ)w(ξ) dξ. R3

We aim to prove that Φ defines a unitary transformation from L2 (Ω) onto L2 (R3 ), with inverse Φ∗ . Note that §1 gives the estimate |u(x, ξ)| ≤ 1 + C(ξ) x −1 ,

(2.10)

with C(ξ) locally bounded, but we have obtained no bound on C(ξ) as |ξ| → ∞, so our analysis of Φ and Φ∗ will require some care. The following results on Φ and Φ∗ are elementary. Lemma 2.3. We have (2.11) (2.12)

Φ : C0∞ (Ω) −→ C(R3 ), 3 ∞ ∞ Φ∗ : L∞ comp (R ) −→ L (Ω) ∩ C (Ω),

and (2.13)

3 (Φ∗ w, v) = (w, Φv), for v ∈ C0∞ (Ω), w ∈ L∞ comp (R ).

2. Eigenfunction expansions

229

We also note that (2.7) gives √   (2.14) ϕ( −Δ)v = Φ∗ ϕ(|ξ|)Φv , for v ∈ C0∞ (Ω), ϕ ∈ C0∞ (R). Using these results, we will be able to establish the following. Proposition 2.4. If v ∈ C0∞ (Ω), then Φv ∈ L2 (R3 ) and Φv L2 (R3 ) = v L2 (Ω) .

(2.15)

Consequently, Φ has a unique extension to an isometric map Φ : L2 (Ω) −→ L2 (R3 ),

(2.16)

and Φ∗ has a unique continuous extension to a continuous map Φ∗ : L2 (R3 ) −→ L2 (Ω),

(2.17) the adjoint of (2.16).

Proof. Given ϕ ∈ C0∞ (R), v ∈ C0∞ (Ω), we have  (2.18)

  ϕ(|ξ|)Φv, Φv) = Φ∗ ϕ(|ξ|)Φv, v √ = (ϕ( −Δ)v, v).

(by (2.13)) (by (2.14))

In other words,  (2.19)

2  √  ϕ(|ξ|) Φv(ξ) dξ = ϕ( −Δ)v, v .

R3

Now let ϕ  1. The monotone convergence theorem applies, so  (2.20)

Φv(ξ) 2 dξ = (v, v).

R3

This proves the proposition. In order to prove that (2.16) is surjective—hence unitary—we will need to know that Φ∗ in (2.17) is injective. Before proving this, it will be useful to establish the following. Proposition 2.5. For any even ϕ ∈ Co (R) (i.e., ϕ continuous and ϕ(t) → 0 as |t| → ∞), and for any w ∈ L2 (R3 ), (2.21)

√   Φ∗ ϕ(|ξ|)w = ϕ( −Δ)Φ∗ w.

230 9. Scattering by Obstacles

Proof. It suffices to establish this identity for w ∈ C0∞ (R3 ). For such w, we have (1 − Δ)Φ∗ w(x) = (2π)−3/2



  u(x, ξ) ξ 2 w(ξ) dξ = Φ∗ ξ 2 w ,

R3

the left side a priori a distribution on Ω. By (2.17), we know that Φ∗ ( ξ 2 w) ∈ L2 (Ω). The integral above clearly belongs to C ∞ (Ω) and vanishes on ∂Ω. Thus Φ∗ w(x) belongs to the domain of Δ∗c , where D(Δc ) = {u ∈ C ∞ (Ω) : u = 0 on ∂Ω, supp u bounded}. It follows from Proposition 2.6 of Chap. 8 that Δ is essentially self-adjoint on D(Δc ), so we conclude that Φ∗ w(x) belongs to the domain of Δ, namely, to H01 (Ω) ∩ H 2 (Ω). An inductive argument shows that Φ∗ w belongs to the domain of each selfadjoint operator (1 − Δ)K and   (1 − Δ)K Φ∗ w(x) = Φ∗ ξ 2K w . Replacing w by ξ −2K w, we deduce   Φ∗ ξ −2K w = (1 − Δ)−K Φ∗ w, for all w ∈ C0∞ (R3 ). From this we get   Φ∗ |ξ|2j ξ −2K w = Δj (1 − Δ)−K Φ∗ w. Consequently, the identity (2.21) is valid for any ϕ(t) = t2j t −2K , j < K. Now the space of finite linear combinations of such ϕ is dense in the space of even elements of Co (R), with the sup norm, by the Stone-Weierstrass theorem, so (2.21) holds in general. We also have the following dual result. Proposition 2.6. For v ∈ L2 (Ω), ϕ ∈ Co (R) even, we have (2.22)

  √ Φ ϕ( −Δ)v (ξ) = ϕ(|ξ|)(Φv)(ξ).

Proof. Since, by Proposition 2.4, Φ and Φ∗ are L2 -continuous and adjoints of each other, this follows directly from (2.21). We now prove the asserted unitarity of Φ and Φ∗ . Proposition 2.7. The map Φ∗ is injective on L2 (R3 ). Hence the maps (2.16) and (2.17) are unitary and are inverses of each other.

Exercises

231

Proof. By Proposition 2.5, if w ∈ ker Φ∗ , then ϕ(|ξ|)w ∈ ker Φ∗ , for any ϕ ∈ C0∞ (R). Hence if ker Φ∗ is nonzero, it contains an element with compact support. Let w denote such an element. Then  (2.23) 0 = u(y, ξ)ϕ(|ξ|)w(ξ) dξ, for all y ∈ Ω, R3

for any continuous ϕ, the integral being absolutely convergent. This being the case, we can take ϕ(|ξ|) = g(x, y, |ξ|).

(2.24)

Also, we can use ϕ(|ξ|) = ∂g(x, y, |ξ|)/∂|y|, and we can also replace u(y, ξ) by ∂u(y, ξ)/∂|y|. Consequently, for all r > R0 , such that K ⊂ {x ∈ R3 : |x| ≤ R0 }, we have  (2.25)

 |y|=r

  ∂u ∂g (x, y, |ξ|) − g(x, y, |ξ|) (y, ξ) dS (y) dξ w(ξ) u(y, ξ) ∂|y| ∂|y| = 0,

for all x ∈ R3 . In the limit r → ∞, this gives  (2.26)

0=

w(ξ)e−ix·ξ dξ, for all x ∈ R3 .

In other words, the Fourier transform of w vanishes identically. This implies w = 0 and completes the proof. If we replace u(x, ξ) = u+ (x, ξ) by u− (x, ξ), given by (1.43)–(1.45), we can define the operator Φ− by (2.27)

(Φ− v)(ξ) = (2π)−3/2

 v(y)u− (y, ξ) dy. Ω

The arguments as above show that Φ− provides a unitary operator from L2 (Ω) onto L2 (R3 ), and the intertwining property (2.22) also holds for Φ− . The relation between Φ− and Φ is important in scattering theory; often we denote Φ by Φ+ to emphasize this.

Exercises √ 1. If ϕ ∈ C0∞ (R) is even, show that the Schwartz kernel of ϕ( −Δ) is given by  (2.28) Kϕ (x, y) = (2π)−3 u(x, ξ)u(y, ξ)ϕ(|ξ|) dξ. R3

232 9. Scattering by Obstacles In particular, (2.29)

Kϕ (x, x) = (2π)−3



|u(x, ξ)|2 ϕ(|ξ|) dξ.

R3 2

2. Show that (2.29) is also valid for ϕ(λ) = ϕt (λ) = e−tλ , given t > 0. (Hint: Let ϕj ∈ C0∞ (R), ϕj ϕt .) 3. Show that the heat kernel Ht (x, y) on Ω × Ω of etΔ , with Dirichlet boundary condition, has the pointwise bound 2

Ht (x, y) ≤ (4πt)−3/2 e−|x−y|

/4t

.

Deduce that, for each x ∈ Ω,  2 (2π)−3 |u(x, ξ)|2 e−t|ξ| dξ ≤ (4πt)−3/2 , R3

and hence



(2.30)

|u(x, ξ)|2 dξ ≤ C R3 .

|ξ|≤R

4. Deduce that (2.28) and (2.29) remain valid for even ϕ ∈ S(R), indeed, for continuous even ϕ satisfying |ϕ(λ)| ≤ C λ −4−ε , ε > 0. 5. Verify that (2.26) follows from (2.25). (Hint: If e−iy·ξ is substituted for u(y, ξ) in (2.25), Green’s formula applies. If v(y, ξ) is substituted, use the asymptotic behavior to show that the inner integral tends to 0 as r → ∞.) 6. Produce results parallel to those of this section for H = −Δ + V , given V ∈ C0∞ (R3 ), real, u(x, ξ) as in Exercise 9 of §1. Show that Φ : Hc → L2 (R3 ) is unitary, where Hc is the orthogonal complement of the set of eigenfunctions of H (with negative eigenvalue, if any). To what extent does k = 0 cause a problem?

3. The scattering operator In §2 we produced the two unitary operators Φ± : L2 (Ω) −→ L2 (R3 ),

(3.1) defined for f ∈ C0∞ (Ω) by (3.2)

−3/2



(Φ± f )(ξ) = (2π)

u± (y, ξ)f (y) dy. Ω

From these one constructs the unitary operator (3.3)

S = Φ+ Φ∗− : L2 (R3 ) −→ L2 (R3 ),

3. The scattering operator

233

√ called the scattering operator. Recall that Φ+ and Φ− intertwine ϕ( −Δ) on L2 (Ω) with multiplication by ϕ(|ξ|) on L2 (R3 ), for ϕ ∈ Co (R). It follows that S commutes with such ϕ(|ξ|): Sϕ(|ξ|) = ϕ(|ξ|)S.

(3.4)

From the definition (3.3) we see that S is uniquely characterized by the property   S ϕ(|ξ|)u− (y, ·) = ϕ(|ξ|)u+ (y, ·), for all y ∈ Ω,

(3.5)

for all ϕ ∈ C0∞ (R). We will relate the operator S to “wave operators” in §5. We aim to establish the following formula for S, in terms of the scattering amplitude a(ω, θ, k) defined in §1. Proposition 3.1. For g ∈ C0∞ (R3 ), we can write (Sg)(ξ) = S(k) g(kω),

(3.6)

ξ = kω,

ω ∈ S2,

where, for each k ∈ R+ , S(k) is a unitary operator on L2 (S 2 ) given by S(k)f (ω) = f (ω) +

(3.7)

k 2πi

 a(ω, θ, k)f (θ) dθ. S2

Proof. Let (3.8)

w(y, kω) = u+ (y, kω) − u− (y, kω) = v+ (y, kω) − v− (y, kω).

The assertion above is equivalent to the integral identity (3.9)

k w(y, kω) = − 2πi

 a(ω, θ, k)u+ (y, kθ) dθ. S2

In order to prove this, note that, since w(x, kω) = 0 for x ∈ ∂K, Green’s theorem gives, for R > |y|, (3.10) w(y, kω) =

  w(x, kω) SR

 ∂G ∂w (y, x, k)−G(y, x, k) (x, kω) dS (x), ∂|x| ∂|x|

where SR = {x : |x| = R}. Now let R → ∞. Using the asymptotic behavior (1.34) and (1.38) for G(y, x, k) and its radial derivative (with x and y interchanged and ω replaced by θ) and the asymptotic behavior, for |x| = R → ∞, x = Rθ,

234 9. Scattering by Obstacles

e−ikR eikR a(−ω, θ, k) − a− (ω, θ, k), R R ∂w eikR e−ikR (x, kω) ∼ ik a(−ω, θ, k) + ik a− (ω, θ, k), ∂|x| R R w(x, kω) ∼

(3.11)

with (3.12)

a− (ω, θ, k) = a(ω, θ, k),

as in (1.46)–(1.47), we see that the integrand in (3.10) is asymptotic to (3.13)

2ik a− (ω, θ, k)u+ (y, kω) + o(R−2 ), 4πR2

(the terms involving e2ikR canceling out), so passing to the limit R → ∞ gives (3.9) and proves the proposition. We can rewrite the formula (3.7) as (3.14)

S(k) = I +

k A(k), 2πi

with  (3.15)

A(k)f (ω) =

a(ω, θ, k)f (θ) dθ. S2

Note that unitarity of S(k) on L2 (S 2 ) is equivalent to the identity (3.16)

1 k A(k)∗ − A(k) = A(k)∗ A(k), 2i 4π

that is, to the integral identity (3.17)

k 1 a(θ, ω, k) − a(ω, θ, k) = 2i 4π

 a(η, ω, k)a(η, θ, k) dη. S2

The special case of this where ω = θ is known as the optical theorem: (3.18)

Im a(ω, ω, k) = −

k 4π



|a(η, ω, k)|2 dη.

S2

It is useful to know integral identities for the scattering amplitude. We note one that follows from the characterization

3. The scattering operator

v(rθ, kω) ∼ r−1 eikr a(−ω, θ, k),

(3.19)

235

r→∞

and the integral identity (a consequence of Green’s identity) (3.20)

v(x, kω) =

  v(y, kω) ∂K

 ∂v ∂g (x, y, k) − g(x, y, k) (y, kω) dS (y). ∂νy ∂ν

We evaluate the integrand on the right as x = rθ, r → ∞. Using (1.36), that is, (3.21)

g(x, y, k) ∼ −(4πr)−1 eikr e−ikθ·y ,

x = rθ, r → ∞,

we find from (3.19) and (3.20) that  ∂ −ikθ·y 1 e eikω·y dS (y) a(ω, θ, k) = − 4π ∂ν ∂K

(3.22) +

1 4π



e−ikθ·y

∂K

∂ v(y, −kω) dS (y). ∂ν

The first term on the right side of (3.22) can be written as (3.23)

ik 4π

 ∂K

    ik ˆ K k(ω − θ) , θ·A ν(y) · θ eik(ω−θ)·y dS (y) = 4π

where, for ξ ∈ R3 , (3.24)

ˆ K (ξ) = A



ν(y) eiξ·y dS (y).

∂K

ˆ K (ξ) clearly extends to an entire analytic function of ξ ∈ C3 . For The function A 3 ξ ∈ R tending to infinity, one can (typically) find the asymptotic behavior of ˆ K (ξ) via the stationary phase method. Note that A ˆ K (0) = 0. A

(3.25)

One way of writing the last term in (3.22) is the following. For any real k, or more generally for Im k ≥ 0, define the Neumann operator N (k) on f ∈ H 1 (∂K) to be the value of ∂v/∂ν in L2 (∂K), where v is the unique solution to the scattering problem (1.1)–(1.3). Define the functions eξ on ∂K by (3.26)

eξ (y) = eiy·ξ ,

Then the last term in (3.22) is

y ∈ ∂K.

236 9. Scattering by Obstacles

 1  N (k)ekω , ekθ L2 (∂K) . 4π

(3.27)

Consequently, the formula for the scattering amplitude can be written as (3.28)

a(ω, θ, k) =

    ik ˆ K k(ω − θ) + 1 N (k)ekω , ekθ 2 θ·A . L (∂K) 4π 4π

We will investigate the Neumann operator further in §7. We can produce a variant of the formula (3.22) by using G(x, y, k) instead of g(x, y, k) in (3.20). We then get  ∂G (3.29) v(x, kω) = − e−ikω·y (x, y, k) dS (y). ∂νy ∂K

Using the limiting behavior for G(x, y, k) as |x| → ∞, which follows from (1.34), we have  ∂u 1 (y, kθ) dS (y). (3.30) a(ω, θ, k) = − eikω·y 4π ∂ν ∂K

If we write u(y, kθ) = e−ikθ·y + v(y, kθ), this becomes a sum of two terms. The first is identical to the first term in (3.22), while the second differs from the second term in (3.22) precisely by the replacement of (ω, θ) by (−θ, −ω). From this observation, we can derive the following identity, called the reciprocity relation: (3.31)

a(ω, θ, k) = a(−θ, −ω, k).

  ˆ K k(ω − θ) = 0. Since ω + θ and To see this, it suffices to show that k(ω + θ) · A ω − θ are orthogonal for unit ω and θ, this is equivalent to the observation that (3.32)

ˆ K (ξ) is parallel to ξ, for ξ ∈ R3 , A

and this follows easily from Green’s theorem.

Exercises 1. Show that (3.5) follows from   u− (x, ξ)ϕ(|ξ|)u− (y, ξ) dξ = u+ (x, ξ)ϕ(|ξ|)u+ (y, ξ) dξ, which in turn follows from (2.28). ˆ K (ξ), and then on the reci2. Fill in the details on the identities (3.25) and (3.32) for A procity relation (3.31). What is the intuitive content of (3.31)? 3. If you set S− = Φ− Φ∗+ , obtain an analogue of (3.7), with a(ω, θ, k) replaced by a− (ω, θ, k).

4. Connections with the wave equation

237

 4. In case f = −e−ikx·ω ∂K , with corresponding scattered wave v, show that the flux Φ studied in Exercise 1 of §1 is given by  |v(x, kω)|2 dS (x) σ(ω, k) = lim πk r→∞

 = πk

|x|=r

|a(−ω, θ, k)|2 dθ.

S2

We call σ(ω, k) the scattering cross section. Using the optical theorem and the reciprocity relation (3.31), show that σ(ω, k) = −4π 2 Im a(ω, ω, k). 5. Generalizing (3.22), show that, for f ∈ H s (∂K), BK (k)f (rθ) ∼ r−1 eikr AK (k)f (θ) + o(r−1 ),

(3.33) as r → ∞, where (3.34)

AK (k)f (θ) =

1 4π



   e−ikθ·y ik ν(y) · θ f (y) + N (k)f (y) dS (y).

∂K

6. Make a parallel study of the scattering operator for H = −Δ + V, V ∈ C0∞ (R3 ), realvalued, using results from the exercises in §§1 and 2. To begin, use the unitary operators Φ± : Hc → L2 (R3 ) to construct S = Φ+ Φ∗− . Show that, parallel to (3.22),  1 a(−ω, θ, k) = − V (y)u(y, kω)e−ikθ·y dy, 4π or equivalently, (3.35)

a(ω, θ, k) = −



π 1/2   1 V k(θ − ω) − V (y)v(y, −kω)e−ikθ·y dy. 2 4π

4. Connections with the wave equation The initial-value problem for the wave equation on R × Ω, with Dirichlet boundary conditions on R × ∂K, is of the following form: ∂2u − Δu = 0, ∂t2

(4.1) (4.2)

u(0, x) = f (x),

ut (0, x) = g(x),

for t ∈ R, x ∈ Ω, with (4.3)

u(t, x) = 0, for x ∈ ∂K.

238 9. Scattering by Obstacles

As we know, given f ∈ H01 (Ω), g ∈ L2 (Ω), there is a unique solution u belonging to C(R, H01 (Ω)) ∩ C 1 (R, L2 (Ω)) to this problem, given in terms of functions of the self-adjoint operator Δ on L2 (Ω), with domain H01 (Ω) ∩ H 2 (Ω), as (4.4)

u(t, x) = (cos tΛ)f (x) + (Λ−1 sin tΛ)g(x),

where Λ = (−Δ)1/2

(4.5)

is the unique nonnegative, self-adjoint square root of −Δ. Recall that the domain of Λ is precisely D(Λ) = H01 (Ω). Alternatively, we can write     u f (4.6) = U (t) , ut g where U (t) is the one-parameter group of operators on H01 (Ω) ⊕ L2 (Ω) given by  (4.7)

U (t) =

 cos tΛ Λ−1 sin tΛ . −Λ sin tΛ cos tΛ

Using either of the unitary operators (4.8)

Φ± : L2 (Ω) −→ L2 (R3 ),

we can write (4.9)

(cos tΛ)f = Φ−1 ± cos t|ξ|Φ± f, −1 (Λ−1 sin tΛ)g = Φ−1 sin t|ξ|Φ± g. ± |ξ|

Note that Φ± also provide isomorphisms (4.10)

Φ± : H01 (Ω) −→ L2 (R3 , ξ 2 dξ).

The group U (t) is not a uniformly bounded group of operators on the Hilbert space H01 (Ω)⊕L2 (Ω). Indeed, with f = 0, we see from (4.4) that the best uniform estimate on u(t, ·) L2 (Ω) is (4.11)

u(t, ·) L2 (Ω) ≤ |t| g L2 (Ω) .

There is another Hilbert space on which U (t) naturally acts as a group of unitary operators, namely the space (4.12)

E = H ⊕ L2 (Ω),

where H is the completion of H01 (Ω) with respect to the norm given by

4. Connections with the wave equation

(4.13)

f 2H = Λf 2L2 (Ω) =



239

|∇f (x)|2 dx .

Ω

(Recall that f 2H 1 (Ω) = f 2L2 (Ω) + Λf 2L2 (Ω) .) If we equip H with this norm, 0 then Φ± extend to unitary operators Φ± : H −→ L2 (R3 , |ξ|2 dξ).

(4.14)

Since unitary operators are special, it is natural to use the Hilbert space (4.12) rather than H01 (Ω) ⊕ L2 (Ω). We will denote an element of E by f, g ; f ∈ H, g ∈ L2 (Ω). When U (t) is applied, this is treated as a column vector, as in (4.6); we will also use the column vector notation for elements of E when convenient. Elements of H need not belong to L2 (Ω), though they do belong to L2loc (Ω). In fact, if B is a bounded subset of Ω, the estimate u L2 (B) ≤ CB u H

(4.15)

can be established by the argument used to prove Proposition 5.2 in Chap. 4, provided K has nonempty interior. Since clearly B |∇u|2 dx ≤ Ω |∇u|2 dx , we hence have  u H . u H 1 (B) ≤ CB

(4.16)

Further estimates are given in the exercises. The unitarity of U (t) on E reflects the conservation of total energy, given by (4.17)

E(u(t)) =

u, ut 2E

 =



 |∇x u(t, x)|2 + |ut (t, x)|2 dx .

Ω

There is also the notion of local energy, given as follows. For a bounded subset B of Ω, set    |∇x u(t, x)|2 + |ut (t, x)|2 dx . (4.18) EB (u(t)) = B

Using the absolute continuity of the spectrum of Δ on L2 (Ω) established in §2, or more precisely, the absolute continuity of the spectrum of a related operator specified below, we will establish the following result on local energy decay. Proposition 4.1. Given f, g ∈ E, u, ut = U (t) f, g , we have (4.19)

EB (u(t)) −→ 0, as |t| −→ ∞,

240 9. Scattering by Obstacles

for any bounded B ⊂ Ω. Before starting the proof of this proposition, we will make some further comments on the infinitesimal generator of the unitary group U (t) on E. This is a skew-adjoint operator, and it has the form  B=

(4.20)

 0 I , −A 0

where, for f ∈ D(A) ⊂ H, (4.21)

Af = −Δf

in the distributional sense. Then B 2 is a self-adjoint operator of the form (4.22)



2

−B =

 A1 0 , 0 A2

where A1 is self-adjoint on H, A2 is self-adjoint on L2 (Ω), and they both satisfy (4.21), on their respective domains. Note that the unitary operators Φ± : H ⊕ L2 (Ω) −→ L2 (R3 , |ξ|2 dξ) ⊕ L2 (R3 ) intertwine (4.20) with multiplication (on each factor) by |ξ|2 and 2 3 2 4 2 3 4 D(Φ± B 2 Φ−1 ± ) = L (R , |ξ| ξ dξ) ⊕ L (R , ξ dξ).

In particular, the operators A1 and A2 have only absolutely continuous spectrum. Let (4.23)

1/2

Lj = Aj

be their unique nonnegative, self-adjoint square roots. Both L1 and L2 are intertwined via Φ± with multiplication by |ξ|, so we can identify them, denoting them by L, and if u, ut = U (t) f, g , we have (4.24)

u(t) = (cos tL)f + (L−1 sin tL)g, ut (t) = (−L sin tL)f + (cos tL)g.

We now begin the proof of Proposition 4.1. Since U (t) is unitary and EB (u(t)) ≤ E(u(t)) = u, ut 2E , we see that it suffices to prove the proposition for f, g in a dense subset of E. In particular, we will take (4.25)

f ∈ D(L1 ) ⊂ H,

g ∈ D(L2 ) ⊂ L2 (Ω).

4. Connections with the wave equation

241

Lemma 4.2. If f and g satisfy (4.25), then, as |t| → ∞, u(t) −→ 0 ut (t) −→ 0

(4.26)

weakly in D(L1 ) and weakly in D(L2 ).

Proof. Fix w0 ∈ D(L1 ), w1 ∈ D(L2 ). Note that Φ± f ∈ L2 (R3 , |ξ|2 ξ 2 dξ),

(4.27)

and so on, so using the images under Φ± to justify the inner-product calculations, and noting that, by (4.27), Lf ∈ H01 (Ω),

(4.28)

L2 f ∈ L2 (Ω),

Lg ∈ L2 (Ω)

(and similarly for w0 , w1 ), we obtain  (4.29)

u(t), w0

 D(L1 )

    = Lu(t), Lw0 E + u(t), w0 E     = L2 u(t), L2 w0 L2 + Lu(t), Lw0 L2 .

To examine each term, write (with j = 1 or 2)  j    L u(t), Lj w0 L2 = (Lj cos tL)f + (Lj−1 sin tL)g, Lj w0 L2  ∞ = (cos tλ) d(Fλ Lj f, Lj w0 ) 0  ∞ (sin tλ) d(Fλ Lj−1 g, Lj w0 ), +

(4.30)

0

where Fλ is the spectral measure of L2 . In light of (4.28) and the absolute continuity of Fλ , it follows that d(Fλ Lj f, Lj w0 ) and d(Fλ Lj−1 g, Lj w0 ) are finite measures on R that are absolutely continuous with respect to Lebesgue measure. Hence (4.30) is the Fourier transform of an L1 -function on R. Thus the Riemann– Lebesgue lemma implies that this tends to 0 as |t| → ∞. Similarly, 

(4.31)

ut (t), w1

 D(L2 )

    = Lut (t), Lw1 L2 + ut (t), w1 L2 .

This time, to examine each term, write (with j = 0 or 1)  (4.32)

Lj ut (t), Lj w1

 L2

  = (−Lj+1 sin tL)f + (Lj cos tL)g, Lj w1 L2  ∞ =− (sin tλ) d(Fλ Lj+1 f, Lj w1 )  0∞ (cos tλ) d(Fλ Lj g, Lj w1 ). + 0

242 9. Scattering by Obstacles

Again the Riemann–Lebesgue lemma applies, and the proof of Lemma 4.2 is complete. To derive local energy decay from this, we reason as follows. For any R < ∞, set ΩR = {x ∈ Ω : |x| < R}.

(4.33)

Then, for f ∈ H, if ιR f = f Ω , by (4.16) we have R

ιR f H 1 (ΩR ) ≤ CR f H .

(4.34)

Similarly, for any f ∈ D(L1 ),  f D(L1 ) . ιR f H 2 (ΩR ) ≤ CR

(4.35)

Thus, restricted to ΩR , u(t) is bounded in H 2 (ΩR ) and ut (t) is bounded in H 1 (ΩR ), for t ∈ R, given the hypothesis (4.25) on the initial data. Thus these two families of functions on ΩR are compact in H 1 (ΩR ) and L2 (ΩR ), respectively, by Rellich’s theorem. The weak convergence to zero of (4.26) hence implies the strong convergence to zero: (4.36)

u(t) −→ 0 in H 1 (ΩR ),

ut (t) −→ 0 in L2 (ΩR ),

as |t| → ∞, whenever f and g satisfy (4.25). Proposition 4.1 is hence proved on the dense set given by (4.25), and as we remarked before, that proves it in general. Instead of representing f, g ∈ E as a pair of functions, L2 with respect to different weights, via Φ± , it is often convenient to use the following construction, of Lax–Phillips. Namely, for f, g ∈ C0∞ (Ω), define Ψ± f, g on R × S 2 by    k2 f f (x)U± (x, k, ω) dx Ψ± (k, ω) = 3/2 g 4π Ω (4.37)  ik + 3/2 g(x)U± (x, k, ω) dx . 4π Ω

This is the same as the (formally computed) E-inner product 

(4.38)

 f, g , U± (·, k, ω), ikU± (·, k, ω) E ,

times 2−1/2 (2π)−3/2 . Note that eikt U± (x, k, ω) solves the wave equation, with Cauchy data U± (x, k, ω), ikU± (x, √ k, ω) . In terms of the operators Φ± , studied before, we can write (4.37) as 1/ 2 times (4.39)

k 2 (Φ± f )(kω) + ik(Φ± g)(kω), 2

k (Φ∓ f )(kω) + ik(Φ∓ g)(kω),

for k > 0, for k < 0.

4. Connections with the wave equation

243

Note that f ∈ H ⇔ Φ± f ∈ L2 (R3 , |ξ|2 dξ) ⇔ |ξ|2 Φ± f ∈ L2 (R3 , |ξ|−2 dξ), or, switching to polar coordinates, (4.40)

f ∈ H ⇐⇒ k 2 (Φ± f )(kω) ∈ L2 (R+ × S 2 , dk dω).

Similarly, (4.41)

g ∈ L2 (Ω) ⇐⇒ k(Φ± g)(kω) ∈ L2 (R+ × S 2 , dk dω).

Therefore, for f, g ∈ E, the quantity (4.39) belongs to (4.42)

L2 (R × S 2 , dk dω) = L2 (R, N ),

with (4.43)

N = L2 (S 2 ).

We can now establish the following. Proposition 4.3. For each choice of sign, Ψ± provides a unitary map of E onto L2 (N ). Proof. It is clear that the restrictions of Ψ± to H ⊕ 0 and to 0 ⊕ L2 (Ω) are both isometries, by the arguments leading to (4.40) and (4.41). Also, it is easy to see that the images of these spaces under Ψ± are mutually orthogonal, so Ψ± is an isometry of E into L2 (R, N ). To show that it is surjective, we show how to solve for f, g ∈ E the pair of equations (4.44)

Φ± (Lf + ig) = u0 ,

Φ∓ (Lf − ig) = u1 ,

for arbitrary u0 , u1 ∈ L2 (R3 ). Inverting the unitary operators Φ± and Φ∓ , we reduce this to a trivial system for Lf +ig and Lf −ig, easily solved for f ∈ H, g ∈ L2 (Ω), since L : H → L2 (Ω) is an isomorphism. This proves the proposition. The maps Ψ± intertwine the evolution group U (t) with a simple multiplication operator: Proposition 4.4. We have, for ϕ ∈ L2 (R, N ), (4.45)

−ikt ϕ(k, ω). Ψ± U (t) Ψ−1 ± ϕ(k, ω) = e

Proof. This follows directly from the intertwining properties of Φ± , given (4.39) and the following computation:

244 9. Scattering by Obstacles

    k 2 Φ± u(t) (kω) + ik Φ± ut (t) (kω)

= k 2 (cos kt)Φ± f + (k −1 sin kt)Φ± g

+ ik −k(sin kt)Φ± f + (cos kt)Φ± g

(4.46)

= k 2 e−ikt Φ± f + ik e−ikt Φ± g, for k > 0, with a similar computation for k < 0. The unitary maps discussed above are called “spectral representations” for U (t). In §6 we will study related maps, called “translation representations.” Note that in the case K = ∅, the functions U± (x, k, ω) become U0 (x, k, ω) = e−ikω·x , and both spectral representations coincide. We denote this free-space spectral representation by Ψ0 . It is a unitary map of E0 = H0 ⊕L2 (R3 ) onto L2 (R, N ), given in terms of the Fourier transform by Ψ0

(4.47)

  ik k2 f (k, ω) = √ fˆ(kω) + √ gˆ(kω). g 2 2

Here, H0 is the completion of C0∞ (R3 ) with respect to the norm ∇f L2 (R3 ) , mapped unitarily by the Fourier transform onto L2 (R3 , |ξ|2 dξ).

Exercises √ Let ϕ ∈ S(R) be an even function in the following exercises. Let Λ = −Δ, as in kernel of ϕ(Λ), as in (2.28). Let Δ0 be the (4.5), and let Kϕ (x, y) be the Schwartz √ free-space Laplacian on R3 , Λ0 = −Δ0 , and let Kϕ0 (x, y) be the Schwartz kernel of ϕ(Λ0 ), so, parallel to (2.28),  Kϕ0 (x, y) = (2π)−3 e−iξ·(x−y) ϕ(|ξ|) dξ. R3

Let Dϕ (x, y) = Kϕ (x, y) − Kϕ0 (x, y), where Kϕ (x, y) is set equal to 0 if x ∈ K or y ∈ K. 1. Use the formula  ∞ 1 ϕ(t) ˆ cos tΛ dt (4.48) ϕ(Λ) = √ 2π −∞ together with finite propagation speed to show that supp ϕ(t) ˆ ⊂ {|t| ≤ T } =⇒ supp Dϕ (x, y) ⊂ {|x|, |y| ≤ R + T } if K ⊂ BR (0). 2. Use (4.48) to show that, for some J = J(α, β),  ˆ L1 (R) + DtJ ϕ ˆ L1 (R) , |Dxα Dyβ Kϕ (x, y)| ≤ C ϕ for x, y ∈ Ω. 3. Use Exercises 1 and 2 to show that when ϕ ∈ S(R) is even, then Dϕ (x, y) is rapidly decreasing and is the Schwartz kernel of a trace class operator on L2 (R3 ).

5. Wave operators

245

4. Let H1 (Rn ) denote the completion of C0∞ (Rn ) with respect to the norm in (4.13). Show that if n ≥ 3, there is a natural injective map ι : H1 (Rn ) −→ S  (Rn ) and the Fourier transform maps H1 (Rn ) isomorphically onto

 FH1 (Rn ) = u ∈ L1loc (Rn ) : |ξ|u(ξ) ∈ L2 (Rn ) = L2 (Rn , |ξ|2 dξ). 5. Show that, for n ≥ 3,

L2 (Rn , |ξ|2 dξ) ⊂ Lqloc (Rn , dξ),

provided 1 ≤ q < 2n/(n + 2). Conclude that if n ≥ 3, any u ˆ ∈ FH1 (Rn ) can 2 n be written as a sum of an element of L (R ) and a compactly supported element of Lq (Rn ), given q ∈ [1, 2n/(n + 2)). Show that L2 (R2 , |ξ|2 dξ) is not contained in L1loc (R2 ). 6. Let ψσ (ξ) be the Fourier transform of x −σ . Show that if q ∈ [1, 2), then g ∈ Lqcomp (Rn ) =⇒ ψσ ∗ g ∈ L2 (Rn ), provided σ ≥ (2 − q)n/2q. (Hint: Interpolate between easy cases.) 7. Show that if n ≥ 3 and σ > 1, then (4.49)

H1 (Rn ) ⊂ L2 (Rn , x −2σ dx ).

Note that this extends the estimate (4.15) in several ways. 8. Show that if n ≥ 3, (4.50)

H1 (Rn ) ⊂ L2n/(n−2) (Rn ).

Show that this result implies (4.49). Reconsider this problem after reading §2 of Chap. 13.

5. Wave operators In this section we examine the asymptotic behavior of the unitary group U (t) on E, as t → ±∞. More precisely, we show that, as t → ±∞, U (t)Mϕ U0 (−t) f, g

converges to a limit, W± f, g ; the operators W± are called wave operators, and they are easily seen to be isometries from E0 into E. Here, E is the space constructed in §4 for Ω = R3 \ K, E0 that for the region Ω0 = R3 , and U0 (t) the “free-space” evolution operator for R3 ; Mϕ is multiplication by a function ϕ ∈ C ∞ (R3 ), equal to zero in a neighborhood of K, and equal to 1 outside a bounded set. We will show that W± have as right inverses operators Ω± = Ψ−1 0 Ψ± , where Ψ± are the unitary operators constructed in §4; Ψ0 is the corresponding operator constructed for Ω0 = R3 . Since Ω± are unitary, it will follow from this that the wave operators are also unitary. We begin with the following observation, a simple consequence of Huygens’ principle. Suppose f and g are in C0∞ (R3 ), supported in BR = {x ∈ R3 : |x| < R}. Then, for |t| > R,

246 9. Scattering by Obstacles

(5.1)

U0 (t) f, g = 0, for |x| < |t| − R.

This follows directly for the formula for the fundamental solution to the wave equation on R × R3 , which, recall from Chap. 3, is R(t, x) =

(5.2)

δ(|x| − |t|) . 4πt

Consequently, if K ⊂ BR and if f and g are supported in BR0 , then (5.3) U (s)U0 (−s) f, g = U (R + R0 )U0 (−R − R0 ) f, g , for s > R + R0 , with a similar identity for s < −R − R0 . We can insert an Mϕ between the two unitary factors on the left if ϕ(x) = 1 for |x| ≥ R, without altering anything. It follows that (5.4)

W± f, g = lim U (−t)Mϕ U0 (t) f, g

t→±∞

exists, for f, g in the dense subset of E0 consisting of compactly supported functions. Consequently, the limits exist on all of E0 , and the operators W± , called wave operators, are isometries from E0 into E. A major result, established below, is that these operators are actually unitary, from E0 onto E. In fact, consider the following operators: Ω± = Ψ−1 0 Ψ± : E −→ E0 .

(5.5)

By Proposition 4.3 we know Ω± are unitary. We aim to establish the following result. Proposition 5.1. We have (5.6)

Ω+ W+ = I and Ω− W− = I on E0 .

In order to prepare to prove this, we introduce the following set of initial data for the wave equation. If R is sufficiently large that K ⊂ BR , set  D0+ (R) = f, g ∈ C0∞ (R3 ) ⊕ C0∞ (R3 ) : U0 (t) f, g = 0,  (5.7) for t > 0, |x| < R + t . In particular, f and g vanish near K, and we can regard f, g as an element of E0 or of E, and (5.8) Clearly,

f, g ∈ D0+ (R) =⇒ U0 (t) f, g = U (t) f, g , for t > 0.

5. Wave operators

247

U0 (t)D0+ (R) ⊂ D0+ (R), for t > 0,

(5.9)

though not for t < 0. Also, by the argument involving Huygens’ principle discussed above, it is clear that  U0 (t)D0+ (R) is dense in E0 . (5.10) t 0. We will show that, for t large, this can be dominated by a small quantity. Indeed, an examination of u(t), ut (t) = U0 (t) f, g via the formula (5.2) for the Riemann function shows that, for t large and positive, ∇x u(t, x) is approximately radial, and ut (t, x) ∼ ur (t, x). Thus (5.15) is equal to (5.16)

   ∂V + + (ik)ut (t, x)V + dx + o(1), ur (t, x) ∂r

Ω

as t → +∞. In light of the radiation condition for V+ , the two terms in this integral cancel out, up to a remainder that vanishes as t → +∞; this proves the lemma. In view of (5.11), we now know that (5.17)

Ω+ W+ = I

on D0+ (R).

248 9. Scattering by Obstacles

Now it follows easily from the definition that W± U0 (t) = U (t)W± , for all t,

(5.18)

and from Proposition 4.4 it follows that Ω± U (t) = U0 (t)Ω± , for all t.

(5.19)

Given that (5.17) holds when applied to U0 (t) f, g , provided this belongs to D0+ (R), we deduce that (5.20)

Ω+ W+ f, g = f, g , for f, g ∈ U0 (−t)D0+ (R), t > 0;

in other words, (5.21)

Ω+ W+ = I

on

 t>0

U0 (−t)D0+ (R).

In light of (5.10), this implies that Ω+ W+ = I on E0 , establishing the first identity in (5.6). The second identity is proved in the same fashion, and Proposition 5.1 is done. The unitarity of Ω± then gives the following result, known as the completeness of the wave operators. Corollary 5.3. The wave operators W± are unitary from E0 onto E. We have the identities (5.22)

W± = Ψ−1 ± Ψ0 .

Note that (5.6) implies the surjectivity of Ω± , hence of Ψ± , since the invertibility of Ψ0 is obvious (just the Fourier inversion formula). Thus the proof of Proposition 5.1 contains an alternative proof of Proposition 4.3, and hence of Proposition 2.7. The operator   Ψ+ Ψ−1 (5.23) S1 = W+−1 W− = Ψ−1 − Ψ0 , 0 a unitary operator on E0 , is often called the scattering operator. In view of the simple nature of Ψ0 , it is equally convenient to call the unitary operator on L2 (R, N ): (5.24)

S = Ψ+ Ψ−1 − ,

also a scattering operator. Note that, if we make the identification L2 (R, N ) = L2 (R+ × S 2 ) ⊕ L2 (R− × S 2 ),

5. Wave operators

249

and follow with the natural unitary map L2 (R± × S 2 ) → L2 (R3 ) involving polar coordinates, we can write   0 Φ+ Φ−1 − . (5.25) S= 0 Φ− Φ−1 + The operator S = Φ+ Φ−1 − is the scattering operator studied in §3; the other operator, Φ− Φ−1 + = S− , appears in Exercise 2 of §3. Another consequence of the unitarity of the wave operators is the following nontrivial variant of (5.10). Proposition 5.4. Pick R so that K ⊂ BR . Then  (5.26) U (t)D0+ (R) is dense in E. t R + R0 . This is equivalent to U0 (−t)JU (t) f3 , g3 = W+−1 f3 , g3 ,

(5.33)

for f3 , g3 = W+ f1 , g1 , t > R + R0 . This gives (5.31) on a dense subset of E, hence on all of E in view of the uniform boundedness of U (t) and U0 (t).

Exercises The following exercises deal with the existence and completeness of Schr¨odinger wave operators: W± f = lim

(5.34)

t→±∞

eitH e−itH0 f,

where H0 = −Δ, H = −Δ + V , acting on functions on Rn . 1. Show that W± ∈ L(L2 (Rn )) exists provided that, for each f ∈ C0∞ (Rn ),  ∞ (5.35) V e−itH0 f  dt < ∞. 0

itH −itH0

t

isH

f = 0 e V e−isH0 f ds.) Note that when W± exist, they are (Hint: e e isometries (i.e., W± f  = f  for all f ∈ L2 (Rn )). 2. Show that f ∈ C0∞ (Rn ) implies eitH0 f L∞ ≤ C t −n/2 . Deduce that W± exists if 2 V ∈ L2 (Rn ). (Hint: eitΔ δ(x) = (4πit)−n/2 e−|x| /4it .) 3. Show that if q = 2/(1 − θ) ∈ [2, ∞), then f ∈ C0∞ (Rn ) implies eitH0 f Lq (Rn ) ≤ C t −nθ/2 . Deduce that W± exists if V ∈ Lr (Rn ), with r < n. In particular, W± exists provided |V (x)| ≤ C x −σ , σ > 1. 4. Show that, for any f, g ∈ L2 (Rn ), (g, e−itH0 f ) → 0 as |t| → ∞. Use this to show that if gj is an eigenfunction of H, then (e−itH gj , eitH0 f ) → 0 as |t| → ∞, for all f ∈ L2 (Rn ). Hence, for W± given by (5.34), R(W± ) ⊂ Hc . 5. Suppose V ∈ C0∞ (R3 ), so we have Φ± by Exercise 5 of §2. Let Φ0 be the incommutes with multiplication by verse Fourier transform. Show that Φ± W± Φ−1 0 2 eis|ξ| , for all s ∈ R. Hence it commutes with ϕ(|ξ|) for all ϕ ∈ Co (R). (Hint: W± = eisH W± e−isH0 .) 6. When the conditions of Exercise 1 hold, show that  ∓∞ εe±εt eitH e−itH0 f dt W± f = lim ε0

0

and hence that (5.36)

  (W± − I)f, g = lim

ε0



∓∞ 0

  i eitH V e−itH0 f, g e±εt dt.

6. Translation representations and the Lax–Phillips semigroup Z(t)

251

7. Choosing the + sign, show that the integral on the right side of (5.36) is equal to  −∞     2 i(Φ+ g)(ξ)V (x) e−it(H0 −|ξ| +iε) f (x) u+ (x, ξ) dx dξ dt 0 (5.37)      Φ+ g (ξ)V (x) (H0 − |ξ|2 + iε)−1 f (x) u+ (x, ξ) dx dξ. = (Hint: Use Φ+ to intertwine eitH with eitH0 .) 8. If V ∈ C0∞ (R3 ), show that the limit of (5.37) as ε  0 is equal to 1 4π



  e−ik|x−y| Φ+ g (ξ)V (x) f (y)u+ (x, ξ) dy dx dξ, |x − y|

provided f ∈ C0∞ (R3 ) and (Φ+ g)(ξ) is supported on |ξ| ∈ [a, b] ⊂⊂ (0, ∞). Here k = |ξ|. Using (1.58), write this as  (Φ+ g)(ξ) v+ (y, ξ) f (y) dy dξ = −(Φ+ f, Φ+ g) + (Φ0 f, Φ+ g). − 9. Using the previous exercises, show that, given V ∈ C0∞ (R3 ) (real-valued), we have (W± f, g) = (Φ0 f, Φ± g) for all f, g ∈ L2 (R3 ), hence W± = Φ−1 ± Φ0 . Deduce the completeness of the wave operators: R(W± ) = Hc . Compare arguments in Chap. 5 of [Si], dealing with a larger class of potentials. Completeness for a nearly maximal class of potentials to which Exercise 3 applies is treated in Chap. 13 of [RS]. Long-range potentials are treated in Chap. 3 of [Ho].

6. Translation representations and the Lax–Phillips semigroup Z(t) From the “spectral representations” Ψ± : E → L2 (R, N ) defined in §4, which, as shown in Proposition 4.4, intertwine U (t) with multiplication by e−ikt , we construct “translation representations,” unitary operators T± : E −→ L2 (R, N ),

(6.1)

by taking the Fourier transform with respect to k: (6.2)



     ∞ f f eiks Ψ± (s, ω) = (2π)−1/2 (k, ω) dk . g g −∞

Consequently, Proposition 4.4 implies

252 9. Scattering by Obstacles

(6.3)

T± U (t)T±−1 f (s, ω) = f (s − t, ω).

The operators T± are useful for exposing various features of U (t), and we explore this in the current section. We begin with a look at the free-space translation representation T0 , a unitary map from E0 onto L2 (R, N ) given by using Ψ0 in (6.2). We can produce an explicit formula for T0 using the formula (4.47) for Ψ0 , which we recall is   f 1/2 Ψ0 (6.4) 2 g (kω). (k, ω) = k 2 fˆ(kω) + ikˆ g The formula for T0 is expressed naturally in terms of the Radon transform, which is defined (initially for f ∈ S(R3 )) by  (6.5)

f (y) dS (y),

Rf (s, ω) = y·ω=s

for s ∈ R, ω ∈ S 2 . Note that the Fourier transform can be expressed as  ∞ −3/2 ˆ (6.6) f (kω) = (2π) e−iks Rf (s, ω) ds. −∞

Thus, taking the inverse Fourier transform in k, we have  ∞ 1/2 (6.7) Rf (s, ω) = (2π) eiks fˆ(kω) dk . −∞

In light of this, we see that taking the Fourier transform with respect to k of (6.4) gives  

1 2 f (6.8) T0 −∂s Rf (s, ω) + ∂s Rg(s, ω) . (s, ω) = 4π g The unitarity of T0 gives rise to the inversion formula  1 k(x · ω, ω) dω, f (x) = 2π S2  (6.9) 1 g(x) = − ∂s k(x · ω, ω) dω, 2π S2

for f, g in terms of (6.10)

k(s, ω) = T0

  f (s, ω). g

6. Translation representations and the Lax–Phillips semigroup Z(t)

253

This result is related to the Radon inversion formula,  1 (6.11) f (x) = ∂s2 Rf (x · ω, ω) dω, 8π 2 S2

which can be deduced from (6.9), or directly from (6.6) and the Fourier inversion formula. In view of (6.3), for T0 , we see that the solution to the free-space wave equation utt − Δu = 0 with initial data f, g can be written as  1 k(x · ω − t, ω) dω, (6.12) u(t, x) = 2π S2

where k(s, ω) is given by (6.10). More fully, by (6.8), (6.13)

u(t, x) =

1 8π 2





−∂s2 Rf (x · ω − t, ω) + ∂s Rg(x · ω − t, ω) dω.

S2

Note that if f and g are supported in BR0 = {|x| < R0 }, then, by (6.5), Rf (s, ω) and Rg(s, ω) vanish for |s| > R0 . Therefore, Rf (x·ω −t, ω) and Rg(x·ω −t, ω) vanish for |x| < |t| − R0 . Thus from (6.13) we rederive the Huygens principle, that u(t, x) vanishes for |x| < |t| − R0 in this case. Use of T0 and T± will augment arguments involving the Huygens principle made in §5. We introduce the space (6.14)

D+ (R) = { f, g ∈ E0 : U0 (t) f, g = 0 for t > 0, |x| < R + t}.

Note that D0+ (R), defined by (5.7), consists of the elements of D+ (R) that are smooth and compactly supported. Similarly, set (6.15)

D− (R) = { f, g ∈ E0 : U0 (t) f, g = 0 for t < 0, |x| < R + |t|}.

For R = 0, we denote these spaces simply by D+ and D− , respectively. From (6.12) it is clear that if T0 f, g (s, ω) is supported in s ≥ R (resp., s ≤ −R), then f, g belongs to D+ (R) (resp., D− (R)). Furthermore, the converse result is true: Proposition 6.1. The transformation T0 : E0 → L2 (R, N ) maps D+ (R) (resp. D− (R)) onto the space of functions in L2 (R, N ) supported in [R, ∞) (resp., supported in (−∞, −R]), for any R ≥ 0. In particular, D+ and D− are orthogonal complements of each other in E0 . In order to prove this proposition, it suffices to demonstrate that if f, g ∈ E0 belongs to D+ , then k(s, ω) = T0 f, g vanishes for s < 0. This comes down to showing that, if k ∈ L2 (R, N ) and if the integral (6.12) vanishes for t > |x|, then

254 9. Scattering by Obstacles

k(s, ω) = 0 for s < 0. Applying a mollifier, we can suppose k ∈ C ∞ (R, N ). Since T0 clearly commutes with rotations, it suffices to prove this for k(s, ω) of the form k(s, ω) = K(s)ϕ(ω), where ϕ is an eigenfunction of the Laplace operator on S 2 . So suppose (6.16)

u(t, x) =

1 2π

 K(x · ω − t)ϕ(ω) dω S2

vanishes for t > |x|. Since this implies Dxα u(t, 0) = 0 for t > 0, for all α, we have  |α| (6.17) 0 = ∂t K(−t) ω α ϕ(ω) dω, t > 0, S2

for all α. Since, by the Stone-Weierstrass theorem, {ω α } has dense linear span in C(S 2 ), there exists α such that the integral in (6.17) is nonvanishing. This implies |α| that ∂t K(−t) = 0 for t > 0, so K(t) coincides with a polynomial in t for t < 0. Since K ∈ L2 (R), this implies K(t) = 0 for t < 0, and the proposition is proved. Now we look at the maps T± : E → L2 (R, N ), in the presence of an obstacle K, which we suppose is contained in a ball BR . Note that D± (R) can be regarded as subspaces both of E0 and of E. Lemma 5.2 (specifically (5.13)), which was important in the last section, immediately implies the following. Proposition 6.2. We have (6.18)

T+ = T0 on D0+ (R) and T− = T0 on D0− (R).

The potential usefulness of this is indicated by the next result. Proposition 6.3. The properties (6.3) and (6.18) uniquely characterize T+ and T− as continuous linear maps. Proof. Equations (6.3) and (6.18) specify T+ on U (t)D0+ (R) for all t ∈ R. By Proposition 5.4, the union of these spaces is dense in E, so the result follows for T+ . The proof for T− is similar. Note that we can set T± = T0 on D± (R) and since U (t) = U0 (t), for t ≥ 0 on D+ (R) and for t ≤ 0 on D− (R), we can extend T± so that (6.3) holds. The uniqueness result above then implies T± = T± , so we have (6.19)

T+ = T0 on D+ (R) and T− = T0 on D− (R),

sharpening (6.18). If we use the translation representations T± in place of the spectral representations Ψ± , the scattering operator S defined by (5.24) is replaced by the unitary operator on L2 (R, N ):

6. Translation representations and the Lax–Phillips semigroup Z(t)

255

Sˆ = T+ T−−1 .

(6.20)

The operator Sˆ clearly commutes with translations. It also possesses the following important property. Proposition 6.4. We have     Sˆ : L2 (−∞, −R], N −→ L2 (−∞, R], N .

(6.21)

Proof. T−−1 maps L2 ((−∞, −R], N ) onto D− (R), which is orthogonal to D+ (R), as a consequence of Proposition 6.1. Since T+ maps D+ (R) onto L2 ([R, ∞), N ) and is unitary, it must map D− (R) into the orthogonal complement of L2 ([R, ∞), N ); this proves (6.21). Now the action of S on L2 (R, N ) is given by multiplication by a unitary operator-valued function S(k), similar to the action of S in terms of S(k) discussed in §3. The action of Sˆ on L2 (R, N ) is then given by convolution by an ˆ operator-valued tempered distribution S(s), the Fourier transform of S(k). From ˆ (6.21) we conclude that S(s) is supported in the half-line (−∞, 2R]. It follows that S(k) extends to be a holomorphic, operator-valued function in the half-space Im k > 0, a fact that can also be seen directly from an analysis of the scattering amplitude a(ω, θ, k), in view of the relation established in §3. We will study the meromorphic continuation of these objects into the lower half-plane in §7. We now look at a semigroup of operators, introduced by P. Lax and R. Phillips, defined as follows. Fixing R such that K ⊂ BR , set  ⊥ K = D+ (R) ⊕ D− (R) ,

(6.22)

the orthogonal complement in E. For t ≥ 0, define Z(t) = PK U (t)PK ,

(6.23)

where PK is the orthogonal projection of E onto K. Proposition 6.5. Z(t) is a strongly continuous semigroup of operators on K, so Z(t + s) = Z(t)Z(s), for t, s ≥ 0.

(6.24)

Proof. If fj , gj ∈ K, then U (t) f1 , g1 ∈ D+ (R) for t ≥ 0, and furthermore  ⊥ U (−s) f2 , g2 ∈ D− (R) for s ≥ 0. Hence, for s, t ≥ 0, (6.25)



   U (−s) f2 , g2 , PK U (t) f1 , g1 E = U (−s) f2 , g2 , U (t) f1 , g1 E .

Thus PK U (s)PK U (t)PK = PK U (s + t)PK , which implies (6.24). The strong continuity is obvious.

256 9. Scattering by Obstacles

We note that the Lax–Phillips semigroup Z(t) can also be expressed as (6.26)

Z(t) = P+ U (t)P−

(t ≥ 0),

 ⊥ where P± is the orthogonal projection of E onto D± (R) . To see this, note that PK = P+ P− = P− P+ . Since U (t) leaves D+ (R) invariant, P+ U (t)P+ = P+ U (t), for t ≥ 0. Similarly, P− U (t)P− = U (t)P− , for t ≥ 0, so

(6.27)

PK U (t)PK = P− P+ U (t)P+ P− = P− P+ U (t)P− = P+ P− U (t)P− = P+ U (t)P− .

Since Z(t) is a strongly continuous semigroup on K, it has a generator C, whose resolvent is given by (6.28)

−1

(λ − C)

 =



0

e−λt Z(t) dt,

Re λ > 0.

The following result gives important spectral information on Z(t). Proposition 6.6. For any T ≥ 2R, λ > 0, (6.29)

(λ − C)−1 Z(T ) is compact.

We can derive this from the following result, of independent interest. Given ρ ∈ C0∞ (R+ ), let  ∞ (6.30) Z(ρ) = ρ(t)Z(t) dt. 0

Define U (ρ) and U0 (ρ) similarly.   Proposition 6.7. If ρ ∈ C0∞ (2R, ∞) , then (6.31)



Z(ρ) = P+ U (ρ) − U0 (ρ) P− .

Proof. Since it is easy to see that (6.32)

P+ U0 (t)P− = 0, for t ≥ 2R,

this is clear from the formula (6.26).

6. Translation representations and the Lax–Phillips semigroup Z(t)

257

Now  to prove  (6.29), it suffices to show that Z(ρ)  ∞ is compact for any ρ ∈ C0∞ (2R, ∞) , since the operator (6.29) is equal to 0 e−λt Z(t + T ) dt, which is a norm limit of such Z(ρ). We show that, for such ρ, U (ρ) − U0 (ρ) is compact, from E to E0 . Indeed, if ρ is supported in [2R, T ], then, by finite propagation speed,

(6.33) U (ρ) − U0 (ρ) f, g is supported in |x| ≤ 2R + T, for any f, g ∈ E. Also we have, for such ρ, by integrating by parts, and elliptic regularity, (6.34)

U (ρ) : E → C ∞ (Ω),

U0 (ρ) : E → C ∞ (R3 ).

The compactness of U (ρ) − U0 (ρ) then follows, by Rellich’s theorem. We note that complementing (6.33), we also have, for any f, g ∈ E, (6.35)



U (ρ) − U0 (ρ) f, g depends only on f, g B

R+T

.

For any nonzero α ∈ C in the spectrum of the operator (6.29) (for a fixed λ > 0, T ≥ 2R), this compact operator has an associated finite-dimensional, generalized α-eigenspace Vα . Z(t) clearly preserves Vα , for t ≥ 0, and the spec trum of Z(t) V consists of eμj t , where, for each such α, μj is a finite set of α complex numbers, each satisfying (λ − μj )−1 eμj t = α. We call the set of all such μj , as α ranges over the nonzero elements of the spectrum of (6.29), scattering characters. It is a fact that this set coincides precisely with the spectrum of the generator C of Z(t), but we will not make explicit use of this and we do not include a proof. (See [LP1].) By the analysis above, the set of scattering characters μj can be characterized as follows: (6.36)

point spec Z(t) = {eμj t : μj scattering character}.

In §7 we relate the set of scattering characters to the set of scattering poles. We end this section with some comments on the semigroup Z(t) in the translation representation, that is, we look at (6.37)

Z+ (t) = T+ Z(t)T+−1 ,

acting on K+ ⊂ L2 (R, N ), where (6.38)

K+ = T+ (K).

Note that f, g belongs to K if and only if

258 9. Scattering by Obstacles

(6.39)

supp T+ f, g ⊂ (−∞, R] and supp T− f, g ⊂ [−R, ∞),

ˆ given in view of Proposition 6.1 and (6.19). Recalling the scattering operator S, by (6.20), we see that (6.40)

     K+ = f ∈ L2 (−∞, R], N : Sˆ−1 f ∈ L2 [−R, ∞), N .

By (6.37) and (6.3) we have, for f ∈ K+ , (6.41)

Z+ (t)f (s, ω) = f (s − t, ω), 0,

for s ≤ R, for s ≥ R.

Exercises 1. Prove the Radon inversion formula (6.11) from the definition (6.5) and the Fourier inversion formula. 2. Consider a first-order, constant-coefficient PDE ∂u = A(Dx )u, ∂t

u(0, x) = f (x),

where A(Dx ) is an  ×  matrix. Assume the principal symbol A1 (ξ) has  distinct imaginary roots for ξ ∈ R3 \ 0. Express the solution in terms of the Radon transform. When can you deduce Huygens’ principle?

7. Integral equations and scattering poles In §1 we established results on the existence and uniqueness of solutions to the scattering problem

(7.1)

(Δ + k 2 )v = 0 on Ω, v = f on ∂K,

∂v − ikv −→ 0, as r → ∞. r ∂r

As in (1.19), let us denote the solution operator to (7.1) by (7.2)

v = B(k)f.

We established the proof that B(k) is uniquely defined, for k ∈ R, via the limiting absorption principle in §1; related is the elementary fact that such a solution operator is also uniquely defined for complex k such that Im k > 0, since k 2 belongs to the resolvent set for the Laplace operator on Ω (with Dirichlet boundary condition) for Im k > 0. The limiting absorption principle implies that B(k)

7. Integral equations and scattering poles

259

is strongly continuous in {k ∈ C : Im k ≥ 0}; of course, it is holomorphic on {k : Im k > 0}. Here we will show that v = B(k)f can be obtained as the solution to an integral equation over ∂K. Use of such integral equations is a convenient tool for a number of investigations in scattering theory. We use it here to show that B(k) has a meromorphic continuation to an operator-valued function on C, with some poles in {k : Im k < 0}. These poles are known as scattering poles and provide fundamental objects for study in scattering theory. The integral equations applying to (7.1) will be obtained from a study of the following operators, called single- and double-layer potentials, respectively:  f (y) g(x, y, k) dS (y)

S (k)f (x) =

(7.3)

∂K

and  (7.4)

D (k)f (x) =

f (y) ∂K

∂g (x, y, k) dS (y), ∂νy

where, as in §1,  −1 ik|x−y| g(x, y, k) = 4π|x − y| e .

(7.5)

For f ∈ C ∞ (∂K), or even for f ∈ L1 (∂K), the functions (7.3) and (7.4) are well ◦



defined and smooth for x ∈ R3 \ ∂K = Ω ∪ K, where K is the interior of K. For such v, x ∈ ∂K, we denote by v+ (x) the limit from the exterior region Ω, v− (x) ◦

the limit from the interior region K, and by ∂v/∂ν+ and ∂v/∂ν− their normal ◦

derivatives, in the direction pointing into Ω, taken as limits from Ω and from K, respectively. By the methods used to treat layer potentials in §11 of Chap. 7, one derives the following results:

(7.6)

S (k)f+ (x) = S (k)f− (x) = G(k)f (x), 1 1 D (k)f± (x) = ± f (x) + N (k)f (x), 2 2

where, for x ∈ ∂K,  (7.7)

G(k)f (x) =

f (y) g(x, y, k) dS (y) ∂K

and

260 9. Scattering by Obstacles

 N (k)f (x) = 2

(7.8)

f (y)

∂K

∂g (x, y, k) dS (y). ∂νy

Note that, for |x − y| ≤ 1, g(x, y, k) has an estimate of the form |g(x, y, k)| ≤ Ck |x − y|−1 .

(7.9)

We have for ∇y g the poorer estimate |∇y g(x, y, k)| ≤ Ck |x − y|−2 , but the normal derivative ∂g/∂νy has a weaker singularity on ∂K × ∂K, of the same kind as g: ∂g (7.10) (x, y, k) ≤ C|x − y|−1 , for x, y ∈ ∂K. ∂νy It follows that G(k) and N (k) are compact operators on L2 (∂K), for each k ∈ C, with holomorphic dependence on k. We will first consider the possibility of obtaining the solution v to the scattering problem in the form v = B(k)f = D (k)g

(7.11)

on Ω,

where g (whose dependence on k we suppress) satisfies the identity 

(7.12)

 I + N (k) g = 2f

on ∂K.

We will establish the following result. Proposition 7.1. The operator I+N (k) is invertible on L2 (∂K) for all Im k > 0, and for all real k, except for k = λj , where −λ2j is an eigenvalue for Δ on the ◦

interior region K, with Neumann boundary condition on ∂K. Proof. Since N (k) is compact, it suffices to consider whether I + N (k) is injective. Suppose therefore that (7.13)



 I + N (k) g = 0,

and consider v = D (k)g on R3 \ ∂K. On Ω, v satisfies (7.1), with f = 0 (for real k, and it is also exponentially decaying as |x| → ∞ if Im k > 0), so the uniqueness result implies that v = 0 on Ω. Thus ∂v/∂ν+ = 0. Now an analysis of the double-layer potential (7.4), parallel to that for (11.39) of Chap. 7, shows that, in general, (7.14)

∂D(k)f ∂D (k)f = ∂ν+ ∂ν−

on ∂K.

Hence, for v = D (k)g, with (7.13) satisfied, we have

7. Integral equations and scattering poles

(7.15)

261

∂v = 0 on ∂K. ∂ν−

Thus v satisfies the homogeneous Neumann boundary condition, together with the PDE (7.16)

(Δ + k 2 )v = 0



on K.

Since, by (7.7), the jump of v across ∂K is g(x), and since v+ = 0, we deduce ◦

that v− = −g, so v is not identically zero in K if g = 0. The spectrum of the Laplace operator Δ on K, with Neumann boundary condition, is a discrete subset of {λ2j } of R− , so the proposition is proved. The extension of B(k) to a neighborhood of the real line in C, including the exceptional points λj defined above, is neatly accomplished by considering the following alternative integral equation. Namely, we look for a solution v to the scattering problem of the form (7.17)

v = B(k)f = D (k)g + iηS (k)g in Ω,

where g is to be determined as a function of f . Here η is a real constant; we can take η = ±1. In this case, we require that g satisfy the identity (7.18)



I + N (k) + 2iηG(k) g = 2f.

Proposition 7.2. For a given real η = 0, the operator I + N (k) + 2iηG(k) is invertible on L2 (∂K), for all k such that (7.19)

Im k ≥ 0 and η Re k ≥ 0.

Proof. Again it suffices to check injectivity. Suppose g ∈ L2 (∂K) satisfies (7.20)



I + N (k) + 2iηG(k) g = 0,

and let (7.21)

v = D (k)g + iηS (k)g in R3 \ ∂K.

Then v satisfies (7.1) (for k real, also with exponential decay for Im k > 0) on Ω, with f = 0, so our familiar uniqueness result implies v = 0 on Ω, hence v+ = 0 and ∂v/∂ν+ = 0 on ∂K. Hence, as before, by (7.6)–(7.8), (7.22)

v− = −g on ∂K.

262 9. Scattering by Obstacles

Similarly, ∂v/∂ν− is equal to the jump of ∂v/∂ν across ∂K. To calculate this jump, we use (7.14) for D (k)g, and for iηS (k)g, we use the identity (7.23)

 ∂S (k)g 1 (x) = N # (k)g ∓ g , ∂ν± 2

where (7.24)

N # (k)g(x) = 2

 g(y)

∂K

∂g (x, y, k) dS (y), ∂νx

x ∈ ∂K.

Consequently, complementing (7.14), we have (7.25)

∂S (k)g ∂S (k)g − = −g on ∂K. ∂ν+ ∂ν−

Therefore, for v given by (7.21), we have (7.26)

∂v = −iηg on ∂K. ∂ν−

Hence, on the interior region, v satisfies (7.27)



(Δ + k 2 )v = 0 on K,

∂v − iηv = 0 on ∂K. ∂ν ◦

Given that η = 0, we claim that this implies v = 0 on K. Indeed, Green’s identity implies (7.28)

∇v 2L2 (K) − k 2 v 2L2 (K) = −iη v 2L2 (∂K) .

Taking the imaginary part of this identity, we have the following. If k = λ + iμ, (7.29)

2λμ v 2L2 (K) = −η v 2L2 (∂K) .

Under the hypotheses (7.19), the coefficients on the two sides of (7.29) have opposite signs, so v = 0 on ∂K. In view of (7.22), this implies g = 0, so this proposition is proved. Taking η = ±1, we have I + N (k) ± 2iG(k) invertible in the first (resp., second) closed quadrant in C, hence invertible in a neighborhood of such a quadrant. Thus B(k) is extended to an operator-valued function holomorphic on a neighborhood of the closed upper half-plane Im k ≥ 0. We next show that, in fact, B(k) has a continuation to a meromorphic operatorvalued function on C. This is an immediate consequence of the following result.

7. Integral equations and scattering poles

263

Proposition 7.3. The operator I + N (k) is invertible on L2 (∂K) for all k ∈ C except for a discrete set, and (I + N (k))−1 is a meromorphic function on C. This result in turn is a special case of the following elementary general result. For more general results in the area of analytic Fredholm theory (including the multiparameter setting), see Theorem 7.9 and Propositions 7.10–7.11 in Appendix A, Outline of functional analysis. Proposition 7.4. Let O be a connected open set in C. Suppose C(z) is a compact, operator-valued, holomorphic function of z ∈ O. Suppose that I + C(z) is invertible at some point p0 ∈ O. Then I + C(z) is invertible except at most on a discrete set in O, and (I + C(z))−1 is meromorphic on O. Proof. The operator I + C(z) fails to be invertible at a point z ∈ O if and only if the compact operator C(z) has −1 in its spectrum. For a given z0 ∈ O, let γ be a small circle about −1, disjoint from the spectrum of C(z0 ). For z in a small neighborhood U of z0 , we can form the projection-valued function (7.30)

1 P (z) = 2πi



 −1 λ − C(z) dλ.

γ

For z ∈ U, this is a projection of finite rank (say ); using P (z0 ) we can produce a family of isomorphisms of the range R(P (z)) with R(P (z0 )), and then C(z)P (z) can be treated as a holomorphic family of × matrices. This proposition in the case of × matrices is easy, via determinants. By hypothesis, −1 is not identically an eigenvalue for this family, so (I + C(z))−1 P (z) is a meromorphic function on U. Clearly, (I + C(z))−1 (I − P (z)) is a holomorphic function on U, so this establishes the proposition. Corollary 7.5. The solution operator B(k) for (7.1) has a meromorphic continuation to C; all its poles are in the lower half-plane Im k < 0. This follows from the formula (7.31)



−1 B(k) = 2D (k) I + N (k) ,

except at the real points k = λj , from Proposition 7.1, together with the formula (7.32)





−1 B(k) = 2 D (k) + iηS (k) I + N (k) + 2iηG(k) ,

for η = ±1, which defines B(k) as holomorphic on a neighborhood of the real axis.

264 9. Scattering by Obstacles

The poles of B(k) are called scattering poles. It follows immediately from (7.31) that the set of scattering poles is contained in the set of poles of [I + N (k)]−1 within the lower half-plane Im k < 0. In fact, these two sets coincide; this is a consequence of the following. Lemma 7.6. If Im k = 0, then D (k) : L2 (∂K) → L2loc (Ω) is injective. Proof. The argument used in the proof of Proposition 7.1 shows that if g ∈ L2 (∂K) and D (k)g = 0 on Ω, then g = v ∂K where v ◦ is an eigenfunction for K



Δ on K, with Neumann boundary condition on ∂K, and with eigenvalue −k 2 . Since the spectrum of this elliptic operator is real and nonpositive, the lemma is proved. Proposition 7.7. The set of scattering poles is precisely equal to the set of poles

−1 k, for I + N (k) , such that Im k < 0. Proof. If [I + N (k)]−1 has a pole of order m at k = kj , Im kj < 0, then there is an element h ∈ L2 (∂K) such that, with nonzero hm ∈ L2 (∂K), (7.33)



−1

I + N (k) h = (k − kj )−m hm + (k − kj )hm−1 + · · · .

Since D (kj )hm = bm = 0 in L2loc (Ω), it follows that, for k near kj , (7.34)

  B(k)h = (k − kj )−m bm + O (k − kj )−m+1 ,

k → kj ,

so B(k) is singular at kj . We also have the following characterization of scattering poles. Proposition 7.8. A complex number kj is a scattering pole if and only if there is a nonzero v ∈ C ∞ (Ω) satisfying (7.35)

(Δ + kj2 )v = 0 on Ω,

v = 0 on ∂K,

of the form v = D (kj )g,

(7.36) for some g ∈ L2 (∂K).

Proof. We know that, for Im kj ≥ 0, v satisfying (7.35)–(7.36) must vanish on Ω. On the other hand, if Im kj < 0, we know that kj is a scattering pole if and only if I + N (kj ) has nonzero kernel. We claim that, for Im kj < 0, (7.37)

  D (kj ) : ker I + N (kj ) −→ {v satisfying (7.35)–(7.36)},

isomorphically. Indeed, surjectivity is obvious, and injectivity follows from Lemma 7.6. This proves Proposition 7.8.

7. Integral equations and scattering poles

265

The condition (7.36) can be viewed as an extension of the radiation condition, which we initially defined for real k. A sharper result is given in Proposition 7.13 below. It is clear that the Green function G(x, y, k), defined in §1 by (1.26)–(1.30), has a meromorphic extension in k, with poles confined to the set of scattering poles defined above. Indeed, we can write (7.38)

G(x, y, k) = g(x, y, k) − B(k)γy,k (x),

where g(x, y, k) is given by (1.5) for all k ∈ C, and γy,k is the restriction of g(x, y, k) to x ∈ ∂K. Similarly, the “eigenfunctions” u+ (x, kω), defined by (1.32)–(1.33), have such a meromorphic continuation in k, and so do the scattering amplitude a(ω, θ, k) and the scattering operators S(k) and S(k). We will explore these last objects further at the end of this section. First we consider another integral-equation approach to the scattering problem (7.1). As another alternative to (7.11), it is of interest to obtain solutions to the scattering problem in the form (7.39)

v = B(k)f = S (k)g on Ω,

where g satisfies the integral equation G(k)g = f on ∂K.

(7.40)

The operator-valued function G(k) is defined by (7.6). As we have noted, G(k) is compact on L2 (∂K). In fact, analysis done in Chap. 7 shows that G(k) is a pseudodifferential operator of order −1 on ∂K, and examination of its symbol shows that it is elliptic. The principal symbol of G(k) is positive on S ∗ (∂K). Consequently, for each k ∈ C, each real s, (7.41)

G(k) : H s (∂K) −→ H s+1 (∂K) is Fredholm, of index zero.

In analogy with Proposition 7.1, we have the following result: Proposition 7.9. The operator G(k) : H s (∂K) → H s+1 (∂K) is invertible for all k such that Im k > 0, and for all real k, except for k = μj such that −μ2j is an ◦

eigenvalue of Δ on the interior region K, with Dirichlet boundary condition on ∂K. Proof. In view of (7.41), it suffices to check the injectivity of G(k). This goes as in the proof of Proposition 7.1. Setting v = S (k)g on R3 \ ∂K, uniqueness as before yields v = 0 on Ω if g ∈ ker G(k), Im k ≥ 0. Then v− = 0 on ∂K, by (7.6), while by (7.25) ∂v/∂ν− = g on ∂K, so if g = 0 then v K = 0 is ◦

an eigenfunction for Δ on K, with Dirichlet boundary condition and with eigenvalue −k 2 .

266 9. Scattering by Obstacles

In addition to (7.41), we obtain from the analysis of G(k) as a pseudodifferential operator that its principal symbol is independent of k, hence (7.42)

G(k) − G(0) = D(k) : H s (∂K) −→ H s+2 (∂K).

By Proposition 7.9, G(0) is invertible. Then (7.43)

G(0)−1 G(k) = I + G(0)−1 D(k) : H s (∂K) −→ H s (∂K)

is holomorphic in k, and G(0)−1 D(k) : H s (∂K) −→ H s+1 (∂K);

(7.44)

in particular, this operator is compact on H s (∂K), for each s ≥ 0. Since Proposition 7.9 implies that the operator (7.43) is invertible for Im k > 0, we can apply the general operator result of Proposition 7.4, to obtain: Proposition 7.10. The operator-valued function G(k)−1 : H s+1 (∂K) −→ H s (∂K)

(7.45)

has a meromorphic continuation to C, with poles contained in Im k < 0 together with the set μj of real numbers specified in Proposition 7.9. In view of (7.39), the set of poles of (7.45) satisfying Im k < 0 contains the set of scattering poles, and B(k) = S (k) G(k)−1 ,

(7.46)

where G(k)−1 is regular. In fact, in parallel with the proofs of Lemma 7.6 and Proposition 7.7, we easily obtain the following: Proposition 7.11. If Im k = 0, then S (k) : L2 (∂K) → L2loc (Ω) is injective. Therefore, the set of scattering poles is precisely equal to the set of poles for G(k)−1 such that Im k < 0. Furthermore, a complex number kj is a scattering pole if and only if there is a nonzero v ∈ C ∞ (Ω) satisfying (7.35), of the form v = S (kj )g,

(7.47)

for some g ∈ L2 (∂K). More precisely, for Im kj < 0, (7.48)

S (kj ) : ker G(kj ) −→ {v satisfying (7.35) and (7.47)},

isomorphically. From the formula (7.7) for G(k), we see that

7. Integral equations and scattering poles

267

G(k)∗ = G(−k).

(7.49)

We therefore have the following: Corollary 7.12. The set of scattering poles is symmetric about the imaginary axis. We can also obtain a characterization of the set of scattering poles which is more satisfactory than that of Proposition 7.8 or the last part of Proposition 7.11. Proposition 7.13. A complex number k is a scattering pole if and only if there is a nonzero v ∈ C ∞ (Ω) satisfying (7.35), of the form v = D (k)g1 + S (k)g2 on Ω,

(7.50) for some gj ∈ L2 (∂K).

Proof. For v of the form (7.50), note that (7.51)

v+ =

 1 I + N (k) g1 + G(k)g2 on ∂K. 2

In particular, if v+ = 0 on ∂K and k is not a scattering pole, but Im k = 0, then g1 = −2(I + N (k))−1 G(k)g2 . Now we know that, for Im k > 0, v+ = 0 on ∂K implies v = 0 on Ω, so we have the identity (7.52)

 −1 2D (k) I + N (k) G(k) − S (k) = 0,

for Im k > 0, as a map from L2 (∂K) to C ∞ (Ω). This identity continues analytically to the lower half-plane Im k < 0, excluding the scattering poles, and implies that if v is of the form (7.50), v+ = 0 on ∂K, and k is not a scattering pole, then v = 0 on Ω. Given the results of Propositions 7.8 and 7.11 when k = kj is a scattering pole, this finishes the proof. We can obtain a few more conclusions from (7.52), which we write as (7.53)

D (k)M (k) = S (k) on Ω,

valid for all k ∈ C at which I + N (k) is invertible, with  −1 (7.54) M (k) = 2 I + N (k) G(k). First, using the injectivity of D (k) for Im k < 0, as in the proof of Proposition 7.7, we see that M (k) has an analytic continuation to all Im k < 0, including the set of scattering poles. The only poles of M (k) are at the real numbers λj of Proposition 7.1. Also, M (k) is invertible, except at the real numbers μj of Proposition 7.9; in particular, M (k) is invertible at all the scattering poles. Therefore, when k = kj is a scattering pole, M (kj ) gives an isomorphism from ker

268 9. Scattering by Obstacles

G(kj ), in (7.48), to ker (I + N (kj )), in (7.37). Furthermore, any v of the form (7.50), with k = kj , can be written both in the form (7.36) and in the form (7.47) (with different g’s). Another calculation using the representation of the solution to the scattering problem by a single-layer potential (7.39)–(7.40), produces an analysis of the Neumann operator N (k), which we define as follows, first for Im k ≥ 0. For f ∈ C ∞ (∂K), let v be the solution to the scattering problem (7.1), v = B(k)f , and define ∂v on ∂K. (7.55) N (k)f = ∂ν+ By elliptic regularity estimates, we can deduce that, for s ≥ 1, (7.56)

N (k) : H s (∂K) −→ H s−1 (∂K).

We produce a formula for N (k) using the representation v = S (k)g, g = G(k)−1 f , valid for Im k > 0. From the formula (7.23) for ∂S (k)g/∂ν± , we see that (7.57)

N (k) =

 1 # N (k) − I G(k)−1 , 2

for Im k > 0. This identity continues analytically to the complement of the set of poles of G(k)−1 in C. Note that, complementing (7.49), N # (k) = N (−k)∗ ,

(7.58)

so (7.57) can also be written as (7.59)

N (−k) =



∗ −1 2(N (k) − I)−1 G(k) .

By the analysis of the scattering problem for Im k ≥ 0, we know that N (k) is a strongly continuous function of k, with values in the Banach space L(H s (∂K), H s−1 (∂K)), for Im k ≥ 0. Thus N (k) does not have poles on the real axis; such singularities are therefore removable on the right side of (7.57). The poles of G(k)−1 on the real axis must be canceled by a null space of N # (k) − I, for k = λj . The occurrence of these real poles of G(k)−1 makes (7.57) a tool of limited value in analyzing the Neumann operator N (k) for real k. We can produce another formula for N (k), first for Im k > 0, by using the representation (1.4) for v = B(k)f , that is, (7.60)

B(k)f = D (k)f − S (k)N (k)f.

7. Integral equations and scattering poles

269

Evaluating this on ∂K, we have f=

(7.61)

 1 I + N (k) f − G(k)N (k)f, 2

which implies (7.62)

N (k) =

  1 G(k)−1 N (k) − I , 2

for Im k > 0. Of course, this identity also continues analytically to all k ∈ C outside the set of poles of G(k)−1 . Comparing (7.62) with (7.57), we see that N (k) and N # (k) are related by the identity (7.63)

N (k)G(k) = G(k)N # (k),

for all k ∈ C. Also, comparing (7.62) with (7.59), we see that (7.64)

N (−k) = N (k)∗ ;

in particular, N (k) is self-adjoint when k is purely imaginary. Furthermore, in view of (7.60) and (7.57), we see that the set of poles of N (k) coincides exactly with the set of scattering poles. Note that the factor (1/2)(N (k) − I) in (7.62) arises from evaluating D (k)f ◦

on ∂K as a limit from the interior region K, by (7.7). Thus the analogue of the ◦

identity (7.53) which is valid on K is obtained by replacing M (k) by N (k)−1 . Equivalently, (7.65)



D (k) = S (k)N (k) on K,

where N (k) is the exterior Neumann operator defined above. So far we have not established that there actually are scattering poles. We will show that in fact there are infinitely many scattering poles on the negative imaginary axis, for any nonempty smooth obstacle K, by a study of G(k). We begin with the following result: Lemma 7.14. For real s ≥ 0, G(is) is positive-definite. Proof. Given g ∈ L2 (∂K), set v = S (is)g on R3 \ ∂K. Then Green’s theorem gives, for s > 0, (Δv, v)L2 (Ω) + dv 2L2 (Ω) = − (7.66) (Δv, v)L2 (K) + dv 2L2 (K) =



 G(is)g

∂K

G(is)g ∂K

∂v dS , ∂ν+

∂v dS . ∂ν−

270 9. Scattering by Obstacles

Recall from (7.25) that ∂v/∂ν− − ∂v/∂ν+ = g, so adding the identities above gives

  G(is)g, g L2 (∂K) = s2 v 2L2 (Ω) + v 2L2 (K) (7.67) + ∇v 2L2 (Ω) + ∇v 2L2 (K) , for s > 0, which proves the lemma in this case. Since we know that G(0) is invertible, this is also positive-definite. To proceed with the demonstration that G(is)−1 is singular for infinitely many negative real s, we set (7.68)

n(s) = # negative eigenvalues of G(is),

for s < 0. Our next claim as follows: Lemma 7.15. As s  −∞, n(s) → ∞.   Proof. We will show that G(is)g1 , g2 defines a negative definite inner product on a vector space V whose dimension can be taken large with |s|. Then the lemma follows, by the variational characterization of the spectrum of G(is). Pick points p, q ∈ K such that |p − q| is maximal. Then, for any N , you can pick pj near p and qj near q, for 1 ≤ j ≤ N , such that (7.69)

    min |pj − qj | : 1 ≤ j ≤ N > max |pj − qk | : j = k .

Put small disjoint disks Dj about pj , Dj about qj , all of the same area, within ∂K, and define functions gj ∈ L2 (∂K) by (7.70)

gj = 1 on Dj ,

−1 on Dj ,

0 elsewhere.

Then {gj : 1 ≤ j ≤ N } is a set of orthogonal functions, all of the same norm. Let V be the linear span of these gj . With V so fixed, of dimension N, a simple calculation gives (7.71)

  G(is)gj , gj < −γ < 0,   G(is)gj , gk 0, (7.74)

S(k) = I +

k A(k), 2πi

where, for such k, A(k) is a compact operator on L2 (S 2 ), given by a smooth integral kernel. Such S(k) is Fredholm of index zero. Thus, for Im k > 0, S(k) fails to be invertible if and only if it has a nonzero kernel. Furthermore, this happens if and only if S(k)∗ has a nonzero kernel. We are now prepared to establish the following result. Proposition 7.17. A complex number μ is a scattering character if and only if iμ is a pole of S(k). Proof. μ is a scattering character if and only if there exists a nonzero f ∈ K+ such that Z+ (t)f = eμt f , for t ≥ 0. By (6.41), this implies (7.75)

f (s, ω) = e−μs ϕ(ω), 0,

for s ≤ R, for s > R,

for some nonzero ϕ ∈ L2 (S 2 ). By (6.40), such an f belongs to K+ if and only if Sˆ∗ f is supported in [R, ∞). By the Paley–Wiener theorem, we can deduce that this will hold if and only if S(k)∗ fˆ(k) is holomorphic in Im k < 0. Now (7.76)

ϕ(ω) , fˆ(k) = (2π)−1/2 ik + μ

272 9. Scattering by Obstacles

which has a pole at k = iμ, so this analyticity holds if and only if ϕ belongs to the kernel of S(k)∗ , for k = iμ. This establishes the proposition. For a rich treatment of a great deal more material on scattering poles, we recommend the monograph [DZ].

Exercises 1. Verify that G(k), defined by (7.7), is an elliptic pseudodifferential operator of order −1 on ∂K. Compute its principal symbol. 2. Justify (7.69). The following exercises deal with an integral-equation attack on the scattering problem for H = −Δ + V on R3 . Assume V ∈ C0∞ . We use (1.57), that is, (I − V(k))v = R(k)f, where V(k) = R(k)(V v) and  R(k)v(x) = −

v(y)g(x, y, k) dy,

with g(x, y, k) = (4π|x − y|)−1 eik|x−y| . 3. Show that, for Im k ≥ 0, σ > 1, R(k) : L2comp (R3 ) −→ L2 (R3 , x −2σ dx ) is compact. 4. Show that, for Im k ≥ 0, k2 not a negative eigenvalue of −Δ + V , and σ > 1, I − V(k) : L2 (R3 , x −2σ dx ) −→ L2 (R3 , x −2σ dx ) is injective, hence invertible. (Hint: If u = V(k)u = R(k)(V u), show that u satisfies the hypotheses for the uniqueness result of Exercise 6 in §1 when k ∈ R. When Im k > 0, the argument is easier.) 5. Fix κ ∈ (0, ∞). Show that, for Im k > −κ, R(k) : L2comp (R3 ) −→ L2 (R3 , e−2κ|x| dx ) is compact. Also show that I − V(k) : L2 (R3 , e−2κ|x| dx ) −→ L2 (R3 , e−2κ|x| dx ) is holomorphic in {k : Im k > −κ}, and invertible for Im k ≥ 0, k2 ∈ / point spec H. Deduce that its inverse has a meromorphic continuation.

8. Trace formulas; the scattering phase   In Proposition 6.7 we showed that, for any ρ ∈ C0∞ (2R, ∞) , the operator ∞ Z(ρ) = 0 ρ(t)Z(t) dt is compact. Recall that the proof used the identity

8. Trace formulas; the scattering phase

(8.1)

273

 

Z(ρ) = P+ U (ρ) − U0 (ρ) P− , for ρ ∈ C0∞ (2R, ∞) .

We then saw that U (ρ)−U0 (ρ) has a smooth, compactly supported integral kernel. It follows that the operator (8.1) is not only compact, but in fact trace class. By a theorem of V. Lidskii, which we will prove in Appendix A at the end of this chapter, it follows that the trace Tr Z(ρ) is equal to the sum of the eigenvalues of Z(ρ), counted with multiplicity. Thus we have Tr Z(ρ) =

(8.2)



ρˆ(iμj )

where the sum is over the set of scattering characters, characterized by (6.36). In view of Proposition 7.17, we can write Tr Z(ρ) =

(8.3)



ρˆ(zj ),

poles

where {zj } is the set of poles of the scattering operator S(k) (counted with multiplicity). Using (8.1), we will establish the following formula for Tr Z(ρ), which then sheds light on the right side of (8.3).   Proposition 8.1. For ρ ∈ C0∞ (2R, ∞) , we have

(8.4)

Tr Z(ρ) = Tr U (ρ) − U0 (ρ)   √

= 2 Tr ρ(t) cos t −Δ − cos t −Δ0 dt,

where Δ is the Laplacian on Ω = R3 \ K, with Dirichlet boundary condition, and Δ0 the Laplacian on R3 . Proof. Using the facts that Tr AB = Tr BA and that P+ P− = P− P+ , we see from (8.1) that, for ρ ∈ C0∞ (2R, ∞) , Tr Z(ρ) = Tr P− [U (ρ) − U0 (ρ)]P+ . Now for any t ≥ 0, U (t) = U0 (t) on D+ , so [U (ρ)−U0 (ρ)]P+ = U (ρ)−U0 (ρ). Similarly, P− [U (ρ) − U0 (ρ)] = U (ρ) − U0 (ρ), so we have the first identity in (8.4). The second identity is elementary. Combining (8.3) and (8.4), we have the identity  (8.5)

Tr

 √

1 ρˆ(zj ), ρ(t) cos t −Δ − cos t −Δ0 dt = 2 poles

valid for any ρ ∈ C0∞ ((2R, ∞)). This identity has been extended to all ρ ∈ C0∞ (R+ ), by R. Melrose [Me1], using a more elaborate argument.

274 9. Scattering by Obstacles

Note that (8.4) is equal to the trace of ϕ

(8.6)

√

   −Δ − ϕ −Δ0 ,

with ϕ(λ) = ρˆ(λ) + ρˆ(−λ). It is useful to note that, for any even ϕ ∈ S(R), the operator (8.6), given by an integral formula such as in the last line of (8.4) with ρ = ϕ, ˆ has a Schwartz kernel that is smooth and rapidly decreasing at infinity, so that (8.6) is of trace class for this more general class of functions ϕ. (See Exercises 1–3 from §4.) Recall from (2.7) that if ϕ ∈ C0∞ (R), then (8.7)

ϕ

√

 

 −Δ v(x) = (2π)−3

u+ (x, ξ)u+ (y, ξ)v(y)ϕ(|ξ|) dy dξ, R3 Ω

where u+ (x, ξ) are the generalized eigenfunctions of Δ on Ω defined by (1.32)–(1.33). It follows that, for such ϕ, the trace of (8.6) is equal to −3



lim (2π)

(8.8)

R→∞

with

 τR (ξ) =

(8.9)

ϕ(|ξ|)τR (ξ) dξ, R3



 |u+ (x, ξ)|2 − 1 dx ,

BR

where we set u+ (x, ξ) = 0, for x ∈ K = R3 \ Ω, and BR = {x : |x| ≤ R}. In order to evaluate (8.9), we will calculate |u+ (x, ξ)|2 dx over ΩR = {x ∈ Ω : |x| ≤ R} via Green’s theorem. Note that since (Δ + k 2 )u+ = 0, for |ξ| = k, we have   ∂u+ (8.10) (Δ + k 2 ) = −2ku+ , |ξ| = k. ∂k Hence, via Green’s theorem, we have     2 ∂ u+ 1 ∂u+ ∂u+ |u+ |2 dx = − u+ − dS 2k ∂ν∂k ∂k ∂ν (8.11)

ΩR

∂ΩR

1 = 2k



|x|=R



∂ 2 u+ ∂u+ ∂u+ u+ − ∂r∂k ∂k ∂r

 dS ,

since u+ = 0 on ∂K. We want to evaluate the limit of (8.11) as R → ∞. Extending (1.41), we can write (8.12)

u+ (rθ, kω) = e−ikr(θ·ω) + eikr B(r, θ, ω, k),

8. Trace formulas; the scattering phase

275

with (8.13)

B ∼ r−1 a(−ω, θ, k) + r−2 a2 (−ω, θ, k) + · · · ,

r → ∞,

where a(ω, θ, k) is the scattering amplitude and aj are further coefficients. Differentiating (8.12) yields the following (unfortunately rather long) formula for the integrand in (8.11): ∂ 2 u+ ∂u+ ∂u+ − ∂k ∂r ∂r∂k ∂B ∂B + irB ∂k ∂r   2 ∂(rB) ∂B ∂B ∂ B + −iB + −B ∂r ∂r ∂k ∂r∂k  + 2kr|B|2 − krBeikr[(θ·ω)+1] [(θ · ω) − 1]

= 2kr(θ · ω)2 + i(θ · ω) − 2kiB

(8.14)

 − krBe−ikr[(θ·ω)+1] [(θ · ω) − (θ · ω)2 ]   ∂(rB) ∂2B ∂B [(θ · ω) − 1] − i − + eikr[(θ·ω)+1] ik ∂k ∂r ∂r∂k   ∂B + i(θ · ω)B . + e−ikr[(θ·ω)+1] −ir(θ · ω) ∂r A primary tool in the analysis of the integral of this quantity over |x| = R will be the stationary phase method, which was established in Appendix B of Chap. 6. We make some preliminary simplification of (8.14), using the fact that (8.11) is clearly real valued. Also, we can throw out some terms in (8.14) that contribute 0 in the limit R → ∞, after being integrated over |x| = R. This includes all the terms in the first set of curly brackets above. Also, a stationary phase evaluation of the last two terms in the third set of curly brackets yields a 0 contribution in the limit R → ∞. Thus, we can replace (8.14) by the real part of

(8.15)

∂B ∂B + irB 2kr(θ · ω)2 − 2kiB ∂k ∂r   + 2kr|B|2 + krBeikr[θ·ω+1] (1 − θ · ω)2   ∂B ∂B ikr[θ·ω+1] +e [θ · ω − 1] + irθ · ω − iθ · ωB . ik ∂k ∂r

3 The first term on the  right side of (8.15) integrates to 2k times (4/3)πR , exactly canceling out |x|≤R dx . The contribution of the second and third terms to (8.11) is, in the limit R → ∞,

276 9. Scattering by Obstacles

−ia

(8.16)

i ∂a − |a|2 , ∂k 2k

integrated with respect to θ.

We can neglect the second term in (8.16), since it is imaginary. Terms in (8.15) appearing with a factor e±ikr[(θ·ω)+1] have an asymptotic behavior as R → ∞ given by the stationary phase method, upon integration with respect to θ. The leading part in the terms within the first set of brackets is seen to be (upon taking the real part) (8.17)

  4π 2k  Im a(−ω, −ω, k) , |a(−ω, θ, k)|2 dθ + r k S2

which cancels, by the optical theorem, (3.18). This cancelation is necessary since, if (8.17) were nonzero, one would get an infinite contribution to (8.11) as R → ∞. What gives a finite contribution to (8.11) is the θ-integral of the next leading term in this part of (8.15); the contribution to (8.11) one gets from this, as R → ∞, is (again upon taking the real part)  (aa2 + aa2 )(−ω, θ, k)dθ (8.18)

S2

+

8π 4π Im a2 (−ω, −ω, k) + Re a(−ω, −ω, k). k k

The rest of the terms in (8.15) also give a finite contribution to (8.11) as R → ∞, via stationary phase, namely −1/2k times (8.19)



∂a 4π − a, at θ = −ω, ∂k k

plus a term containing an oscillatory factor e−2ikr , which disappears after integration with respect to ξ. This disappearance is guaranteed, since the limit in (8.8) as R → ∞ does exist. Putting together (8.16)–(8.19), we arrive at a computation of (8.9). All these contributions are expressed in terms of the scattering amplitude a, except for (8.18), which involves also the coefficient a2 appearing in (8.13). Now a2 is related to a in a simple fashion, because (Δ + k 2 )(eikr B) = 0. Expressing Δ in polar coordinates gives a sequence of relations among the coefficients in the expansion of B as r → ∞. In particular, we get (8.20)

2ik a2 (ω, θ, k) = Δ2 a(ω, θ, k),

where Δ2 denotes the Laplace operator on the sphere {|θ| = 1}, applied to the second argument of a(ω, θ, k). It follows that aa2 (−ω, θ, k)dθ is purely imaginary, so the integral in (8.18) vanishes. In concert with the reciprocity formula (3.31), we can deduce that 4ik a2 (ω, ω, k) = Δ2 a(ω, ω, k) + Δ1 a(−ω, −ω, k).

8. Trace formulas; the scattering phase



Hence 4ik

277

 a2 (ω, ω, k) dω =

(Δ1 + Δ2 )a(ω, ω, k) dω = 0.

This disposes of the middle term in (8.18), upon integration with respect to ω. Thus, in addition to (8.16) and (8.19), the last term in (8.18) remains. Consequently, we have k2 lim R→∞ (2π)3





 |u+ (x, kω)|2 − 1 dx dω

BR ×S 2

(8.21)

   −ik 2 1 ∂a = Re (−ω, θ, k) dθ − a a(−ω, −ω, k) 3 (2π) ∂k (2π)2 S2

S2



 k ∂a (−ω, −ω, k) dω. 2 (2π) ∂k

On the other hand, −(1/2πi)S(k)(dS ∗ /dk ) has integral kernel

(8.22)

−1  −1 −k ∂a (θ, ω, k) a(ω, θ, k) + 2πi 2πi 2πi ∂k  k + 2 a(ω, τ, k)a(τ, θ, k) dτ 4π S2   k2 ∂a + 2 (τ, ω, k) dτ . a(ω, τ, k) 4π ∂k S2

Noting that the trace of −(1/2πi)S(k)(dS ∗ /dk ) must also be real, one sees that (8.21) is equal to the trace of this operator, which proves the following: Proposition 8.2. For even ϕ ∈ C0∞ (R), (8.23)

  √

T r ϕ( −Δ) − ϕ( −Δ0 ) = −

0



ϕ(k)s (k) dk ,

with (8.24)

s (k) =

    1 1 Tr S(k)∗ S  (k) = − Tr S(k)S  (k)∗ , 2πi 2πi

where S(k) is the scattering operator (3.7). An equivalent characterization of (8.24) is s (k) = ds(k)/dk , with (8.25)

s(k) =

1 1 log det S(k) = arg det S(k). 2πi 2π

278 9. Scattering by Obstacles

The quantity s(k) is called the scattering phase. It is real, for k ∈ R, since the scattering operator is unitary. To give yet another formulation, if we set D(k) = det S(k),

(8.26) then

s (k) =

(8.27)

1 D (k) . 2πi D(k)

By both (8.24) and (8.27) it is clear that s (k) extends from k ∈ R to a meromorphic function in the plane, with poles coinciding precisely with the poles of the scattering operator and their complex conjugates. For complex k, one replaces (8.24) by   1 Tr S(k)∗ S  (k) . s (k) = 2πi As stated, Propositions 8.1 and 8.2 apply in disjoint situations, but note that the left side of (8.23) is defined for any even ϕ ∈ S(R) and defines a continuous linear functional of such ϕ. Thus the right side of (8.23) is well defined, at least in a distributional sense; in particular, we have s ∈ S  (R). Also, replacing ρ by its even part on the left side of (8.5) leaves this quantity unchanged. We deduce the following.   Proposition 8.3. Let ρ ∈ C0∞ (2R, ∞) . Then  ∞ 1 ρˆ(zj ) = − ϕ(k)s (k) dk , 2 0 poles

with

1 ρˆ(k) + ρˆ(−k) . 2 Equivalently, with s(k) = −s(−k) for k ∈ R,  ∞  ρˆ(zj ) = ρˆ (k)s(k) dk , (8.28) ϕ(k) =

poles

−∞

the integral interpreted a priori in the sense of tempered distributions. In view of (8.27), this identity can be thought of as a “formal” consequence of the residue calculus, but a rigorous proof seems to require arguments as described above. It can be proved that the integral above is actually absolutely convergent. Indeed, it has been shown that s(k) has the asymptotic behavior (8.29)

s(k) = C(vol K)k 3 + O(k 2 ), as k → ∞, in R.

9. Scattering by a sphere

279

This was established for K strictly convex by A. Majda and J. Ralston [MjR], and for K starshaped by A. Jensen and T. Kato [JeK]. We outline a proof for the starshaped case in the exercises (with a weaker remainder estimate). The result (8.29) was extended to “nontrapping” K by V. Petkov and G. Popov [PP] and finally to general smooth K by Melrose [Me3]. Also, results of Melrose [Me1] extend (8.28) to all ρ ∈ C0∞ (R+ ).

Exercises 1. Use the formula (8.24) to establish the following formula for s (k):    2  ∂u+  s (k) = C (x · ν) (x, kθ) dθ dS (x) ∂ν ∂K S 2  (8.30)   (x · ν)N (k)ekθ , N (k)ekθ L2 (∂K) dθ. =C S2

2. Conclude that if K is starshaped, so one can arrange x · ν > 0, then s(k) is monotone. 3. Set s(k) = −s(−k), k ∈ R, as in Proposition 8.3. Show that if K is starshaped, s (k) is a positive function that defines a tempered distribution on R and hence that s(k) has a polynomial bound in k: |s(k)| ≤ C k M . 4. Write (8.23) in the form √  √ 1 Tr ϕ( −Δ) − ϕ( −Δ0 ) = 2



∞ −∞

ϕ (k)s(k) dk ,

for even ϕ ∈ S(R). If K is starshaped, Exercise 3 implies that the integral on the right 2 is absolutely convergent. Use ϕ(k) = ϕt (k) = e−tk and the results on heat kernel asymptotics of Chap. 7 to deduce that  ∞ 2 e−tk ks(k) dk = (4πt)−3/2 vol K + o(t−3/2 ), (8.31) t −∞

as t  0. 5. Show that Karamata’s Tauberian theorem (established in §3 of Chap. 8) applies to (8.31) to yield s(k) = C(vol K)k3 + o(k3 ), k → ∞. Evaluate C.

9. Scattering by a sphere In this section we analyze solutions to problems of scattering by the unit sphere S 2 ⊂ R3 , starting with the scattering problem (9.1)

(Δ + k 2 )v = 0 on Ω, v = f on S 2 , r(∂r v − ikv ) → 0, as r → ∞,

280 9. Scattering by Obstacles

where Ω = {x ∈ R3 : |x| > 1}, the complement of the unit ball. We start by considering real k. This problem can be solved by writing the Laplace operator Δ on R3 in polar coordinates, Δ = ∂r2 + 2r−1 ∂r + r−2 ΔS ,

(9.2)

where ΔS is the Laplace operator on the sphere S 2 . Thus v in (9.1) satisfies r2 ∂r2 v + 2r∂r v + (k 2 r2 + ΔS )v = 0,

(9.3)

for r > 1. In particular, if {ϕj } is an orthonormal basis of L2 (S 2 ) consisting of eigenfunctions of ΔS , with eigenvalue −λ2j , and we write v(rω) =

(9.4)



vj (r)ϕj (ω),

r ≥ 1,

j

then the functions vj (r) satisfy r2 vj (r) + 2rvj (r) + (k 2 r2 − λ2j )vj (r) = 0,

(9.5)

r > 1.

As in (1.14), this is a modified Bessel equation, and the solution satisfying the radiation condition r(vj (r) − ikv j (r)) → 0 as r → ∞ is of the form vj (r) = aj r−1/2 Hν(1) (kr), j

(9.6) (1)

where Hν (λ) is the Hankel function, which arose in the proof of Lemma 1.2. We recall from (6.33) of Chap. 3 the integral formula 2 1/2 ei(z−πν/2−π/4)  ∞ s ν−1/2 −s ν−1/2 1 − e s ds. (9.7) Hν(1) (z) = πz 2iz Γ(ν + 12 ) 0 This is valid for Re ν > −1/2 and −π/2 < arg z < π. Also, in (9.6), νj is given by (9.8)

1 1/2 . νj = λ2j + 4

The coefficients aj in (9.6) are determined by the boundary condition vj (1) = (f, ϕj ), so   f, ϕj (9.9) aj = (1) . Hνj (k) Using these calculations, we can write the solution operator B(k) to (9.1), v = B(k)f , as follows. Introduce the self-adjoint operator

9. Scattering by a sphere

281

1 1/2 A = −ΔS + , 4

(9.10) so

Aϕj = νj ϕj .

(9.11) Then (9.12)

B(k)f (rθ) = r−1/2 κ(A, k, kr)f (θ), (1)

(1)

where κ(ν, k, kr) = Hν (kr)/Hν (k) and, for each k, r, κ(A, k, kr) is regarded as a function of the self-adjoint operator A. For convenience, we use the notation (9.13)

B(k)f (rθ) = r−1/2

(1)

HA (kr) (1)

HA (k)

f (θ),

θ ∈ S2.

Similar families of functions of the operator A will arise below. Taking the r-derivative of (9.13), we have the following formula for the Neumann operator:  1 f (θ). − N (k)f (θ) = k (1) 2 HA (k) 

(9.14)

(1)

HA (k)

We also denote the operator on the right by kQ(A, k) − 1/2, with (1)

Q(ν, k) =

(9.15)

Hν (k) (1)

Hν (k)

.

We will want to look at the Green function and scattering amplitude, but first we derive some properties of the operators (9.13) and (9.14) which follow from the special nature of the operator defined by (9.10). The analysis of the spectrum of the Laplace operator on S 2 given in Chap. 8 shows that (9.16)

  1 spec A = m + : m = 0, 1, 2, . . . . 2 (1)

Now, as shown in Chap. 3, Hm+1/2 (λ) and the other Bessel functions of order m + 1/2 are all elementary functions of λ. We have (9.17) where

(1)

Hm+1/2 (λ) =

2λ 1/2 π

hm (λ),

282 9. Scattering by Obstacles

hm (λ) = −i(−1)m

(9.18)

1 d m eiλ

λ dλ λ

= λ−m−1 pm (λ)eiλ and pm (λ) is a polynomial of order m in λ, given by pm (λ) = i−m−1

m k  i (m + k)! m−k λ 2 k!(m − k)!

k=0

(9.19)

= im−1 λm + · · · +

1 (2m)! . m!

2m i

Consequently, (9.20)

 hm (kr)  1 pm (kr) 1 = r−m−1 eik (r−1) r− 2 κ m + , k, kr = 2 hm (k) pm (k)

and 

(9.21)

1 kQ m + , k 2





1 = ik − m + 2

 +k

pm (k) . pm (k)

Each polynomial pm (λ) has m complex zeros {ζm1 , . . . , ζmm }, by the fundamental theorem of algebra, and the collection of all these ζmj is clearly the set of scattering poles for S 2 . Note that (9.21) can be written as (9.22)

    m  1 1 kQ m + , k = ik − m + (k − ζmj )−1 . +k 2 2 j=1

We now look at the expression for the Green kernel G(x, y, k) for the operator (Δ + k 2 )−1 , for k real. Thus we look for a solution to (9.23)

(Δ + k 2 )u = f on Ω,

u = 0 on ∂K,

satisfying the radiation condition at infinity, given f ∈ C0∞ (Ω). If we write  (9.24) f (rθ) = fj (r)ϕj (θ), j

using the eigenfunctions ϕj as before, and  (9.25) u(rθ) = uj (r)ϕj (θ), j

then the functions uj (r) satisfy

9. Scattering by a sphere

(9.26)

r2 uj (r) + 2ruj (r) + (k 2 r2 − λ2j )uj (r) = r2 fj (r),

283

r > 1,

together with the boundary condition uj (1) = 0 and, as a consequence of the radiation condition, r(uj (r)−iku j (r)) → 0 as r → ∞. We will write the solution in the form  ∞ Gνj (r, s, k)fj (s)s2 ds, (9.27) uj (r) = 1

where the kernel Gν (r, s, k) remains to be constructed, as the Green kernel for the ordinary differential operator (9.28)

  d2 2 d λ2 2 + k − 2 , Lν = 2 + dr r dr r

 ν=

1 λ + 4 2

1/2 ,

that is, (9.29)

Lν gν (·, s, k) = s−2 δs

on (1, ∞),

satisfying the boundary condition of vanishing at r = 1, together with the radiation condition as r → ∞. This operator is self-adjoint on the space L2 ([1, ∞), r2 dr), and Gν (r, s, k) satisfies the symmetry condition (9.30)

Gν (r, s, k) = Gν (s, r, k).

Thus it suffices to specify Gν (r, s, k) for r > s. Since Gν (·, s, k) is annihilated by Lν for r > s and satisfies the radiation condition, we must have (9.31)

Gν (r, s, k) = cν (s, k)r−1/2 Hν(1) (rk), for r > s,

for some coefficient cν (s, k) that remains to be determined. In view of the symmetry (9.30), cν (·, k) satisfies the same sort of modified Bessel equation, and so is a (1) linear combination of s−1/2 Jν (sk) and s−1/2 Hν (sk). The boundary condition gives cν (s, k) = 0 at s = 1, so we can write (9.32)



Jν (k) (1) cν (s, k) = bν (k)s−1/2 Jν (sk) − (1) Hν (sk) , Hν (k)

where the coefficient bν (k) remains to be determined. This can be done by plugging (9.32) into (9.31), using (9.30) to write Gν (r, s, k) for r < s, and examining the jump in the first derivative of gν with respect to r across r = s. Achieving (9.29) then specifies bν (k) uniquely. A straightforward calculation shows that bν (k) is the following constant, independent of ν and k, in view of the Wronskian relation:

284 9. Scattering by Obstacles

(9.33)

bν (k) = b =

sk (1) Jν (sk)Hν (sk)



(1) Jν (sk)Hν (sk)

=

π . 2i

To summarize, Gν (r, s, k) is given by

(9.34)



Jν (k) (1) b(rs)−1/2 Jν (sk) − (1) Hν (sk) Hν(1) (rk), Hν (k)

Jν (k) (1) Hν (rk) Hν(1) (sk), b(rs)−1/2 Jν (rk) − (1) Hν (k)

r ≥ s, r ≤ s.

In light of this, we can represent the Green kernel for the solution to (9.23) satisfying the radiation condition as follows. Using the Schwartz kernel theorem, we can identify an operator on functions on Ω with a (generalized) function of r, s with values in the space of operators on functions on the sphere S 2 . With this identification, we have G(x, y, k) =

(9.35)

1 GA (r, s, k), 4π

where |x| = r, |y| = s, and A is given by (9.10). This is also the formula for the resolvent kernel of (Δ + k 2 )−1 , for Im k > 0. The formula (9.34) for Gν (r, s, k), as a sum of two terms, corresponds to the decomposition (1.30) for G(x, y, k), that is, G(x, y, k) = g(x, y, k) + h(x, y, k),

(9.36) where, as in (1.5), (9.37)

g(x, y, k) =

eik |x−y| . 4π|x − y|

Now recall from Proposition 1.6 how we can obtain the eigenfunctions (9.38)

u(x, ξ) = e−ix·ξ + v(x, ξ)

from the asymptotic behavior of G(x, y, k) as |y| → ∞, via (9.39)

h(x, rω, k) =

eikr v(x, kω) + O(r−2 ), 4πr

r → ∞,

proved in (1.37). We therefore have (9.40) where we set

v(rθ, kω) = lim se−iks hA (r, s, k), s→∞

9. Scattering by a sphere

(9.41)

hν (r, s, k) = 4π b(rs)−1/2

Jν (k) (1)

Hν (k)

285

Hν(1) (sk)Hν(1) (rk).

As before we identify a function of (θ, ω) with an operator on C ∞ (S 2 ), with A acting on functions of θ. To evaluate the limit in (9.40), we can use (9.42)

Hν(1) (λ) =

2 1/2 ei(λ−πν/2−π/4) + o(λ−1/2 ), πλ

λ → ∞,

which can be deduced from the integral formula (9.7). We obtain v(rθ, kω) = V(A, r, k),

(9.43) where (9.44)

V(ν, r, k) = 2π 2 i

2 1/2 Jν (k) Hν(1) (rk). e−(1/2)πi(ν+1/2) (1) πrk Hν (k)

We can now evaluate the scattering amplitude, which satisfies (9.45)

a(−ω, θ, k) = lim re−ikr v(rθ, kω), r→∞

according to (1.41). Using (9.42)–(9.44), we have (9.46)

a(−ω, θ, k) =

4π JA (k) −πiA e . k H (1) (k) A

In other words, if the right side is Ξ(A), then  Ξ(A)f (θ) =

a(−ω, θ, k)f (ω) dω.

Now, as shown in the study of harmonic analysis on spheres, in (4.44) of Chap. 8, (9.47)

e−πiA f (ω) = −i f (−ω),

f ∈ L2 (S 2 ),

so we can write (9.48)

a(ω, θ, k) = −

4πi JA (k) . k H (1) (k) A

Recall that the scattering amplitude a(ω, θ, k) is related to the scattering operator S(k) by (9.49)

S(k) = I +

k A(k), 2πi

286 9. Scattering by Obstacles

where a(ω, θ, k) is the kernel of A(k), by (3.14)–(3.15). In other words, A(k) is the operator on the right side of (9.48). Therefore, the scattering operator itself has the form (2)

S(k) = −

(9.50)

HA (k) (1)

HA (k)

in view of the identity (9.51)

Hν(1) (λ) + Hν(2) (λ) = 2 Jν (λ).

We also note that (1)

Hν(2) (k) = Hν (k),

(9.52)

for ν and k real, so (9.50) explicitly displays the unitarity of the scattering operator, for real k. The investigation of scattering by a sphere can be carried further, based on (1) these formulas. For example, qualitative information on the zeros of Hν (λ) yields qualitative information on the scattering poles. Some of the most delicate (1) results on such scattering make use of the uniform asymptotic behavior of Hν (λ) as ν and λ both tend to ∞. A treatment of this in a modern spirit, touching on more general approaches to diffraction problems, is given in [T2] and [MT1], and, in more detail, in Appendix C of [MT2]. Also, [Nus] gives a lengthy analysis of scattering by a sphere, from a more classical perspective.

Exercises (1)

1. Derive from (9.7) that Hm+1/2 (z) = (2z/π)1/2 z −m−1 pm (z)eiz , with (−i)m+1 pm (z) = m!



∞ 0

s m e−s sm z − ds. 2i

Show that this yields (9.19). 2. From the material on Bessel functions developed in Chap. 3, show that there is the Wronskian identity Hν(1) (λ)Hν(2) (λ) − Hν(1) (λ)Hν(2) (λ) = (1)

C , λ

and evaluate C. Using this, prove that Hν (λ) is not zero for any λ ∈ (0, ∞), ν ∈ (0, ∞). 3. Use results on the location of scattering poles from §7 to show that (9.13) and (9.14) (1) imply Hν (z) has zeros only in Im z < 0, for ν = m + 1/2, m = 0, 1, 2, . . . . It is known that this property holds for all ν ∈ [0, ∞). See [Wat], p. 511. There it

Exercises

287

is stated in terms of the zeros of Kν (z), which is related to the Hankel function by (1) Kν (z) = (πi/2)eπiν/2 Hν (iz). 4. A formula of Nicholson (see [Olv], p. 340, or [Wat], p. 444) implies  ∞ 8 K0 (2z sinh t) cosh 2νt dt, Jν (z)2 + Yν (z)2 = 2 π 0 for Re z > 0. Here K0 (r) is Macdonald’s function, the ν = 0 case of the function mentioned in Exercise 3; cf. (6.50)–(6.54) in Chap. 3. K0 (r) is a decreasing function of r ∈ (0, ∞), and hence, for fixed ν > 0, Jν (x)2 + Yν (x)2 is a decreasing function of x ∈ R+ . Show that this implies that B(k, r) : L2 (S 2 ) −→ L2 (S 2 ), defined by B(k, r)f (θ) = B(k)f (rθ), has operator norm ≤ r−1/2 , for r ≥ 1. Consequently, B(k)f L2 (Ω,|x|−4 dx ) ≤ f L2 (S 2 ) .

(9.53)

Using the integral formula (6.50) of Chap. 3, show that rK0 (r) is decreasing on R+ , (1) hence that |r1/2 Hν (r)| is decreasing on R+ , for fixed ν > 0. Use this to show that −1 B(k, r) ≤ r for r ≥ 1, and sharpen (9.53). 5. Let A = {x ∈ R3 : 1 < |x| < 2}. With u = B(k)f , use Δu = −k2 u and estimates derivable from Chap. 5, in concert with Exercise 4, to show that B(k)f H 2 (A) ≤ Cf H 3/2 (S 2 ) + Ck2 f L2 (S 2 ) .

(9.54) Deduce that (9.55) 6. Show that

N (k)f H 1/2 (S 2 ) ≤ Cf H 3/2 (S 2 ) + k2 f L2 (S 2 ) .     k Q(m + 1/2, k) ≤ C |k| + m + 1 ,

for k ∈ R, m ≥ 0. Deduce that, for s ∈ R, k ∈ R, (9.56)

N (k)f H s (S 2 ) ≤ Cs f H s+1 (S 2 ) + Cs |k| · f H s (S 2 ) .

Compare this with the bound on N (k) derived in the previous exercise. (Hint: Consider uniform asymptotic expansions of Bessel and Hankel functions, discussed in [Erd] and in Chap. 11 of [Olv]. Compare a related analysis in [T2].) 7. Suppose an obstacle K is contained in the unit ball B1 = {|x| < 1}. Show that the solution to the scattering problem (1.1)–(1.3) is uniquely characterized on Ω1 = (R3 \ K) ∩ B1 as the solution to (9.57)

(Δ + k2 )v = 0 on Ω1 ,

v = f on ∂K,

∂v = N (k)v on S 2 , ∂r

where N (k) is given by (9.14). 8. Derive the formulas of this section, particularly the formula analogous to (9.50) for S(k), in the case of scattering by a sphere of radius R, centered at p ∈ R3 , displaying explicitly the dependence of the various quantities on R and p.

288 9. Scattering by Obstacles 9. It follows from (9.46)–(9.48) that the scattering amplitude for S 2 satisfies (9.58)

a(ω, θ, k) = a(θ, ω, k)

and a(ω, θ, k) = a(−ω, −θ, k).

Demonstrate these identities directly, for ∂K = S 2 . How much more generally do they hold? Compare (3.31). 10. Suppose v ∈ C ∞ (R3 ) solves (Δ + k2 )v = 0. Show that v(rθ) has the form  v(rθ) = vj (r)ϕj (θ), j

where ϕj is an eigenfunction of ΔS , as in (9.4), and vj (r) = bj jνj −1/2 (kr). (Hint: vj (r) solves (9.5) and does not blow up as r → 0.) Deduce that, for some coefficients β , eikr ω·θ =

(9.59)

∞ 

β j (kr) P (ω · θ),

=0

where P (t) are the Legendre polynomials, defined in (4.36) of Chap. 8. As shown in (4.49) of Chap. 8, this formula holds with β = (2 + 1)i , so (9.60)

eist =

∞  (2 + 1) i j (s) P (t),

s ∈ R,

t ∈ [−1, 1].

=0

11. As noted in §4 of Chap. 8, the identity (9.59), with β = (2 + 1)i , is equivalent to the assertion that eikr ω·θ is the integral kernel of (9.61)

Ξkr (A) = 4π e(1/2)πi(A−1/2) jA−1/2 (kr).

Show that this is in turn equivalent to the r = 1 case of (9.43)–(9.44). (Hint: Use (9.47).) 12. Derive explicit formulas for scattering objects (e.g., S(k)), in the case of the quantum scattering problem for H = −Δ + V , when V (x) = b, 0,

for |x| ≤ R, otherwise.

Keep track of the dependence on b and R. If you fix R = 1 and let b decrease from b = 0 to the first value b = −β0 , below which −Δ + V has a negative eigenvalue, what happens to some of the scattering poles?

10. Inverse problems I By “inverse problems” we mean problems of determining a scatterer ∂K in terms of information on the scattered waves. These problems are of practical interest. One might be given observations of the scattered wave v(x, kω), for x in a region not far from ∂K, k belonging to some restricted set of frequencies (maybe a single frequency). Or one might have only the far field behavior, defined by the

10. Inverse problems I

289

scattering amplitude a(ω, θ, k), which we recall is related to v(x, kω) by (10.1)

v(rθ, kω) ∼

eikr a(−ω, θ, k), r

r → ∞.

In this section we examine the question of what scattering data are guaranteed to specify ∂K uniquely, at least if the data are measured perfectly. It is useful to begin with the following explicit connection between the scattered wave v(x, kω) and the scattering amplitude. Proposition 10.1. If K ⊂ BR (0), then, for r ≥ R, (10.2)

a(−ω, θ, k) = −ik −1 e(1/2)πi(A−1/2) hA−1/2 (kr)−1 g(θ),

where g = gr,ω,k is given by g(θ) = v(rθ, kω).

(10.3)

As in §9, hA−1/2 (kr)−1 is regarded as a family of functions of the self-adjoint operator A defined by (9.10). Recall that hm (λ) is given by (9.17)–(9.18). Proof. This result follows easily from (9.12), which implies (10.4)

B(k)f (rθ) =

hA−1/2 (kr) f (θ), hA−1/2 (k)

for f ∈ C ∞ (S 2 ). To prove (10.2)–(10.3), we can suppose without loss of generality that R = 1 and apply (10.4) with f (θ) = v+ (θ, kω) to get v+ (rθ, kω) =

hA−1/2 (kr) f (θ); hA−1/2 (k)

f (θ) = v(θ, kω).

Now compare the asymptotic behavior of both sides as r → ∞. For the left side we have (10.1), while the behavior of the right side is governed by (10.5)

hm (kr) ∼ im−1

eikr kr

by (9.18)–(9.19), so (10.2) follows. Now we can invert the operator in (10.2), to write (10.6)

v(rθ, kω) = ik e−(1/2)πi(A−1/2) hA−1/2 (kr)a(−ω, θ, k),

where the operator acts on functions of θ. The operator hA−1/2 (kr) is an unbounded operator on L2 (S 2 ); indeed, it is not continuous from C ∞ (S 2 ) to D (S 2 ), which has consequences for the inverse problem, as we will see in §11.

290 9. Scattering by Obstacles

F IGURE 10.1 Two Obstacles

Suppose now that K1 and K2 are two compact obstacles in R3 giving rise to scattered waves which both agree with v(x, kω) in some open set O in R3 \ (K1 ∪ K2 ). In other words vj (x, kω) = v(x, kω) for x ∈ O, where the functions vj are solutions to (10.7)

(Δ + k 2 )vj = 0 on R3 \ Kj ,

vj = −e−ikx ·ω on ∂Kj ,

satisfying the radiation condition. We suppose the sets Kj have no “cavities”; that is, each Ωj = R3 \ Kj has just one connected component. In this case, possibly the complement of K1 ∪ K2 is not connected; cf. Fig. 10.1. We will let U denote the unbounded, connected component of this complement, and consider R3 \ U,  2 , so K1 ⊂ K  2 . This is illustrated in Figs. 10.1 and 10.2. which we denote by K  2 \K1 . We assume O ⊂ U. Let R be any connected component of the interior of K (Switch indices if K2 ⊂ K1 .) The functions v1 and v2 described above agree on U, since they are real analytic and agree on O. Thus u1 and u2 agree on U, where u j (x) = vj (x) + e−ikx ·ω . Since each uj vanishes on ∂Kj , it follows that u = u1 R vanishes on ∂R, so (10.8)

(Δ + k 2 )u = 0 on R,

u = 0 on ∂R.

In fact, u ∈ H01 (R). However, u does not vanish identically on R. In particular, u provides an eigenfunction of Δ on each connected component R of the interior  2 \ K1 , with Dirichlet boundary condition (and with eigenvalue −k 2 ) if u of K is not identically zero, and if the symmetric difference K1  K2 has nonempty interior. Now, there are circumstances where we can obtain bounds on (10.9) dim ker (Δ + k 2 ) H 1 (R) = d(k); 0

for example, if we know the obstacle is contained in a ball BR . We then have the following uniqueness result:

10. Inverse problems I

291

F IGURE 10.2 Filled Obstacle

Proposition 10.2. Let k ∈ (0, ∞) be fixed. Suppose Σ = {ω } is a subset of S 2 whose cardinality is known to be greater than d(k)/2. (If ω and −ω both belong to Σ, do not count them separately.) Then knowledge of v+ (x, kω ) for x in an open set O uniquely determines the obstacle K. Hence knowledge of a(−ω , θ, k) for θ ∈ S 2 uniquely determines K. Proof. If K were not uniquely determined, there would be a nonempty set R such as described above. The corresponding u (x) = v(x, kω )+eikx ·ω , together with their complex conjugates, which are all eigenfunctions on R, must be linearly independent. Indeed, any linear dependence relation valid on R must continue on all of R3 \ (K1 ∩ K2 ); but near infinity, u (x) = eikx ·ω + O(|x|−1 ) guarantees independence. We make a few complementary remarks. First, a(−ω, θ, k) is analytic in its arguments, so for any given ω, k, it is uniquely determined by its behavior for θ in any open subset of S 2 . Next, for k small enough, we can say that d(k) = 0, so uniqueness holds in that case, for a single ω = ω . Note that even when k 2 is an eigenvalue of −Δ on R, it would be a real coincidence for a corresponding eigenfunction to happen to continue to R3 \ (K1 ∩ K2 ) with the appropriate behavior at infinity. It is often speculated that knowledge of a(−ω, θ, k), for θ ∈ S 2 (or an open set) and both k and ω fixed, always uniquely determines the obstacle K. This remains an interesting open problem. Furthermore, suppose a(−ω, θ, k) is known on θ ∈ S 2 , for a set {ω } ⊂ 2 S and a set {km } ⊂ R+ . Then one has uniqueness provided card{ω } > min d(km )/2. In particular, if {km } consists of an interval I (of nonzero length), then min d(km ) = 0, so knowledge of a(−ω, θ, k) for θ ∈ S 2 , k ∈ I, and a single ω uniquely determines K. All of these considerations are subject to the standing assumption made throughout this chapter on the smoothness of ∂K. There are interesting cases of non-smooth obstacles, not equal to the closure of their interiors, to which the proof of Proposition 10.2 would not apply. We will discuss this further in §12.

292 9. Scattering by Obstacles

We also mention that the method used to prove Proposition 10.2 is ineffective when one has the Neumann boundary condition. A uniqueness result in that case, using a different technique, can be found in [CK2]; see also [Isa]. One study that sheds light on the inverse problem is the linearized inverse problem. Here, given an obstacle K, denote by BK (k) the solution operator (7.1)–(7.2) and by SK (k) the scattering operator (3.7), with corresponding scattering amplitude aK (ω, θ, k), as in (3.14)–(3.15). We want to compute the “derivative” with respect to K of these objects, and study their inverses. More precisely, if K is given, ∂K smooth, we can parameterize nearby smooth obstacles by a neighborhood of 0 in C ∞ (∂K), via the correspondence that, to ψ ∈ C ∞ (∂K) (real-valued), we associate the image ∂Kψ of ∂K under the map (10.10)

Fψ (x) = x + ψ(x)N (x),

x ∈ ∂K,

where N (x) is the unit outward-pointing normal to ∂K, at x. Then, denote BKψ (k) and aKψ (ω, θ, k) by Bψ (k) and aψ (ω, θ, k). We want to compute ∂ Bsψ (k)f s=0 ∂s and Dψ aK (ω, θ, k) = ∂s asψ (ω, θ, k) s=0 . The following is a straightforward exercise. Proposition 10.3. If f is smooth near ∂K and vψ (x) = ∂s Bsψ (k)f s=0 , for x ∈ R3 \ K, then vψ (x) is uniquely characterized by (10.11)

(10.12)

Dψ BK (k)f =

(Δ + k 2 )vψ = 0 on R3 \ K,

∂v ψ − ikv ψ → 0, as r → ∞, r ∂r ∂f

vψ = ψ(x) N (k)f − on ∂K. ∂ν

Here, N (k) is the Neumann operator, defined by (7.55). In other words, (10.13)

 ∂f  Dψ BK (k)f = BK (k) ψ(x) N (k)f − . ∂ν

The linearized problem is to find ψ. Therefore, for a given smooth obstacle K, granted that the operators BK (k) and N (k) have been constructed (e.g., by methods of §7), we can to some degree reduce the linearized inverse problem for ψ to the following linear inverse problem: Problem. Given (an approximation to) w = B(k)g(x) on |x| = R1 (and assuming that K ⊂ {x : |x| < R1 }), find (an approximation to) g on ∂K.

10. Inverse problems I

293

As for finding BK (k) and N (k) via an integral-equation method, we mention that an integral equation of the form (7.18) is preferable to one of the form (7.12), since it is very inconvenient to deal with the set of values of k for which (7.12) is not solvable. This point is made in many expositions on the subject, such as, [Co]. Solving (7.18) leads to the formula (7.32) for BK (k). We note that when we take f = e−ik ω·x , the solution to the linearized inverse problem is unique: Proposition 10.4. Given K nonempty, smooth, and compact, such that R3 \ K is connected, define (10.14)

LK (k, ω) : H s (∂K) → C ∞ (R3 \ K)

by (10.15)

LK (k, ω)ψ = Dψ BK (k)f,

f (x) = e−ik ω·x .

Then LK (k, ω) is always injective. Proof. By (10.13), our claim is that  ∂f  = 0 on R3 \ K =⇒ ψ = 0 on ∂K. BK (k) ψ(x) N (k)f − ∂ν   Since BK (k)g ∂K = g, the hypothesis in (10.16) implies ψ(x) N (k)f − ∂ν f = 0 on ∂K, so it suffices to show that

(10.16)

(10.17)

N (k)f −

∂f vanishes on no open subset O of ∂K ∂ν

when f (x) = e−ik ω·x . To see this, consider w = B(k)f −e−ik ω·x , which satisfies (10.18)

(Δ + k 2 )w = 0 on R3 \ K,

w = 0 on ∂K.

If N (k)f − ∂ν f = 0 on O, then ∂w = 0 on O. ∂ν

(10.19)

But if O is a nonempty, open subset of ∂K, then (10.18)–(10.19) imply that w is identically zero, by uniqueness in the Cauchy problem for Δ + k 2 . This is impossible, so the proof is complete. Parallel to (10.13), we have

(10.20)

 ∂f  Dψ aK (−ω, θ, k) = AK (k) ψ(x) N (k)f − (θ), ∂ν f (x) = e−ik ω·x ,

294 9. Scattering by Obstacles

where AK (k) is as in (3.33)–(3.34). Note that (10.2) extends readily to the identity (10.21)

AK (k)f = −ik −1 e(1/2)πi(A−1/2) hA−1/2 (kr)−1 BKr (k)f.

In view of the injectivity of the operator acting on BKr (k)f , on the right side of (10.21) we see that, under the hypotheses of Proposition 10.4, we have (10.22)

LK (k, ω) : H s (∂K) −→ C ∞ (S 2 ) injective,

for each k ∈ R, ω ∈ S 2 , where (10.23)

LK (k, ω)ψ = Dψ AK (k)f,

f = e−ik ω·x .

Exercises 1. Fix k ∈ (0, ∞). Show that a given obstacle K is a ball centered at 0 if and only if a(ω, θ, k) = a(R(ω), R(θ), k), for every rotation R : S 2 → S 2 . (Hint: For the “if” part, make use of Proposition 10.2 to compare K and its image under a rotation.) 2. Suppose you are given that K is a ball, but you are not given its radius or its center. How can you determine these quantities from the scattering amplitude? How little information on a(ω, θ, k) will suffice? 3. The set R arising in the proof of Proposition 10.2 might not have smooth boundary, so how do you know that u = u1 R , which vanishes on ∂R, belongs to H01 (R)? 4. Suppose K is known to be contained in the unit cube Q = {x ∈ R3 : 0 ≤ xj ≤ 1}. Let ω ∈ S 2 , k ∈ R be fixed. Show that exact knowledge of a(−ω, θ, k), for θ ∈ S 2 , uniquely determines K, as long as √ |k| < 6 π. Given ω1 , ω2 ∈ S 2 , such that ω1 , ω2 , −ω1 , and −ω2 are distinct, show that a(−ωj , θ, k), for k fixed, θ ∈ S 2 , j = 1, 2, uniquely determines K, as long as |k| < 3π. Can you improve these results? 5. Give a detailed proof of Proposition 10.3.

11. Inverse problems II In this section we describe some of the methods used to determine an obstacle K (approximately) when given a measurement of scattering data, and we deal with some aspects of the “ill-posed” nature of such an inverse problem.

11. Inverse problems II

295

For simplicity, suppose you know that B1 ⊂ K ⊂ BR0 , where Br = {x ∈ R3 : |x| ≤ r}. Suppose you have a measurement of a(−ω, θ, k) on θ ∈ S 2 , with k fixed and ω fixed. One strategy is to minimize (11.1)

Φ(f, K) = A1 (k)f − a(−ω, ·, k) 2L2 (S 2 ) + B1K (k)f + e−ik ω·x 2L2 (∂K) ,

with f and K varying over certain compact sets, determined by a priori hypotheses on the scatterer. This is close to some methods of Angell and Kleinman, Kirsch and Kress, as described at the end of [Co2]. Here we use the following notation: Ar (k) = ABr (k), where AK (k) is as in (10.20), that is, (11.2)

AK (k)f (θ) = lim re−ikr BK (k)f (rθ). r→∞

Also, if K1 is contained in the interior of K2 , then g = BK1 (k)f ∂K defines a 2 bounded operator (11.3)

BK1 K2 (k) : H s (∂K1 ) −→ C ∞ (∂K2 ),

and if either K1 = Br or K2 = Bρ , we use the notation BrK2 (k) or BK1 ρ (k); if both K1 and K2 are such balls, we use the notation Brρ (k). More generally, we might have measurements of a(−ωj , θ, k) on θ ∈ S 2 , for a sequence of directions ωj . Then one might take Φ= (11.4)

N    A1 (k)fj − a(−ωj , ·, k)2 2 2 L (S ) j=1

+

N    B1K (k)fj + e−ik ωj ·x 2 2 L (∂K) j=1

and minimize over (f1 , . . . , fN ; K). One might also consider weighted sums, and perhaps stronger norms. Note that (11.5)

A1 (k)fj = AK (k)B1K (k)fj .

The feasibility of approximating the actual scattered wave by such a function follows from the next lemma, provided K is connected and has connected complement.

296 9. Scattering by Obstacles

Lemma 11.1. If Kj are compact sets in R3 (with connected complement) such ◦

that K1 ⊂ K 2 , then for any k ∈ R the map BK1 K2 (k) is injective. If also K2 is connected, this map has dense range. Proof. If u = BK1 (k)f vanishes on ∂K2 , then u restricted to R3 \ K2 is an outgoing solution to the basic scattering problem (1.1), with K = K2 , so by the uniqueness of solutions to (1.1)–(1.3), we have u = 0 on R3 \ K2 . Then unique continuation forces u = 0 on R3 \ K1 , so the injectivity of (11.3) is established. ◦

As for the second claim, note that if y ∈ K 1 , then (11.6)

|x − y|−1 eik |x−y| = gy (x) ∈ Range BK1 (k).

Thus if f ∈ L2 (∂K2 ) is in the orthogonal complement of the range of BK1 K2 (k), we deduce that  f (y)gx (y) dS (y) (11.7) F (x) = ∂K2 ◦





is zero for x ∈ K 1 , hence for x ∈ K 2 (if K 2 is connected). Also material in §7 implies that F is continuous across ∂K2 and is an outgoing solution of (1.1) on R3 \ K2 . The uniqueness of solutions to (1.1)–(1.3) forces F = 0 on R3 \ K2 . Since, by (7.25), the jump of ∂ν F across ∂K2 is proportional to f, this implies f = 0, proving denseness. If K is not known to be connected, one could use several spherical bodies as domains of fj in (11.4), provided it is known that each connected component of K contains one of them, as can be seen by a variant of the proof of Lemma 11.1. Instead of minimizing (11.4) over (f1 , . . . , fN ; K), an alternative is first to minimize the first term of (11.4), thus choosing fj , within some compact set of functions, and then to pick K to minimize the second term, within some compact set of obstacles. An attack pursued in [Rog] and [MTW] takes a guess Kμ of K, solves (approximately) a linearized inverse problem, given Kμ , and applies an iteration, provided by Newton’s method, to approximate K. See also [Kir]. If one has a measurement of the scattered wave v(x, kω) on the sphere |x| = r, say for k fixed and ω = ω1 , . . . , ωN , instead of a measurement of a(−ω, θ, k), then parallel to (11.4) one might take Φ= (11.8)

N    B1r (k)fj − v(·, kωj )2 2 2 L (S ) j=1

+

r

N    B1K (k)fj + e−ik ωj ·x 2 2 L (∂K) j=1

11. Inverse problems II

297

and minimize over (f1 , . . . , fN ; K). Alternatively, first minimize the first sum over fj (in some compact set of functions) and then minimize the second sum over K (in some compact set of obstacles). In fact, a number of approaches to the inverse problem, when measurements of the scattering amplitude a(−ω, θ, k) are given, start by first constructing an approximation to v(x, kω) on some sphere |x| = r, such that K ⊂ Br , and then proceed from there to tackle the problem of approximating K. Recall the relation established in §10: (11.9)

v(rθ, kω) = ik e−(1/2)πi(A−1/2) hA−1/2 (kr) a(−ω, θ, k),

where the operator acts on functions of θ. As noted there, the operator hA−1/2 (kr) is a seriously unbounded operator on L2 (S 2 ). In fact, this phenomenon is behind the ill-posed nature of recovering the near field behavior v(x, kω) from the far field behavior defined by the scattering amplitude a(−ω, θ, k). As this is one of the simplest examples of an ill-posed problem, we will discuss the following problem. Suppose you know that an obstacle K is contained in a ball BR0 . Fix k ∈ R, ω ∈ S2. Problem A. Given an approximation b(−ω, θ, k) to the scattering amplitude, with (11.10)

a(−ω, ·, k) − b(−ω, ·, k) L2 (S 2 ) ≤ ε,

how well can you approximate the scattered wave v(x, kω), for x on the shell |x| = R1 , given R1 > R0 ? As we have said, what makes this problem difficult is the failure of the operator hA−1/2 (kr) appearing in (11.9) to be bounded, even from C ∞ (S 2 ) to D (S 2 ). Indeed, for fixed s ∈ (0, ∞), one has the asymptotic behavior, as ν → +∞, (11.11)

Hν(1) (s)

 ∼ −i

2 πν

1/2 

2ν es



(see the exercises) and hence (11.12)

hν−1/2 (s) ∼ −i(sν)−1/2



2ν es

ν .

Consequently, an attempt to approximate v+ (x, kω) for x = R1 θ by (11.13)

v0 (θ) = ik e−(1/2)πi(A−1/2) hA−1/2 (kR1 )b(−ω, θ, k)

could lead to nonsense. We will describe a method below that is well behaved. But first we look further into the question of how well can we possibly hope to approximate v(x, kω) on the shell |x| = R1 with the data given.

298 9. Scattering by Obstacles

In fact, it is necessary to have some further a priori bound on v+ to make progress here. We will work under the hypothesis that a bound on v(x, kω) is known on the sphere |x| = R0 : (11.14)

v(R0 θ, kω) L2 (Sθ2 ) ≤ E.

Now, if we are given that (11.10) and (11.14) are both true and we have b(−ω, θ, k) in hand (for ω, k fixed, θ ∈ S 2 ), then we can consider the set F of functions f (θ) such that (11.15)

f − b(−ω, ·, k) L2 (S 2 ) ≤ ε

and such that (11.16)

k hA−1/2 (kR0 )f L2 (S 2 ) ≤ E,

knowing that F is nonempty. We know that a(θ) = a(−ω, θ, k) belongs to F, and that is all we know about a(−ω, θ, k), in the absence of further data. The greatest accuracy of an approximation v1 (θ) to v(R1 θ, kω) that we can count on, measured in the L2 (S 2 ) norm, is (11.17)

v1 (θ) − v(R1 θ, kω) L2 (Sθ2 ) ≤ 2 M (ε, E),

where M (ε, E) is defined as follows. Denote by (11.18)

Tj : F −→ L2 (S 2 ),

j = 0, 1,

the maps (11.19)

Tj f (θ) = ik e−(1/2)πi(A−1/2) hA−1/2 (kRj )f (θ).

Then we set (11.20)

2M (ε, E) = sup { T1 f − T1 g L2 (S 2 ) : f, g ∈ F},

that is, (11.21)

M (ε, E) = sup { T1 f L2 : f L2 ≤ ε and T0 f L2 ≤ E}.

One way to obtain as accurate as possible an approximation to v on |x| = R1 would be to pick any f ∈ F and evaluate T1 f . However, it might not be straightforward to obtain elements of F. We describe a method, from [Mr2] and [MrV], which is effective in producing a “nearly best possible” approximation. We formulate a more general problem. We have a linear equation

11. Inverse problems II

299

Sv = a,

(11.22)

where S is a bounded operator on a Hilbert space H, which is injective, but S −1 is unbounded (with domain a proper linear subspace of H). Given an approximate measurement b of a, we want to find an approximation to the solution v. This is a typical ill-posed linear problem. As a priori given information, we assume that (11.23)

b − a H ≤ ε,

T0 a H ≤ E.

T0 is an auxiliary operator. In the example above, H = L2 (Sθ2 ), and T0 and T1 are given by (11.19). Generalizing (11.20)–(11.21), we have a basic measurement of error:   M (ε, E) = sup T1 f H : f H ≤ ε and T0 f H ≤ E (11.24)   1 = sup T1 f − T1 g H : f, g ∈ F , 2 where (11.25)

F = {f ∈ H : f − b H ≤ ε, T0 f H ≤ E}.

Now, if all one knows about a in (11.22) is that it belongs to F, then the greatest accuracy of an approximation v1 to the solution v of (11.22) one can count on is (11.26)

v1 − v H ≤ 2M (ε, E).

This recaps the estimates in (11.14)–(11.21). Now we proceed. An approximation method is called nearly best possible (up to a factor γ) if it yields a v1 ∈ H such that (11.27)

v1 − v H ≤ 2γM (ε, E).

We now describe one nearly best possible method for approximating v, in cases where T0 is a self-adjoint operator, with discrete spectrum accumulating only at +∞. Then, pick an orthonormal basis {uj : 1 ≤ j < ∞} of H, consisting of eigenvectors, such that (11.28)

T0 uj = αj uj ,

αj  +∞.

When T0 is given by (11.19), this holds as a consequence of (11.12). It is essential that the αj be monotonic, so the eigenvectors need to be ordered correctly. Now let (11.29)

f = P b,

P u =

  j=1

(u, uj )uj .

300 9. Scattering by Obstacles

Now let N be the first such that f − b H ≤ 2ε.

(11.30) We then claim that

T0 fN H ≤ 2E.

(11.31) This can be deduced from:

Lemma 11.2. If the set F defined by (11.25) is nonempty, and if M + 1 is the first j such that αj > E/ε, then (11.32)

fM − b H ≤ 2ε and T0 fM H ≤ 2E.

Proof of lemma. The key facts about M are the following: (11.33)

PM g ≤ ε =⇒ T0 PM g ≤ E, (T0 (1 − PM )h ≤ E =⇒ (1 − PM )h ≤ ε.

We are given that there exists f such that f − b ≤ ε and T0 f ≤ E.

(11.34)

The first part of (11.34) implies PM f − fM ≤ ε, which via the first part of (11.33) yields the second part of (11.32). The second part of (11.34) implies T0 (1 − PM )f ≤ E, which by the second part of (11.33) gives (1 − PM )f ≤ ε. Since f − b ≤ ε, this yields the first part of (11.32). Having the lemma, we see that N ≤ M , so T0 fN ≤ T0 fM , giving (11.31). Then (11.30) and (11.31), together with (11.23), yield (11.35)

fN − a H ≤ 3ε and T0 (fN − a) H ≤ 3E.

We have established: Proposition 11.3. Under the hypotheses (11.23), if we set vN = T1 fN , where N is the smallest such that (11.30) holds, we have (11.36)

vN − v H ≤ 3 M (ε, E).

Hence this method of approximating v is nearly best possible. Note that the value of the estimate E of (11.14) does not play an explicit role in the method described above for producing the approximation vN ; it plays a role in estimating the error vN − v. The method described above provides a technique for solving a certain class of ill-posed problems. Other related problems involve the analytic continuation of

11. Inverse problems II

301

functions and solving backwards heat equations. Further discussions of this and other techniques, can be found in papers of K. Miller [Mr1], [Mr2], and references given there. We now turn to the task of estimating M (ε, E), for our specific problem, defined by (11.18)–(11.21). Thus, with R0 < R1 , we want to estimate hA−1/2 (kR1 ) L2 , given that f L2 ≤ ε and hA−1/2 (kR0 )f L2 ≤ E. If we also assume that k lies in a bounded interval, this is basically equivalent to estimating A−1/2 e−βA AA f L2 , given that A−1/2 e−αA AA f L2 ≤ E,

f L2 ≤ ε,

where e−α = 2/eR0 and e−β = 2/eR1 , so α < β. We can get a hold on this using the inequality (11.37)

√ x 2x ,

ν −1/2 e−βν ν ν ≤ e−(β−α)x (ν −1/2 e−αν ν ν ) +

valid for ν ≥ 1/2, 0 < x < ∞, to write (11.38)

T1 f L2 ≤ C inf

x∈R+



e−γx E + xx ε ,

where γ = β − α, given f ≤ ε, T0 f ≤ E. While picking x to minimize the last quantity is not easy, we can obtain a reasonable estimate by picking x=

(11.39)

log E/ε , log log E/ε

in which case e−γx =

ε γα(E/ε) E

,

εxx = E

ε β(E/ε) E

,

with α(E/ε) = 1/(log log E/ε) and β(E/ε) = (log log log E/ε)/(log log E/ε). Consequently, (11.40)

M (ε, E) ≤ CE



ε γα(E/ε) E

+

ε β(E/ε)  E

.

As for the exponents that appear in (11.40), note the following values (to three digits):

302 9. Scattering by Obstacles

ε/E 10−2 10−3 10−4 10−5 10−6 10−7

α(E/ε) .655 .517 .450 .409 .381 .360

β(E/ε) .277 .341 .359 .366 .368 .368

The close agreement of the last two figures in the right column is due to the fact that f (y) = (log log log y)/(log log y) achieves its maximum value of 1/e at e y = ee ≈ 3.81 × 106 and is very slowly varying in this region. As for the close agreement of the two figures corresponding to ε = 10−7 , note that log log log e ee = 1. An estimate similar to (11.40) is also given in [Isa]. Even though the analysis in (11.13)–(11.39) does not directly deal with the problem of describing ∂K given an approximation b(ω, θ, k) to the scattering amplitude a(ω, θ, k), to some degree it reduces this problem to that of describing ∂K, given the solution u = B(k)f to the scattering problem (1.1)–(1.3) (i.e., to (7.1)), with u(x) evaluated near |x| = R1 , for a certain class of boundary data, namely f (x) = e−kω·x ∂K (where k and ω belong to a specified subset of R and S 2 , respectively). One assumes it given that K ⊂ {x : |x| < R0 }, where R0 < R1 . This reduction is an intermediate step in many studies of inverse problems. Thus Problem A is complemented by: Problem B. Approximate v = B(k)f on |x| = R0 , given (an approximation to) v on |x| = R1 and having some a priori estimate of v on |x| = R0 , but not on a smaller sphere. Rescaling, we can consider the case R0 = 1, R1 = R > 1. By (10.4), we have (11.41)

g = v(θ) and w = v(Rθ) =⇒ g =

hA−1/2 (k) w = CR (k)w, hA−1/2 (kR)

where the last identity is the definition of the unbounded operator CR (k) on L2 (S 2 ). In view of (11.12), we have, for fixed k ∈ (0, ∞), R > 1, (11.42)

hν−1/2 (k) ∼ Rν+1/2 = C eγν , hν−1/2 (kR)

ν → +∞,

where γ = log R > 0. Parallel to (11.14)–(11.21), we consider the problem of estimating CR (k)w in L2 (S 2 ), given a small bound on w L2 (S 2 ) (estimate on observational error) and an a priori bound on CR (k)w in H  (S 2 ), for some > 0. That is, we want to estimate (11.43)

  M (ε, E) = sup CR (k)w L2 : w L2 ≤ ε, CR (k)w H  ≤ E .

11. Inverse problems II

303

Parallel to (11.37)–(11.38), we can attack this by writing (11.44)

eγν ≤ (νx−1 ) eγν + eγx ,

valid for ν, x ∈ (0, ∞). Thus, if g H  = A g L2 , we have (11.45)

  CR (k)w L2 ≤ Ck inf x− E + eγx ε . x>0

We get a decent upper bound by setting x = (1/2γ) log( E/εγ). This yields (11.46)

Eε 1/2 E − M (ε, E) ≤ Ck (2γ) E log + Ck . γε γ

This is bad news; ε would have to be terribly tiny for M (ε, E) to be small. Fortunately, this is not the end of the story. As a preliminary to deriving a more satisfactory estimate, we produce a variant of the “bad” estimate (11.46). Fix ψ ∈ C0∞ (R), supported on [−1, 1], such that ψ(0) = 1. Instead of (11.43), we estimate   (11.47) Mδ (ε, E) = sup ψ(δA)CR (k)w L2 : w L2 ≤ ε, CR (k)w L2 ≤ E . We proceed via (11.48)



ψ(δν)eγν ≤ e−γx ψ(δν)eγν eγν + eγx ψ(δν),

to get (11.49)

  ψ(δA)eγA w L2 ≤ inf C(γ, δ)e−γx Ck E + eγx ε , x>0

where (11.50)

C(γ, δ) = sup ψ(δν)eγν ≤ eγ/δ = R1/δ . ν>0

Using (11.42) again, we have the estimate (11.51)

Mδ (ε, E) ≤ Ck R1/2δ

√ Eε.

Now we do want to be able to take δ small, to make ψ(δA)f close to f , but R1/2δ = eγ/2δ blows up very rapidly as δ  0, so this gives no real improvement over (11.46). Compare (11.51) with the estimate (11.52)

ψ(δA)CR (k)w L2 ≤ Ck R1/δ ε,

304 9. Scattering by Obstacles

when w L2 ≤ ε, involving no use of the a priori estimate CR (k)w L2 ≤ E. We now show that a different technique yields a useful bound on the quantity Mδ (ε, E), when δ lies in the range δ > 1/k. Proposition 11.4. Let R > 1 and α > 1 be fixed. Then there is an estimate (11.53)

ψ(δA)CR (k)w L2 (S 2 ) ≤ C w L2 (S 2 ) , for

α ≤ δ. k

In particular, C is independent of k. Proof. Since ψ(δA) and CR (k) commute, it suffices to show that (11.54)

CR (k)w L2 ≤ C w L2 , for w ∈ Range χ(δA),

given α/k ≤ δ, where χ(λ) is the characteristic function of [0, 1]. Thus (11.55)

0 ≤ A ≤ (kα−1 )I on Range χ(δA).

Equivalently, we claim that an outgoing solution u(r, ω) to the reduced wave equation (Δ + k 2 )u = 0 satisfies (11.56)

u(1, ·) L2 (S 2 ) ≤ C u(R, ·) L2 (S 2 ) , for u(R, ·) ∈ Range χ(δA),

given α/k ≤ δ. Now u satisfies the equation (11.57)

 ∂ 2 u 2 ∂u  2 + k − r−2 L u = 0, + ∂r2 r ∂r

L = A2 −

1 = −ΔS . 4

We can replace u(r, ω) by v(r, ω) = ru(r, ω), satisfying  ∂2v  2 + k − r−2 L v = 0, 2 ∂r

(11.58) and it suffices to establish (11.59)

v(1, ·) L2 (S 2 ) ≤ C v(R, ·) L2 (S 2 ) ,

given v(R, ·) ∈ Range χ(δA), and assuming that v = ru, u an outgoing solution to (11.57); let us denote by Vkδ the vector space of such functions v. It will be convenient to use a family of norms, depending on r and k, given by (11.60)

Nkr (v)2 =

 

1 − (kr)−2 L v, v

 2 −2  ∂v  + k   2 2 , ∂r L (S ) L2 (S 2 )

11. Inverse problems II

305

where v = v(r, ·) ∈ Vkδ . Note that ∂v/∂r = [Nr (k) + r−1 ]v(r, ·), where Nr (k) is the Neumann operator (7.55), for the obstacle {|x| = r}. By (9.56), extended to treat balls of radius r ∈ [1, R], Nr (k)f 2L2 (S 2 ) ≤ C f 2H 1 (S 2 ) + Ck 2 f 2L2 (S 2 )   2   ≤ Ck 2 k −2 Lf, f L2 + f L2 .

(11.61)

Now, by (11.55), (11.62)

0 ≤ (kr)−2 L ≤ α−2 I on Range χ(δA),

given α/k ≤ δ and r ≥ 1. Consequently, if α > 1, we have constants Cj ∈ (0, ∞), independent of k, such that (11.63)

C0 v(r) 2L2 (S 2 ) ≤ Nkr (v(r))2 ≤ C1 v(r) 2L2 (S 2 ) ,

for all v ∈ Vkδ . We now show that, for v ∈ Vkδ , Nkr (v(r))2 = E(r) is a monotonically increasing function of r ∈ [1, R]; this will establish the estimate (11.59) and hence complete the proof of Proposition 11.4. To see this, write

(11.64)

  ∂v

dE = 2 Re 1 − (kr)−2 L ,v dr ∂r  2 2 −2 ∂ v ∂v , + 2 3 (Lv, v) + 2 Re k , k r ∂r2 ∂r

  and use (11.58) to replace k −2 ∂ 2 v/∂r2 by − 1 − (kr)−2 L v. We obtain  2 dE = (kr)−2 Lv, v ≥ 0, dr r

(11.65) and the proof is complete.

We can place the analysis in (11.60)–(11.65) in the following more general context. Suppose (11.66)

∂2v + A(r)v = 0, ∂r2

where each A(r) is positive-definite, all having the same domain. If we set (11.67) then

  Qr (v) = A(r)v, v + ∂r v 2 ,

306 9. Scattering by Obstacles

(11.68)

    d Qr (v) = 2 Re A(r)v, ∂r v + 2 Re ∂r2 v, ∂r v dr     + A (r)v, v = A (r)v, v .

If A (r) can be bounded by A(r), then we have an estimate (11.69)

d Qr (v) ≤ C Qr (v). dr

Of course, if A (r) is positive-semidefinite, we have monotonicity of Qr (v), as in (11.65). This result indicates that, using signals of wavelength λ = 1/k, one can expect to “regularize” inverse problems, to perceive details in an unknown obstacle on a length scale ∼ λ. Further analytical estimates with the goal of making this precise are given in [T4]. This idea is very much consistent with intuition and experience. For example, a well-known statement on the limitations of an optical microscope is that if it has perfect optics, one can use it to examine microscopic detail on a length scale approximately equal to, but not smaller than, the wavelength of visible light. We emphasize that this limitation applies to discerning detail on an obstacle whose diameter is much larger than 1/k. If one has a single obstacle whose diameter is ∼1/k, then one is said to be dealing with an inverse problem in the “resonance region,” and, given some a priori hypotheses on the obstacle, one can hope to make out some details of its structure to a higher precision than one wavelength. This sort of problem is discussed in a number of papers on inverse problems, such as [ACK], [AKR], [JM], and [MTW]. The interest in inverse problems extends beyond the setting of scattering theory. Important inverse problems arise in the setting of bounded domains. For more on such problems, we mention [AK2LT] and [LTU], and references given in these papers.

Exercises 1. Using the power series for Jν (z) given as (6.19) of Chap. 3, show that, for fixed s ∈ (0, ∞), as ν → +∞,

es ν . Jν (s) ∼ (2πν)−1/2 2ν Modify this argument to establish (11.11). 2. Generalize Proposition 11.3 to the case where different Hilbert spaces (or even different Banach spaces) Hj are involved, that is, S : H1 → H2 and Tj : Vj → Hj , j = 0, 1, where Vj ⊂ H2 , a ∈ V0 ∩ V1 .

12. Scattering by rough obstacles In the previous sections we have restricted attention to scattering of waves by compact obstacles in R3 with smooth boundary. Here we extend some of this

12. Scattering by rough obstacles

307

material to the case of compact K ⊂ R3 , which is not assumed to have smooth boundary. We do assume that Ω = R3 \ K is connected. The first order of business is to construct the solution (in a suitable sense) to the problem (12.1)

(Δ + k 2 )v = 0 on Ω = R3 \ K,

v = f on ∂K,

satisfying the radiation condition (12.2)

|rv(x)| ≤ C,

r

∂v ∂r

− ikv → 0, as r → ∞,

given k > 0. Our analysis will use a method from §5 of Chap. 5; we take compact Kj with smooth boundary such that (12.3)







K 1 ⊃⊃ K 2 ⊃⊃ · · · ⊃⊃ K j  K.

Set Ωj = R3 \ Kj , so Ωj  Ω. Let us assume f ∂K is the restriction to ∂K of some f ∈ C02 (R3 ). We will extend the method of proof of Theorem 1.3. For ε > 0, j ∈ Z+ , let wεj ∈ L2 (Ωj ) be the solution to (12.4)

  Δ + (k + iε)2 wεj = hε on Ωj ,

wεj ∂K = 0, j

where (12.5)

  hε = − Δ + (k + iε)2 f, on Ω.

Set wεj = 0 on Kj . Then set vεj = f + wεj , so (12.6)

  Δ + (k + iε)2 vεj = 0 on Ωj ,

vεj ∂Kj = f.

By methods of Chap. 5, §5, for fixed ε > 0, as j → ∞, wεj → wε in H01 (Ω), the domain of (−Δ)1/2 , when Δ is the self-adjoint operator on L2 (Ω) with the Dirichlet boundary condition on ∂Ω = ∂K, and (12.7)

−1  wε = Δ + (k + iε)2 hε ∈ H01 (Ω).

We have wε ∂K = 0 in a generalized sense. It follows that vεj → vε = wε + f in H 1 (Ω), and (12.8)

  Δ + (k + iε)2 vε = 0 on Ω,

vε ∂K = f,

308 9. Scattering by Obstacles

the boundary condition holding in a generalized sense. Furthermore, vε is the unique solution to (12.8) with the property that vε − f ∈ H01 (Ω). If f ∈ C02 (R3 ) is supported in BA = {|x| ≤ A}, then elliptic estimates imply wεj → wε in C ∞ (R3 \ BA ), hence vεj → vε in C ∞ (R3 \ BA ). It follows that, for any fixed A1 > A, if Σ = {|x| = A1 }, then (12.9)

vε (x) =

 

vε (y)

Σ

∂vε (y)  ∂gε − gε dS (y), ∂ν ∂ν

|x| > A1 ,

where gε = g(x, y, k + iε) is given by (1.6). Compare with the identity (1.24). We now state a result parallel to Theorem 1.3. Theorem 12.1. For vε constructed above, we have, as ε  0, a unique limit vε → v = B(k)f,

(12.10)

satisfying (12.1)–(12.2). Convergence occurs in the norm topology of the space L2 (Ω, x −1−δ dx ), for any δ > 0, as well as weakly in H 1 (Ω ∩ {|x| < R}), for any R < ∞, and the limit v satisfies the identity (12.11)

v(x) =

  v(y) Σ

∂g ∂v(y)  −g dS (y), ∂ν ∂ν

|x| > A1 .

More generally, we can replace the boundary condition on vε in (12.8) by vε = fε on ∂K, with fε → f in C02 (R3 ). As in §1, we begin the proof by establishing a uniqueness result. Lemma 12.2. Given k > 0, if v satisfies (12.1)–(12.2) with f = 0, then v = 0. Proof. Here, to say v = 0 on ∂K means χv ∈ H01 (Ω), for some χ ∈ C0∞ (R3 ), chosen so χ(x) = 1 for |x| ≤ A. The proof of Proposition 1.1 works here, with a minor change in the identity (1.8). Namely, write the equation (12.12) (Δ + k 2 )v = 0 on BR \ K, v ∂K = 0, v SR = v in the weak form



(12.13)



− dv, dϕ + k 2 vϕ dx =

BR \K

 v SR

∂ϕ dS , ∂ν

for all ϕ ∈ H 1 (BR \ K) such that ϕ ∂K = 0 and ϕ is smooth near SR . This applies to ϕ = v, yielding  

(12.14) − dv, dv + k 2 vv dx = vv r dS . BR \K

SR

12. Scattering by rough obstacles

309

Also, we can interchange the roles of v and v and subtract the resulting identity from (12.14), obtaining  (12.15) (vv r − vvr ) dS = 0, SR

as in (1.8). The rest of the proof proceeds exactly as in the proof of Proposition 1.1. We continue with the proof of Theorem 12.1. Pick R > A1 and set OR = Ω ∩ {|x| < R}. Parallel to Lemma 1.4, we have Lemma 12.3. Assume vε O is bounded in L2 (OR ) as ε  0. Then the concluR sions of Theorem 12.1 hold. Proof. Fix S ∈ (A1 , R). Elliptic estimates imply that if vε L2 (OR ) is bounded, then (12.16)

vε H 1 (OS ) ≤ C.

Passing to a subsequence, which we continue to denote by vε , we have (12.17)

vε → v weakly in H 1 (OS ).

Also, wε = vε − fε → w = v − f , and for χ ∈ C0∞ (|x| < S) such that χ = 1 on a neighborhood of K, we have χwε → χw in H01 (OS ). Thus v ∂K = f , in our current sense.   Since Δ + (k + iε)2 vε = 0 on OR , elliptic estimates imply that if vε OR is bounded in L2 (OR ), then vε is bounded in C ∞ (A < |x| < R). Thus we obtain (12.11) from (12.9), and (12.11) implies the radiation condition (12.2) and also the convergence vε → v in L2 (Ω, x −1−δ dx ). So far we have convergence for subsequences, but by Lemma 12.2 this limit is unique, so Lemma 12.3 is proved. The proof of Theorem 12.1 is completed by: Lemma 12.4. The hypotheses of Theorem 12.1 imply that the family {vε } is bounded in the space L2 (Ω, x −1−δ dx ), for any δ > 0. The proof is the same as that of Lemma 1.5. The fact that, for each j ∈ Z+ , vεj converges as ε → 0 to a limit vj solving the scattering problem (12.18)

(Δ + k 2 )vj = 0 on Ωj ,

vj = f on ∂Kj ,

plus the radiation condition, is a consequence of Theorem 1.3. The following approximation result is useful. Extend vj to be equal to f on Ω \ Ωj .

310 9. Scattering by Obstacles

Proposition 12.5. For any R ∈ (A, ∞), δ > 0, we have   vj → v in C ∞ (Ω1 ) ∩ L2 Ω, x −1−δ dx .

(12.19)

More generally, we can replace the boundary condition by vj = fj on ∂Kj , where fj → f in C02 (R3 ). Furthermore, vj → v in H 1 (OR ), in norm.

(12.20)

Proof. To establish (12.19), an argument parallel to that used for Lemmas 1.4 and 12.3 shows that it suffices to demonstrate that {vj } is bounded in the space L2 (OR ), and then an argument parallel to that used for Lemmas 1.5 and 12.4 shows that indeed {vj } is bounded in L2 (Ω, x −1−δ dxdx ). Arguments such as those used to prove Theorems 1.3 and 12.1 also show that vj → v weakly in H 1 (OR ). To get the norm convergence stated in (12.20), note that, parallel to (12.14),  (12.21)



2

2

− dvj , dv j + k |vj |

 dx =

OR

vj (∂r v j ) dS . SR

of SR = Since vj → v in L2(OR ) and vj → v in C ∞ on a neighborhood  {|x| = R}, we have OR k 2 |vj |2 dx → OR k 2 |v|2 dx and SR vj (∂r v j ) dS →  v(∂r v) dS . Consequently, SR 

|dvj |2 dx −→

OR



|dv|2 dx ,

OR

so (12.22)

vj H 1 (OR ) −→ v H 1 (OR ) .

This, together with weak convergence, yields (12.20). It is useful to note that if we extend vj to be fj on Kj and extend v to be f on K, then (12.20) can be sharpened to (12.23)

vj −→ v in H 1 (BR ), in norm.

Now, we have a well-defined operator (12.24)

BK (k) : C 2 (K) −→ C ∞ (R3 \ K),

for any k > 0, any compact K ⊂ R3 , extending (1.19). By (12.11), we have asymptotic results on BK (k)f (x), as |x| → ∞, of the same nature as derived

12. Scattering by rough obstacles

311

in §1. In particular, the scattering amplitude aK (−ω, θ, k) is defined as before, in terms of the asymptotic behavior of BK (k)f (rθ), when f (x) = −e−ikx ·ω on ∂K. We next want to discuss the uniqueness of the scatterer, when K is not required to be smooth. A special case of Proposition 10.2 is that if K ⊂ BR is assumed to be smooth and one has fixed k ∈ (0, ∞) and sufficiently many ω ∈ S 2 , then the knowledge that aK (−ω , θ, k) = 0, ∀ θ ∈ S 2 , implies K is empty. The appropriate statement of this result when K is not required to have any smoothness is the following: Proposition 12.6. Given compact K ⊂ BR , fixed k ∈ (0, ∞), and θ ∈ S 2 , then if aK (−ω , θ, k) = 0,

(12.25)

∀ θ ∈ S2,

for a single ω ∈ S 2 , it follows that cap K = 0.

(12.26)

Here “cap K” is the Newtonian capacity of K, which is discussed in detail in §6 of Chap. 11. One characterization is (12.27) cap K = inf



 |∇f (x)|2 dx : f ∈ C0∞ (R3 ), f = 1 on nbd of K .

One can derive the estimate that if f ∈ Lip(R3 ) has compact support and λ > 0, then (12.28)

  cap {x ∈ R3 : |f (x)| ≥ λ} ≤ λ−2 ∇f 2L2 .

See (6.64)–(6.65) of Chap. 11. To prove the proposition, first note that, as in the proof of Proposition 10.2, the hypothesis (12.25) implies (12.29)

u(x, kω ) = e−ik ω ·x ,

 the unbounded, connected component of R3 \ K; we may as well for x ∈ R3 \ K,  = K. Fix ϕ ∈ C ∞ (R3 ) so that ϕ = 1 on a neighborhood of K. suppose that K 0 Then (12.29) implies (12.30)

ϕ(x)e−ik ω ·x ∈ H01 (R3 \ K).

Hence (12.31)

ϕ ∈ H01 (R3 \ K).

312 9. Scattering by Obstacles

We claim this implies cap K = 0. Indeed, take ϕν ∈ C0∞ (R3 \K) so that ϕν → ϕ in H1 -norm. Then fν = ϕ − ϕν ∈ C0∞ (R3 ), fν = 1 on a neighborhood of K, and |∇fν (x)|2 dx → 0. By (12.27), this implies cap K = 0, so the proposition is proved. We now want to compare two nonempty obstacles, K1 and K2 , with identical scattering data a(−ω, θ, k), perhaps for (ω, k) running over some set. Our next step is to push the arguments used in the proof of Proposition 10.2 to show that under certain conditions the symmetric difference K1  K2 has empty interior. After doing that, we will take up the question of whether cap(K1  K2 ) = 0. So, as in the proof of Proposition 10.2, suppose K1 and K2 are two compact obstacles in R3 giving rise to scattered waves vj that agree on an open set O in the unbounded, connected component of R3 \ (K1 ∪ K2 ). In other words, uj = e−ik ω·x + vj (x, kω) has the properties (12.32)

(Δ + k 2 )uj = 0 on R3 \ Kj ,

ϕuj ∈ H01 (R3 \ Kj ),

and vj = uj −e−ik ω·x satisfies the radiation condition. Here, we fix ϕ ∈ C0∞ (R3 ) such that ϕ = 1 on a ball containing K1 ∪ K2 in its interior. We suppose the sets Kj have no cavities, so each Ωj = R3 \ Kj has just one connected component. As before, R3 \ (K1 ∪ K2 ) might not be connected, so let U be its unbounded  = R3 \ U. If K1 = K2 , then either K1 or K2 is a component, and consider K  Let us suppose K1 is. proper subset of K. Note that the functions u1 and u2 agree on U, since they are real analytic and agree on O. Proposition 12.7. For any (ω, k) for which K1 and K2 have identical scattering  data (for all θ), so do K1 and K. Proof. By the uniqueness result, Lemma 12.2, it suffices to show that (12.33)

u = u 1 = u2

on U

has the property that ϕu ∈ H01 (U), for any ϕ ∈ C0∞ (R3 ) equal to 1 on a neigh In turn, this is a consequence of the following general result. borhood of K. Lemma 12.8. Let Ωj be open in Rn , fj ∈ H01 (Ωj ). Let O be a connected component of Ω1 ∩ Ω2 . Then (12.34)

f1 = f2 = f

on O =⇒ f ∈ H01 (O).

Proof. It suffices to assume that the functions fj are real-valued. The hypotheses imply fj+ ∈ H01 (Ωj ) and f1+ = f2+ = f + on O. Thus it suffices to assume in addition that fj ≥ 0 on Ωj . Now we can find gν ∈ C0∞ (Ω1 ) and hν ∈ C0∞ (Ω2 ) such that gν → f1 in H 1 (Ω1 ) and hν → f2 in H 1 (Ω2 ). Hence gν+ → f1 and 1 h+ ν → f2 in H -norm. Now

12. Scattering by rough obstacles

313

ϕν = min (gν+ , h+ ν) O has the properties ϕν ∈ H01 (O),

ϕν → f in H 1 -norm,

so (12.34) is proved.  (which we relabel as K2 ), and we investigate Thus we replace K2 by K whether K1 ⊂ K2 can have identical scattering data, for (k, ω) belonging to some set. We return to the considerations of the functions uj , as in (12.32). (Now, U = Ω2 .) Suppose K2 \ K1 has nonempty interior R. Note that ∂R \ ∂Ω1 ⊂ Ω1 ∩ ∂Ω2 . We claim that w = u1 R has the property w ∈ H01 (R).

(12.35)

This is a consequence of the following general result. Lemma 12.9. Let R ⊂ Ω be open. Then (12.36)

f ∈ H01 (Ω) ∩ C(Ω), f = 0 on ∂R \ ∂Ω =⇒ f R ∈ H01 (R).

Proof. It suffices to assume f is real-valued. Then the hypotheses apply to f + and f − , so it suffices to assume f ≥ 0 on Ω. Take fν ∈ C0∞ (Ω), fν → f in H 1 -norm. Then fν+ → f in H 1 -norm. Also, if we define ηε (s) for s ≥ 0 to be (12.37)

ηε (s) =

0 s−ε

if 0 ≤ s ≤ ε, if s ≥ ε,

and extend ηε to be an odd function, we have ηε (f ) → f in H 1 -norm. Now set (12.38)

  gν = min fν+ , η1/ν (f ) R .

We see that gν ∈ H01 (R) and gν → f in H 1 -norm, so we have (12.36). Now return to w = u1 R . We claim this function satisfies the hypotheses of (12.36), with Ω = Ω1 . To see that w vanishes on ∂R \ ∂Ω1 , we use the fact that u1 = u2 on Ω2 and argue that u2 vanishes pointwise on a dense subset of ∂R \ ∂Ω1 . In fact, a dense subset satisfies an exterior sphere condition (with respect to Ω2 ). Hence, a barrier construction (applied to the harmonic function eky u2 ) gives this fact. Thus we have (12.35). Also, (12.39)

(Δ + k 2 )w = 0

on R.

Of course, w is not identically zero on R, so k 2 must be an eigenvalue of −Δ on R. Hence we have the following parallel to Proposition 10.2. Suppose we have

314 9. Scattering by Obstacles

a bound on (12.40)

dim ker(Δ + k 2 ) H 1 (R) = d(k). 0

Proposition 12.10. Let K1 and K2 be arbitrary compact obstacles, with no cavities. Let k ∈ (0, ∞) be fixed. Suppose Σ = {ω } is a subset of S 2 whose cardinality is known to be greater than d(k)/2. (If ω and −ω both belong to Σ, do not count them separately.) Then (12.41)

aK1 (−ω , θ, k) = aK2 (−ω , θ, k),

∀ ω ∈ Σ, θ ∈ S 2

implies that K1  K2 has empty interior. We next show that under stronger hypotheses we can draw a stronger conclusion. Proposition 12.11. If K1 ⊂ K2 are compact sets without cavities in R3 and (12.42)

aK1 (−ω, θ, k) = aK2 (−ω, θ, k),

∀ ω, θ ∈ S 2 , k ∈ (0, ∞),

then every compact subset of K2 \ K1 is negligible. Proof. What we need to show is that if L is a compact subset of K2 \ K1 , and if β ∈ H −1 (R3 ) is supported on L, then β = 0. By Proposition 12.10, the current hypotheses imply that K2 \ K1 has empty interior. Hence K2 \ K1 ⊂ ∂Ω2 , so K2 \ K1 = Ω1 ∩ ∂Ω2 . Also, as in the considerations above, we have u1 (x, kω) = u2 (x, kω) for all x ∈ Ω2 ; this time, for all kω ∈ R3 \ 0. We claim that this implies, for all compact L ⊂ Ω1 ∩ ∂Ω2 , (12.43)

β ∈ H −1 (R3 ),

supp β ⊂ L =⇒ u1 (·, kω), β = 0.

To see this, we argue as follows (suppressing the parameters k, ω): Pick ϕ ∈ C0∞ (Ω1 ), equal to 1 on a neighborhood of L. Then ϕu2 ∈ H01 (Ω2 ), so we can take a sequence fν ∈ C0∞ (Ω2 ) such that fν → ϕu2 in H 1 (Ω2 )-norm. We can also regard fν as an element of C0∞ (Ω1 ), and of course (fν ) is Cauchy in H01 (Ω1 ). We claim that (12.44)

fν → ϕu1 in H 1 (Ω1 ).

Indeed, we have fν → w for some w ∈ H01 (Ω1 ), and hence w = ϕu2 on Ω2 . We want to show that w = ϕu1 on Ω1 . Since u1 = u2 on Ω2 , we have w = ϕu1 on Ω2 , so (12.45)

supp (w − ϕu1 ) ⊂ Ω1 ∩ K2 ,

A. Lidskii’s trace theorem

315

a set that, in the current setting, is equal to Ω1 ∩ ∂Ω2 . Of course, if Ω1 ∩ ∂Ω2 has three-dimensional Lebesgue measure 0, we can deduce w = ϕu1 . If it has positive measure, we argue as follows. First, the characterization w = lim fν clearly implies w = 0 on Ω1 ∩ K2 . Furthermore, material on regular points discussed in Chap. 11, §6, applied to the harmonic functions eky uj (x), implies that limx→x0 u2 (x, kω) = 0 for all x0 ∈ ∂Ω2 except for a set of interior capacity zero; in the current situation this implies u1 = 0 a.e. on Ω1 ∩ ∂Ω2 . Hence we again have w = ϕu1 , so (12.44) holds. On the other hand, (12.43) follows from (12.44). Having (12.43), for all ξ = kω ∈ R3 \ 0, we deduce that, given F ∈ C0∞ 3 (R \ 0), if we set  (12.46)

g(x) =

u1 (x, ξ)F (ξ) dξ,

then (12.47)

β ∈ H −1 (R3 ), supp β ⊂ L =⇒ g, β = 0.

However, the set of functions of the form (12.46) is dense in H01 (Ω1 ), by the isomorphism (4.10), which continues to hold in this setting. Thus (12.48)

β ∈ H −1 (R3 ), supp β ⊂ L =⇒ β = 0.

As discussed in Chap. 5, this means L is negligible. A consequence of material in §6 of Chap. 11 is that if a compact set is negligible, then its capacity is zero. Thus, by Proposition 12.11 (in conjunction with Proposition 12.7), when (12.43) holds, K1  K2 has “inner capacity” zero; see §6 of Chap. 11 for further discussion of inner capacity.

Exercises 1. Extend results of §§1–6 to obstacles considered here, with particular attention to results needed in the proof of Proposition 12.11. 2. Show that, for any open Ω ⊂ Rn , the map u → |u| is continuous on H 1 (Ω). Use this to justify the limiting arguments made in the proofs of Lemmas 12.8 and 12.9.

A. Lidskii’s trace theorem The purpose of this appendix is to prove the following result of V. Lidskii, which is used for (8.2): Theorem A.1. If A is a trace class operator on a Hilbert space H, then

316 9. Scattering by Obstacles

(A.1)

Tr A =



(dim Vj )λj ,

where {λj : j ≥ 1} = Spec A \ {0} and Vj is the generalized λj -eigenspace of A. We will make use of elementary results about trace class operators, established in §6 of Appendix A, Functional Analysis. In particular, if {uj } is any orthonormal basis of H, then  (A.2) Tr A = (Auj , uj ), the result being independent of the choice of orthonormal basis, provided A is trace class.  To begin the proof, let E = j≤ Vj , and let P = Q1 + · · · + Q denote the orthogonal projection of H onto E . Thus AP = P AP , restricted to E , has spectrum {λj :1 ≤ j ≤ }. We will choose an orthonormal set {uj : j ≥ 1} according to the following prescription: {uj : 1 + dim E−1 ≤ j ≤ dim E } will be an orthonormal basis of R(Q ), with the property that Q AQ (restricted to R(Q )) is upper triangular. That this can be done is proved in Theorem 4.7 of Chap. 1. Note that {uj : 1 ≤ j ≤ dim E } is then an orthonormal basis of E , with respect to which AP  = P AP is upper triangular. It follows that the diagonal entries of P AP E with respect to this basis are precisely  λj , 1 ≤ j ≤ , counted with multiplicity dim Vj . Inductively, we conclude that the diagonal entries of each block Q AQ consist of dim V copies of λ . Let H0 denote the closed linear span of {uj : j ≥ 1}, and H1 the orthogonal complement of H0 in H, and let Rν be the orthogonal projection of H on Hν . We can write A in block form   A0 B (A.3) A= , 0 A1 where Aν = Rν ARν , restricted to Hν . Clearly, A0 and A1 are trace class and, by the construction above plus (A.2), we have (A.4)

Tr A0 =



(dim Vj )λj .

Thus (A.1) will follow if we can show that Tr A1 = 0. If H1 = 0, there is no problem. Lemma A.2. If H1 = 0, then Spec A1 = {0}. Proof. Suppose Spec A1 contains an element μ = 0. Since A1 is compact on H1 , there must exist a unit vector v ∈ H1 such that A1 v = μv. Let H = H0 + (v).

A. Lidskii’s trace theorem

317

Note that Av = μv + w,

w ∈ H0 .

Hence H is invariant under A; let A denote A restricted to H. Of course, H0 is invariant under A, and A restricted to H0 is A0 . Note that both Tμ = A0 − μI (on H0 ) and Tμ = A − μI (on H) are Fredholm operators of index zero, and that Codim Tμ (H) = 1 + Codim Tμ (H0 ). Hence Dim Ker(A − μI) = 1 + Dim Ker(A0 − μI). It follows that the μ-eigenspace of A is bigger than the μ-eigenspace of A0 . But this is impossible, since by construction, for any μ = 0, the μ-eigenspace of A0 is the entire μ-eigenspace of A. Thus the lemma is proved. A linear operator K is said to be quasi-nilpotent provided Spec K = {0}. If this holds, then (I + zK)−1 is an entire holomorphic function of z. The convergence of its power series implies (A.5)

sup |z|j K j < ∞,

∀ z ∈ C,

j

a condition that is in fact equivalent to Spec K = {0}. To prove Theorem A.1, it suffices to demonstrate the following. Lemma A.3. If K is a trace-class operator on a Hilbert space and K is quasinilpotent, then Tr K = 0. To prove Lemma A.3, we use results on the determinant established in §6 of Appendix A, Functional Analysis. Thus, we consider the entire holomorphic function (A.6)

ϕ(z) = det(I + zK),

which is well defined for trace class K. By (6.45) of Appendix A, (A.7)

|ϕ(z)| ≤ Cε eε|z| ,

∀ ε > 0.

Also, by Proposition 6.16 of Appendix A, ϕ(z) = 0 whenever I+zK is invertible. Now, if K is quasi-nilpotent, then, as remarked above, I + zK is invertible for all z ∈ C. Hence ϕ(z) is nowhere vanishing, so we can write (A.8)

ϕ(z) = ef (z) ,

with f (z) holomorphic on C. Now (A.7) implies Re f (z) ≤ Cε + ε|z| for all ε > 0, and a Harnack inequality argument applied to this gives

318 9. Scattering by Obstacles

(A.9)

|Re f (z)| ≤ Cε + ε|z|,

∀ ε > 0,

See Chap. 3, §2, Exercises 13–16. The estimate (A.9) in turn (e.g., by Proposition 4.6 of Chap. 3) implies that Re f is constant, so f is constant, and hence ϕ is constant. But, by (6.41) of Appendix A, we have (A.10)

Tr K = ϕ (0),

so the lemma is proved. Hence the proof of Theorem A.1 is complete. A proof of Lidskii’s theorem—avoiding the first part of the argument given above, and simply using determinants, but making heavier use of complex function theory—is given in [Si2].

References [Ag] S. Agmon, Lectures on Elliptic Boundary Problems, Van Nostrand, New York, 1964. [AK2LT] M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas, and M. Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem, Invent. Math. 158 (2004), 261–321. [ACK] T. Angell, D. Colton, and A. Kirsch, The three dimensional inverse scattering problem for acoustic waves, J. Diff. Eq. 46(1982), 46–58. [AKR] T. Angell, R. Kleinman, and G. Roach, An inverse transmission problem for the Helmholtz equation, Inverse Probl. 3(1987), 149–180. [Ber] A. M. Berthier, Spectral Theory and Wave Operators for the Schr¨odinger Equation, Pitman, Boston, 1982. [Bor] D. Borthwick, Spectral Theory of Infinite-area Hyperbolic Spaces, Birkhauser, Boston, 2007. [Bs] V. Buslaev, Scattered plane waves, spectral asymptotics, and trace formulas in exterior problems, Dokl. Akad. Nauk. SSSR 197(1971), 1067–1070. [Co] D. Colton, The inverse scattering problem for time-harmonic accoustic waves, SIAM Rev. 26(1984), 323–350. [Co2] D. Colton, Partial Differential Equations, an Introduction, Random House, New York, 1988. [CK] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983. [CK2] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, New York, 1992. [CH] R. Courant and D. Hilbert, Methods of Mathematical Physics II, J. Wiley, New York, 1966. [DZ] S. Dyatlov and M. Zworski, Mathematical Theory of Scattering Resonances, Amer. Math. Soc., Providence RI, 2019. [Ei] D. Eidus, On the principle of limiting absorption, Mat. Sb. 57(1962), 13–44. [Erd] A. Erd´elyi, Asymptotic Expansions, Dover, New York, 1956. [HeR] J. Helton and J. Ralston, The first variation of the scattering matrix, J. Diff. Equations 21(1976), 378–394.

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[HiP] E. Hille and R. Phillips, Functional Analysis and Semigroups, Colloq. Publ. Vol. 31, AMS, Providence, RI, 1957. [Ho] L. H¨ormander, The Analysis of Linear Partial Differential Operators, Vols. 3 and 4, Springer, New York, 1985. [Isa] V. Isakov, Uniqueness and stability in multi-dimensional inverse problems, Inverse Probl. 9(1993), 579–621. [JeK] A. Jensen and T. Kato, Asymptotic behavior of the scattering phase for exterior domains, Comm. PDE 3(1978), 1165–1195. [JM] D. Jones and X. Mao, The inverse problem in hard acoustic scattering, Inverse Probl. 5(1989), 731–748. [Kt] T. Kato, Perturbation Theory for Linear Operators, Springer, New York, 1966. [Kir] A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Probl. 9(1993), 81–96. [KZ] A. Kirsch, R. Kress, P. Monk, and A. Zinn, Two methods for solving the inverse scattering problem, Inverse Probl. 4(1988), 749–770. [KV] G. Kristensson and C. Vogel, Inverse problems for acoustic waves using the penalised liklihood method, Inverse Probl. 2(1984), 461–479. [LTU] M. Lassas, M. Taylor, and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. in Analysis and Geometry 11 (2003), 207–221. [LP1] P. Lax and R. Phillips, Scattering Theory, Academic, New York, 1967. [LP2] P. Lax and R. Philips, Scattering theory, Rocky Mountain J. Math. 1(1971), 173–223. [LP3] P. Lax and R. Phillips, Scattering theory for the acoustic equation in an even number of space dimensions, Indiana University Math. J. 22(1972), 101–134. [LP4] P. Lax and R. Phillips, The time delay operator and a related trace formula, in Topics in Func. Anal., I. Gohberg and M. Kac, eds., Academic, New York, 1978, pp. 197–215. [LP5] P. Lax and R. Phillips, Scattering Theory for Automorphic Functions, Princeton University Press, Princeton, NJ, 1976. [MjR] A. Majda and J. Ralston, An analogue of Weyl’s theorem for unbounded domains, Duke Math. J. 45(1978), 513–536. [MjT] A. Majda and M. Taylor, Inverse scattering problems for transparent obstacles, electromagnetic waves, and hyperbolic systems, Comm. PDE 2(1977), 395–438. [Me1] R. Melrose, Scattering theory and the trace of the wave group, J. Func. Anal. 45(1982), 29–40. [Me2] R. Melrose, Polynomial bound on the number of scattering poles, J. Func. Anal. 53(1983), 287–303. [Me3] R. Melrose, Weyl asymptotics for the phase in obstacle scattering, Comm. PDE 11(1988), 1431–1439. [Me4] R. Melrose, Geometric Scattering Theory, Cambridge University Press, Cambridge, 1995. [MT1] R. Melrose and M. Taylor, Near peak scattering and the corrected Kirchhoff approximation for convex bodies, Adv. Math. 55(1985), 242–315. [MT2] R. Melrose and M. Taylor, Boundary Problems for Wave Equations with Grazing and Gliding Rays. Monograph, Preprint, 2010. [Mr1] K. Miller, Least squares methods for ill-posed problems with a prescribed bound, SIAM J. Math. Anal. 1(1970), 52–74. [Mr2] K. Miller, Efficient numerical methods for backward solutions of parabolic problems with variable coefficients, in Improperly Posed Boundary Value Problems, A.Carasso and A.Stone, eds., Pitman, London, 1975.

320 9. Scattering by Obstacles [MrV] K. Miller and G. Viano, On the necessity of nearly-best possible methods for analytic continuation of scattering data, J. Math Phys. 14(1973), 1037–1048. [Mw] C. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math. 15(1962), 349–361. [Mw2] C. Morawetz, Exponential decay of solutions to the wave equation, Comm. Pure Appl. Math. 19(1966), 439–444. [MRS] C. Morawetz, J. Ralston, and W. Strauss, Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math. 30(1977), 447–508. [MTW] R. Murch, D. Tan, and D. Wall, Newton-Kantorovich method applied to twodimensional inverse scattering for an exterior Helmholtz problem, Inverse Probl. 4(1988), 1117–1128. [New] R. G. Newton, Scattering Theory of Waves and Particles, Springer, New York, 1966. [Nus] H. Nussensweig, High frequency scattering by an impenetrable sphere, Ann. Phys. 34(1965), 23–95. [Olv] F. Olver, Asymptotics and Special Functions, Academic, New York, 1974. [PP] V. Petkov and G. Popov, Asymptotic behavior of the scattering phase for nontrapping obstacles, Ann. Inst. Fourier (Grenoble) 32(1982), 111–149. [Rl] J. Ralston, Propagation of singularities and the scattering matrix, in Singularities in Boundary Value Problems, H.Garnir, ed., D.Reidel, Dordrecht, 1981, pp. 169–184. [Rm] A. Ramm, Scattering by Obstacles, D.Reidel, Dordrecht, 1986. [Rau] J. Rauch, Partial Differential Equations, Springer, New York, 1991. [RT] J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Func. Anal. 18(1975), 27–59. [RS] M. Reed and B. Simon, Methods of Mathematical Physics, Academic, New York, Vols. 1,2, 1975; Vols. 3,4, 1978. [Rog] A. Roger, Newton-Kantorovich algorithm applied to an electromagnetic inverse problem, IEEE Trans. Antennas Propagat 29(1981), 232–238. [Si] B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, Princeton, NJ, 1971. [Si2] B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Notes no. 35, Cambridge University Press, Cambridge, 1979. [Stk] I. Stakgold, Boundary Value Problems of Mathematical Physics, Macmillan, New York, 1968. [T1] M. Taylor, Propagation, reflection, and diffraction of singularities of solutions to wave equations, Bull. AMS 84(1978), 589–611. [T2] M. Taylor, Fourier integral operators and harmonic analysis on compact manifolds, Proc. Symp. Pure Math. 35(pt. 2)(1979), 113–136. [T3] M. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981. [T4] M. Taylor, Estimates for approximate solutions to acoustic inverse scattering problems, Inverse problems in wave propagation, 463–499, IMA Vol. Math. Appl. 90, Springer NY, 1997. [VV] V. K. Varadan and V. V. Varadan (eds.), Acoustic, Electromagnetic, and Elastic Wave Scattering − Focus on the T matrix Approach, Pergammon, New York, 1980. [Vas] A. Vasy, Propagation of singularities in three-body scattering, Asterisque #262, 2000.

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[Wat] G. Watson, Theory of Bessel Functions, Cambridge University Press, Cambridge, 1945. [Wil] C. Wilcox, Scattering Theory for the d’Alembert Equation in Exterior Domains, LNM no. 442, Springer, New York, 1975. [Yo] K. Yosida, Functional Analysis, Springer, New York, 1965.

10 Dirac Operators and Index Theory

Introduction The physicist P. A. M. Dirac constructed first-order differential operators whose squares were Laplace operators, or more generally wave operators, for the purpose of extending the Schrodinger–Heisenberg quantum mechanics to the relativistic setting. Related operators have been perceived to have central importance in the interface between PDE and differential geometry, and we discuss some of this here. We define various classes of “Dirac operators,” some arising on arbitrary Riemannian manifolds, some requiring some special geometrical structure, such as a spin structure, discussed in §3, or a spinc structure, discussed in §8. Dirac operators on compact Riemannian manifolds are elliptic and have an index. Evaluating this index, in terms of an integrated “curvature,” is the essence of the famous Atiyah–Singer index theorem. We present a proof of this index formula here, using a “heat-equation” method of proof. Such a proof was first suggested in [McS], but it seemed difficult to carry out, as it required understanding of a coefficient in the asymptotic expansion of the traces of e−tLj , for a pair of positive, second-order, elliptic operators Lj , well below the principal term. Ingenious arguments, beginning with V. Patodi [Pt1, Pt2], led to a proof in Atiyah–Bott–Patodi [ABP]. Later, physicists, motivated by ideas from “supersymmetry,” proposed more direct heat-equation proofs. Such proposals were first made by E. Witten [Wit]; particularly elegant mathematical treatments were given by E. Getzler in [Gt1] and [Gt2]. We present a heat-equation proof in §6, using Getzler’s method of exploiting an analogue of Mehler’s formula for the exponential of the harmonic oscillator Hamiltonian. Our analytical details differ from Getzler’s; instead of introducing a noncommutative symbol calculus as in [Gt1], or the dilation argument of [Gt2], we fit the analysis more into a “classical” examination of heat-equation asymptotics, such as dealt with in Chap. 7. One major achievement of Getzler’s approach is to make the appearance of the (rather subtle) ˆ A-genus of M in the index formula arise quite naturally. We present two specific examples of the index formula here. In §7 we derive the Chern–Gauss–Bonnet formula, giving the Euler characteristic χ(M ) c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  M. E. Taylor, Partial Differential Equations II, Applied Mathematical Sciences 116, https://doi.org/10.1007/978-3-031-33700-0 10

323

324 10. Dirac Operators and Index Theory

of a compact, orientable Riemannian manifold in terms of an integral of the “Pfaffian” applied to its curvature tensor. In §9 (following a discussion in §8 of spinc structures) we discuss the Riemann–Roch formula, a tool for understanding holomorphic (and meromorphic) sections of line bundles over Riemann surfaces (of real dimension 2), which is important in the study of the structure and function theory of Riemann surfaces. These are the simplest applications of the Atiyah–Singer formula; both were established well before the general formula. ˆ From a technical point of view, both have in common that the A-genus of M is effectively discarded. Other examples of the index formula include higher-dimensional Riemann–Roch formulas and signature formulas. Further material on these can be found in [Pal] and [Gil]. There is also an operator associated with “self-dual” connections on bundles over 4-manifolds, whose index plays an important role in the study of the Yang–Mills equations; see [AHS] and [FU]. For a variant, arising from the Seiberg–Witten equations, see [D] and [Mor]. The heat-equation proof of the Chern–Gauss–Bonnet theorem was Patodi’s [Pt1] first step in this circle of results. An exposition of the heat-equation proof of this result due to B. Simon is given in the last chapter of [CS]. Another proof of the Chern–Gauss–Bonnet theorem, celebrating closely physicists’ ideas about supersymmetry, is given in [Rog]. Due to the low dimension, one can give a direct proof of the Riemann–Roch theorem, using techniques of [McS]; such a proof is given in [Kot]. Such a direct approach, with a good bit more effort, could be expected to be effective in other low dimensions (e.g., complex dimension 2); in a sense, the sort of analysis required to accomplish this is what was done in [Ko1]. In §10, we give a direct proof of an index formula for first-order, elliptic differential operators of Dirac type on a 2-manifold M , in terms of a direct calculation of the second term in the expansion of the heat kernel, carried out in §14 of Chap. 7, using the Weyl calculus. We show how this formula yields the Gauss–Bonnet formula and the Riemann–Roch formula. There are also other heat-equation proofs of the index theorem, particularly [Bi1] and [BV]. In [Bi2] the heat equation method is applied to families of operators; see also [Don] and [BiC]. There are several books devoted to expositions of heat-equation proofs of the index theorem, including [BGV, Gil, Mel], and [Roe]. A systematic “blow up” of the original proof of the Atiyah–Singer index theorem has led to the interesting subject of operator K-theory. An introductory exposition is given in Blackadar [Bl]. Further developments are described in [BDT, Con], and [BHS]. In §11 we change course, and produce an index formula for a class of elliptic k × k systems on Euclidean space Rn . We do this for the class of pseudodifferential operators of harmonic oscillator type, introduced in §15 of Chap. 7. The proof here makes no use of heat-equation techniques. It uses some results from topology, particularly results on the homotopy groups of the unitary groups U (k), including the Bott periodicity theorem, results for which we refer to [Mil] for proof. Section 11 can be read independently of the other sections of this chapter.

1. Operators of Dirac type

325

1. Operators of Dirac type Let M be a Riemannian manifold, Ej → M vector bundles with Hermitian metrics. A first-order, elliptic differential operator D : C ∞ (M, E0 ) −→ C ∞ (M, E1 )

(1.1)

is said to be of Dirac type provided D∗ D has scalar principal symbol. This implies σD∗ D (x, ξ) = g(x, ξ)I : E0x −→ E0x ,

(1.2)

where g(x, ξ) is a positive quadratic form on Tx∗ M . Thus g itself arises from a Riemannian metric on M . Now the calculation of (1.2) is independent of the choice of Riemannian metric on M . We will suppose M is endowed with the Riemannian metric inducing the form g(x, ξ) on T ∗ M . If E0 = E1 and D = D∗ , we say D is a symmetric Dirac-type operator. Given ˜ on a general operator D of Dirac type, if we set E = E0 ⊕ E1 and define D C ∞ (M, E) as   ∗ ˜= 0 D , (1.3) D D 0 then D is a symmetric Dirac-type operator. Let ϑ(x, ξ) denote the principal symbol of a symmetric Dirac-type operator. With x ∈ M fixed, set ϑ(ξ) = ϑ(x, ξ). Thus ϑ is a linear map from Tx∗ M = {ξ} into End(Ex ), satisfying ϑ(ξ) = ϑ(ξ)∗

(1.4) and

ϑ(ξ)2 = ξ, ξI.

(1.5)

Here,  ,  is the inner product on Tx∗ M ; let us denote this vector space by V . We will show how ϑ extends from V to an algebra homomorphism, defined on a Clifford algebra Cl(V, g), which we now proceed to define. Let V be a finite-dimensional, real vector space, g a quadratic form on V . We allow g to be definite or indefinite if nondegenerate; we even allow g to be degenerate. The Clifford algebra Cl(V, g) is the quotient algebra of the tensor algebra (1.6)



by the ideal I ⊂

V = R ⊕ V ⊕ (V ⊗ V ) ⊕ (V ⊗ V ⊗ V ) ⊕ · · ·



V generated by

326 10. Dirac Operators and Index Theory

(1.7)

{v ⊗ w + w ⊗ v − 2v, w · 1 : v, w ∈ V },

where ,  is the symmetric bilinear form on V arising from g. Thus, in Cl(V, g), V occurs naturally as a linear subspace, and there is the anticommutation relation (1.8)

vw + wv = 2v, w · 1 in Cl(V, g),

v, w ∈ V.

We will look more closely at the structure of Clifford algebras in the next section. Now if ϑ : V → End(E) is a linear map of the V into the space of endomorphisms of a vector space E, satisfying (1.5), i.e., ϑ(v)2 = v, vI,

(1.9)

v ∈ V,

it follows from expanding ϑ(v + w)2 = [ϑ(v) + ϑ(w)]2 that (1.10)

ϑ(v)ϑ(w) + ϑ(w)ϑ(v) = 2v, wI,

v, w ∈ V.

Then, from the construction of Cl(V, g), it follows that ϑ extends uniquely to an algebra homomorphism (1.11)

ϑ : Cl(V, g) −→ End(E),

ϑ(1) = I.

This gives E the structure of a module over Cl(V, g), or a Clifford module. If E has a Hermitian metric and (1.4) also holds, that is, (1.12)

ϑ(v) = ϑ(v)∗ ,

v ∈ V,

we call E a Hermitian Clifford module. For this notion to be useful, we need g to be positive-definite. In the case where E = E0 ⊕ E1 is a direct sum of Hermitian vector spaces, we say a homomorphism ϑ : Cl(V, g) → End(E) gives E the structure of a graded Clifford module provided ϑ(v) interchanges E0 and E1 , for v ∈ V , in addition to the hypotheses above. The principal symbol of (1.3) has this property if D is of Dirac type. Let us give some examples of operators of Dirac type. If M is a Riemannian manifold, the exterior derivative operator (1.13)

d : Λj M −→ Λj+1 M

has a formal adjoint (1.14)

δ = d∗ : Λj+1 M −→ Λj M,

discussed in Chap. 2, §10, and in Chap. 5, §§8 and 9. Thus we have

1. Operators of Dirac type

327

d + δ : Λ∗ M −→ Λ∗ M,

(1.15) where, with n = dim M ,



Λ M=

n 

Λj M.

j=0

As was shown in Chap. 2, (d + δ)∗ (d + δ) = d∗ d + dd ∗ is the negative of the Hodge Laplacian on each Λj M , so (1.15) is a symmetric Dirac-type operator. There is more structure. Indeed, we have d + δ : Λeven M −→ Λodd M.

(1.16)

If D is this operator, then D∗ = d + δ : Λodd M → Λeven M , and an operator of type (1.3) arises. If M is compact, the operator (1.16) is Fredholm, with index equal to the Euler characteristic of M , in view of the Hodge decomposition. A calculation of this index in terms of an integrated curvature gives rise to the generalized Gauss–Bonnet formula, as will be seen in §7. Computations implying that (1.15) is of Dirac type were done in §10 of Chap. 2, leading to (10.22) there. If we define (1.17)

∧v : Λj V −→ Λj+1 V,

∧v (v1 ∧ · · · ∧ vj ) = v ∧ v1 ∧ · · · ∧ vj ,

on a vector space V with a positive-definite inner product, and then define ιv : Λj+1 V −→ Λj V

(1.18)

to be its adjoint, then the principal symbol of d + δ on V = Tx∗ M is 1/i times ∧ξ − ιξ . That is to say, iM (v) = ∧v − ιv

(1.19)

defines a linear map from V into End(Λ∗C V ) which extends to an algebra homomorphism M : Cl (V, g) −→ End(Λ∗C V ). Given ∧v ∧w = − ∧w ∧v and its analogue for ι, the anticommutation relation (1.20)

M (v)M (w) + M (w)M (v) = 2v, wI

follows from the identity (1.21)

∧v ιw + ιw ∧v = v, wI.

In this context we note the role that (1.21) played as the algebraic identity behind Cartan’s formula for the Lie derivative of a differential form:

328 10. Dirac Operators and Index Theory

LX α = d(α X) + (dα) X;

(1.22)

cf. Chap. 1, Proposition 13.1, and especially (13.51). Another Dirac-type operator arises from (1.15) as follows. Suppose dim M = n = 2k is even. Recall from Chap. 5, §8, that d∗ = δ is given in terms of the Hodge star operator on Λj M by d∗ = (−1)j(n−j)+j ∗ d∗

(1.23)

= ∗d ∗

if n = 2k.

Also recall that, on Λj M , ∗2 = (−1)j(n−j) = (−1)j

(1.24)

if n = 2k.

Now, on the complexification Λ∗C M of the real vector bundle Λ∗ M , define α : ΛjC M −→ Λn−j C M

(1.25) by

α = ij(j−1)+k ∗

(1.26)

on ΛjC M.

It follows that α2 = 1

(1.27) and

α(d + δ) = −(d + δ)α.

(1.28) Thus we can write (1.29)

Λ∗C M = Λ+ M ⊕ Λ− M, with α = ±I on Λ± M,

and we have (1.30)

± = d + δ : C ∞ (M, Λ± ) −→ C ∞ (M, Λ∓ ). DH

+ − is an operator of Dirac type, with adjoint DH . This operator is called Thus DH the Hirzebruch signature operator, and its index is called the Hirzebruch signature of M . Other examples of operators of Dirac type will be considered in the following sections.

1. Operators of Dirac type

329

Both of the examples just discussed give rise to Hermitian Clifford modules. We now show conversely that generally such modules produce operators of Dirac type. More precisely, if M is a Riemannian manifold, Tx∗ M has an induced inner product, giving rise to a bundle Cl (M ) → M of Clifford algebras. We suppose E → M is a Hermitian vector bundle such that each fiber is a Hermitian Clx (M )module (in a smooth fashion). Let E → M have a connection ∇, so (1.31)

∇ : C ∞ (M, E) −→ C ∞ (M, T ∗ ⊗ E).

Now if Ex is a Clx (M )-module, the inclusion Tx∗ → Clx gives rise to a linear map (1.32)

m : C ∞ (M, T ∗ ⊗ E) −→ C ∞ (M, E),

called “Clifford multiplication.” We compose these two operators; set (1.33)

D = i m ◦ ∇ : C ∞ (M, E) −→ C ∞ (M, E).

We see that, for v ∈ Ex , (1.34)

σD (x, ξ)v = m(ξ ⊗ v) = ξ · v,

so σD (x, ξ) is |ξ|x times an isometry on Ex . Hence D is of Dirac type. If U is an open subset of M , on which we have an orthonormal frame {ej } of smooth vector fields, with dual orthonormal frame {vj } of 1-forms, then, for a section ϕ of E,  (1.35) Dϕ = i vj · ∇ej ϕ on U. Note that σD (x, ξ)∗ = σD (x, ξ), so D can be made symmetric by altering it at most by a zero-order term. Given a little more structure, we have more. We say ∇ is a “Clifford connection” on E if ∇ is a metric connection that is also compatible with Clifford multiplication, in that (1.36)

∇X (v · ϕ) = (∇X v) · ϕ + v · ∇X ϕ,

for a vector field X, a 1-form v, and a section ϕ of E. Here, of course, ∇X v arises from the Levi–Civita connection on M . Proposition 1.1. If ∇ is a Clifford connection on E, then D is symmetric. Proof. Let ϕ, ψ ∈ C0∞ (M, E). We want to show that  (1.37) M

Dϕ, ψ − ϕ, Dψ dV = 0.

330 10. Dirac Operators and Index Theory

We can suppose ϕ, ψ have compact support in a set U on which local orthonormal frames ej , vj as above are given. Define a vector field X on U by X, v = ϕ, v · ψ,

v ∈ Λ1 U.

If we show that, pointwise in U , (1.38)

i div X = Dϕ, ψ − ϕ, Dψ,

then (1.37) will follow from the divergence theorem. Indeed, starting with div X =

(1.39)



∇ej X, vj ,

and using the metric and derivation properties of ∇, we have

 ej · X, vj  − X, ∇ej vj 

 ej ϕ, vj · ψ − ϕ, (∇ej vj ) · ψ . =

div X =

Looking at the last quantity, we expand the first part into a sum of three terms, one of which cancels the last part, and obtain (1.40)

div X =



∇ej ϕ, vj · ψ + ϕ, vj · ∇ej ψ ,

which gives (1.38) and completes the proof. If E = E0 ⊕ E1 is a graded Hermitian Cl(M )-module, if E0 and E1 are each provided with metric connections, and if (1.36) holds, then the construction above gives an operator of Dirac type, of the form (1.3). The examples in (1.15) and (1.30) described above can be obtained from Hermitian Clifford modules via Clifford connections. The Clifford module is Λ∗ M → M , with natural inner product on each factor Λk M and Cl(M )-module structure given by (1.19). The connection is the natural connection on Λ∗ M , extending that on T ∗ M , so that the derivation identity (1.41)

∇X (ϕ ∧ ψ) = (∇X ϕ) ∧ ψ + ϕ ∧ (∇X ψ)

holds for a j-form ϕ and a k-form ψ. In this case it is routine to verify the compatibility condition (1.36) and to see that the construction (1.33) gives rise to the operator d + d∗ on differential forms. We remark that it is common to use Clifford algebras associated to negativedefinite forms rather than positive-definite ones. The two types of algebras are simply related. If a linear map ϑ : V → End(E) extends to an algebra homomorphism Cl(V, g) → End(E), then iϑ extends to an algebra homomorphism Cl(V, −g) → End(E). If one uses a negative form, the condition (1.12) that E

2. Clifford algebras

331

be a Hermitian Clifford module should be changed to ϑ(v) = −ϑ(v)∗ , v ∈ V . In such a case, we should drop the factor of i in (1.33) to associate the Dirac-type operator D to a Cl(M )-module E. In fact, getting rid of this factor of i in (1.33) and (1.35) is perhaps the principal reason some people use the negative-definite quadratic form to construct Clifford algebras.

Exercises 1. Let E be a Cl(M )-module with connection ∇. If ϕ is a section of E and f is a scalar function, show that D(f ϕ) = f Dϕ + i(df ) · ϕ, where the last term involves a Clifford multiplication. 2. If ∇ is a Clifford connection on E and u is a 1-form, show that D(u · ϕ) = −u · Dϕ + 2i∇U ϕ + i(Du) · ϕ, where U is the vector field corresponding to u via the metric tensor on M , and D : C ∞ (M, Λ1 ) −→ C ∞ (M, Cl) is given by Du = i



vj · ∇ej u,

with respect to local dual orthonormal frames ej , vj , and ∇ arising from the Levi–Civita connection. 3. Show that D(df ) = iΔf . Note: Compare with Exercise 6 of §2. 4. If D arises from a Clifford connection on E, show that D2 (f ϕ) = f D2 ϕ − 2∇grad f ϕ − (Δf )ϕ.

2. Clifford algebras In this section we discuss some further results about the structure of Clifford algebras, which were defined in §1. First we note that, by construction, Cl(V, g) has the following universal property. Let A0 be any associative algebra over R, with unit, containing V as a linear subset, generated by V , such that the anticommutation relation (1.8) holds in A0 , for all v, w ∈ V ; that is, vw + wv = 2v, w · 1 in A0 . Then there is a natural surjective homomorphism (2.1)

α : Cl(V, g) −→ A0 .

If {e1 , . . . , en } is a basis of V , any element of Cl(V, g) can be written as a polynomial in the ej . Since ej ek = −ek ej + 2ej , ek  · 1 and in particular e2j = ej , ej  · 1, we can, starting with terms of highest order, rearrange each monomial

332 10. Dirac Operators and Index Theory

in such a polynomial so the ej appear with j in ascending order, and no exponent greater than 1 occurs on any ej . In other words, each element w ∈ Cl(V, g) can be written in the form  (2.2) w= ai1 ···in ei11 · · · einn , iν =0 or 1

with real coefficients ai1 ···in . Denote by A the set of formal expressions of the form (2.2), a real vector space of dimension 2n ; we have a natural inclusion V ⊂ A. We can define a “product” A⊗A → A in which a product of monomials (ei11 · · · einn )·(ej11 · · · ejnn ), with each iν and each jμ equal to either 0 or 1, is a linear combination of monomials of such j a form, by pushing each eμμ past the eiνν for ν > μ, invoking the anticommutation relations. It is routine to verify that this gives A the structure of an associative algebra, generated by V . The universal property mentioned above implies that A is isomorphic to Cl(V, g). Thus each w ∈ Cl(V, g) has a unique representation in the form (2.2), and dim Cl(V, g) = 2n if dim V = n. Recall from §1 the algebra homomorphism M : Cl(V, g) → End(Λ∗ V ), defined there provided g is positive-definite (which can be extended to include general g). Then, we can define a linear map (2.3)

˜ : Cl(V, g) −→ Λ∗ V ; M

˜ (w) = M (w)(1), M

˜ (v) = v. Comparing for w ∈ Cl(V, g). Note that if v ∈ V ⊂ Cl(V, g), then M ∗ the anticommutation relations of Cl(V, g) with those of Λ V , we see that if w ∈ Cl(V, g) is one of the monomials in (2.2), say w = ej11 · · · ejnn , all jν either 0 or 1, k = j1 + · · · + jn , then (2.4)

˜ (ej1 · · · ejn ) − ej1 ∧ · · · ∧ ejn ∈ Λk−1 V. M n n 1 1

It follows easily that (2.3) is an isomorphism of vector spaces. This observation also shows that the representation of an element of Cl(V, g) in the form (2.2) is unique. If g is positive-definite and ej is an orthonormal basis of V , the difference in (2.4) vanishes. In the case g = 0, the anticommutation relation (1.8) becomes vw = −wv, for v, w ∈ V , and we have the exterior algebra Cl(V, 0) = Λ∗ V. Through the remainder of this section we will restrict attention to the case where g is positive-definite. We denote v, v by |v|2 . For V = Rn with g its standard Euclidean inner product, we denote Cl(V, g) by Cl(n). It is useful to consider the complexified Clifford algebra Cl(n) = C ⊗ Cl(n), as it has a relatively simple structure, specified as follows.

2. Clifford algebras

333

Proposition 2.1. There are isomorphisms of complex algebras Cl(1) ≈ C ⊕ C,

(2.5)

Cl(2) ≈ End(C2 ),

and Cl(n + 2) ≈ Cl(n) ⊗ Cl(2);

(2.6) hence, with κ = 2k , (2.7)

Cl(2k) ≈ End(Cκ ),

Cl(2k + 1) ≈ End(Cκ ) ⊕ End(Cκ ).

Proof. The isomorphisms (2.5) are simple exercises. To prove (2.6), imbed Rn+2 into Cl(n) ⊗ Cl(2) by picking an orthonormal basis {e1 , . . . , en+2 } and taking (2.8)

ej → i ej ⊗ en+1 en+2 , for 1 ≤ j ≤ n, ej → 1 ⊗ ej , for j = n + 1 or n + 2.

Then the universal property of Cl(n + 2) leads to the isomorphism (2.6). Given (2.5) and (2.6), (2.7) follows by induction. While, parallel to (2.5), one has Cl(1) = R ⊕ R and Cl(2) = End(R2 ), other algebras Cl(n) are more complicated than their complex analogues; in place of (2.6) one has a form of periodicity with period 8. We refer to [LM] for more on this. k It follows from Proposition 2.1 that C2 has the structure of an irreducible Cl(2k)-module, though making the identification (2.7) explicit involves some untangling, in a way that depends strongly on a choice of basis. It is worthwhile to note the following explicit, invariant construction, for V , a vector space of real dimension 2k, with a positive inner product  , , endowed with one other piece of structure, namely a complex structure J. Assume J is an isometry for  , . Denote the complex vector space (V, J) by V, which has complex dimension k. On V we have a positive Hermitian form (2.9)

(u, v) = u, v + iu, Jv.

Form the complex exterior algebra (2.10)

Λ∗C V =

k 

ΛjC V,

j=0

with its natural Hermitian form. For v ∈ V, one has the exterior product v∧ : j+1 j ΛjC V → Λj+1 C V; denote its adjoint, the interior product, by jv : ΛC V → ΛC V. Set

334 10. Dirac Operators and Index Theory

(2.11)

i μ(v)ϕ = v ∧ ϕ − jv ϕ,

v ∈ V,

ϕ ∈ Λ∗C V.

Note that v ∧ ϕ is C-linear in v and jv ϕ is conjugate linear in v, so μ(v) is only R-linear in v. As in (1.20), we obtain (2.12)

μ(u)μ(v) + μ(v)μ(u) = 2u, v · I,

so μ : V → End(Λ∗C V) extends to a homomorphism of algebras (2.13)

μ : Cl(V, g) −→ End(Λ∗C V),

hence to a homomorphism of C-algebras (2.14)

μ : Cl(V, g) −→ End(Λ∗C V),

where Cl(V, g) denotes C ⊗ Cl(V, g). Proposition 2.2. The homomorphism (2.14) is an isomorphism when V is a real vector space of dimension 2k, with complex structure J, V the associated complex vector space. Proof. We already know that both Cl(V, g) and End(Λ∗C V) are isomorphic to End(Cκ ), κ = 2k . We will make use of the algebraic fact that this is a complex algebra with no proper two-sided ideals. Now the kernel of μ in (2.14) would have to be a two-sided ideal, so either μ = 0 or μ is an isomorphism. But for v ∈ V, μ(v) · 1 = v, so μ = 0; thus μ is an isomorphism. We next mention that a grading can be put on Cl(V, g). Namely, let Cl0 (V, g) denote the set of sums of the form (2.2) with i1 + · · · + in even, and let Cl1 (V, g) denote the set of sums of that form with i1 + · · · + in odd. It is easy to see that this specification is independent of the choice of basis {ej }. Also we clearly have (2.15)

u ∈ Clj (V, g), w ∈ Clk (V, g) =⇒ uw ∈ Clj+k (V, g),

where j and k are each 0 or 1, and we compute j + k mod 2. If (V, g) is Rn with its standard Euclidean metric, we denote Clj (V, g) by Clj (n), j = 0 or 1. We note that there is an isomorphism (2.16)

j : Cl(2k − 1) −→ Cl0 (2k)

uniquely specified by the property that, for v ∈ R2k−1 , j(v) = ve2k , where {e1 , . . . , e2k−1 } denotes the standard basis of R2k−1 , with e2k added to form a basis of R2k . This will be useful in the next section for constructing spinors on odd-dimensional spaces. We can construct a finer grading on Cl(V, g). Namely, set (2.17)

Cl[k] (V, g) = set of sums of the form (2.2), with i1 + · · · + in = k.

Exercises

335

Thus Cl[0] (V, g) is the set of scalars and Cl[1] (V, g) is V . If we insist that {ej } be an orthonormal basis of V , then Cl[k] (V, g) is invariantly defined, for all k. In fact, using the isomorphism (2.3), we have ˜ −1 Λk V . Cl[k] (V, g) = M

(2.18) Note that Cl0 (V, g) =



Cl[k] (V, g) and Cl1 (V, g) =

k even



Cl[k] (V, g).

k odd

Let us also note that Cl[2] (V, g) has a natural Lie algebra structure. In fact, if {ej } is orthonormal, (2.19)

[ei ej , ek e ] = ei ej ek e − ek e ei ej = 2(δjk ei e − δj ei ek + δik e ej − δi ek ej ).

The construction (2.17) makes Cl(V, g) a graded vector space, but not a graded algebra, since typically Cl[j] (V, g) · Cl[k] (V, g) is not contained in Cl[j+k] (V, g), as (2.19) illustrates. We can set (2.20)

Cl(k) (V, g) =



 Cl[j] (V, g) : j ≤ k, j = k mod 2 ,

and then Cl(j) (V, g) · Cl(k) (V, g) ⊂ Cl(j+k) (V, g). As k ranges over the even or the odd integers, the spaces (2.20) provide filtrations of Cl0 (V, g) and Cl1 (V, g).

Exercises 1. Let V have an oriented orthonormal basis {e1 , · · · , en }. Set (2.21)

ν = e1 · · · en ∈ Cl(V, g).

Show that ν is independent of the choice of such a basis. ˜ as in (2.3). ˜ (ν) = e1 ∧ · · · ∧ en ∈ Λn V , with M Note: M 2. Show that ν 2 = (−1)n(n−1)/2 . 3. Show that, for all u ∈ V, νu = (−1)n−1 uν. 4. With μ as in (2.11)–(2.14), show that μ(ν)∗ = (−1)n(n−1)/2 μ(ν) and μ(ν)∗ μ(ν) = I. 5. Show that ˜ (νw) = cnk ∗ M ˜ (w), M for w ∈ Cl[k] (V, g), where ∗ : Λk V → Λn−k V is the Hodge star operator. Find the constants cnk . 6. Let D : C ∞ (M, T ∗ ) → C ∞ (M, Cl) be as in Exercise 2 of §1, namely,

336 10. Dirac Operators and Index Theory Du = i



vj · ∇ej u,

where {ej } is a local orthonormal frame of vector fields, {vj } the dual frame. Show that ˜ (Dv) = −i(d + d∗ )v. M 7. Show that End(Cm ) has no proper two-sided ideals. (Hint: Suppose M0 = 0 belongs to such an ideal I and v0 = 0 belongs to the range of M0 . Show that every v ∈ Cm belongs to the range of some M ∈ I, and hence that every one-dimensional projection belongs to I.)

3. Spinors We define the spinor groups Pin(V, g) and Spin(V, g), for a vector space V with a positive-definite quadratic form g; set |v|2 = g(v, v) = v, v. We set (3.1)

Pin(V, g) = {v1 · · · vk ∈ Cl(V, g) : vj ∈ V, |vj | = 1},

with the induced multiplication. Since (v1 · · · vk )(vk · · · v1 ) = 1, it follows that Pin(V, g) is a group. We can define an action of Pin(V, g) on V as follows. If u ∈ V and x ∈ V , then ux + xu = 2x, u · 1 implies (3.2)

uxu = −xuu + 2x, uu = −|u|2 x + 2x, uu.

If also y ∈ V , (3.3)

uxu, uyu = |u|2 x, y = x, y

if |u| = 1.

Thus if u = v1 · · · vk ∈ Pin(V, g) and if we define a conjugation on Cl(V, g) by (3.4)

u ∗ = vk · · · v 1 ,

vj ∈ V,

it follows that (3.5)

x → uxu∗ ,

x ∈ V,

is an isometry on V for each u ∈ Pin(V, g). It will be more convenient to use (3.6)

u# = (−1)k u∗ ,

u = v1 · · · v k .

Then we have a group homomorphism (3.7) defined by

τ : Pin(V, g) −→ O(V, g),

3. Spinors

(3.8)

τ (u)x = uxu# ,

337

x ∈ V, u ∈ Pin(V, g).

Note that if v ∈ V, |v| = 1, then, by (3.2), τ (v)x = x − 2x, vv

(3.9)

is the reflection across the hyperplane in V orthogonal to v. It is easy to show that any orthogonal transformation T ∈ O(V, g) is a product of a finite number of such reflections, so the group homomorphism (3.7) is surjective. Note that each isometry (3.9) is orientation reversing. Thus, if we define (3.10)

Spin(V, g) = {v1 · · · vk ∈ Cl(V, g) : vj ∈ V, |vj | = 1, k even} = Pin(V, g) ∩ Cl0 (V, g),

then τ : Spin(V, g) −→ SO(V, g)

(3.11)

and in fact Spin(V, g) is the inverse image of SO(V, g) under τ in (3.7). We now show that τ is a 2-fold covering map. Proposition 3.1. τ is a 2-fold covering map. In fact, ker τ = {±1}. Proof. Note that ±1 ∈ Spin(V, g) ⊂ Cl(V, g) and ±1 acts trivially on V , via (3.8). Now, if u = v1 · · · vk ∈ ker τ, k must be even, since τ (u) must preserve orientation, so u# = u∗ . Since uxu∗ = x for all x ∈ V , we have ux = xu, so uxu = |u|2 x, x ∈ V . If we pick an orthonormal basis {e1 , . . . , en } of V and write u ∈ ker τ in the form (2.2), each i1 + · · · + in even, since ej uej = u for each j, we deduce that, for each j, u=



(−1)ij ai1 ···in ei1 ···in

if u ∈ ker τ.

Hence ij = 0 for all j, so u is a scalar; hence u = ±1. We next consider the connectivity properties of Spin(V, g). Proposition 3.2. Spin(V, g) is the connected 2-fold cover of SO(V, g), provided g is positive-definite and dim V ≥ 2. Proof. It suffices to connect −1 ∈ Spin(V, g) to the identity element 1 via a continuous curve in Spin(V, g). In fact, pick orthogonal e1 , e2 , and set   γ(t) = e1 · −(cos t)e1 + (sin t)e2 ,

0 ≤ t ≤ π.

If V = Rn with its standard Euclidean inner product g, denote Spin(V, g) by Spin(n). It is a known topological fact that SO(n) has fundamental group Z2 , and

338 10. Dirac Operators and Index Theory

Spin(n) is simply connected, for n ≥ 3. Though we make no use of this result, we mention that one route to it is via the “homotopy exact sequence” (see [BTu]) for S n = SO(n + 1)/SO(n). This leads to π1 SO(n + 1) ≈ π1 SO(n) for n ≥ 3. Meanwhile, one sees directly that SU(2) is a double cover of SO(3), and it is homeomorphic to S 3 . We next produce representations of Pin(V, g) and Spin(V, g), arising from the homomorphism (2.13). First assume V has real dimension 2k, with complex structure J; let V = (V, J) be the associated complex vector space, of complex dimension k, and set (3.12)

S(V, g, J) = Λ∗C V,

with its induced Hermitian metric, arising from the metric (2.9) on V. The inclusion Pin(V, g) ⊂ Cl(V, g) ⊂ Cl(V, g) followed by (2.14) gives the representation

(3.13)

ρ : Pin(V, g) −→ Aut S(V, g, J) .

Proposition 3.3. The representation ρ of Pin(V, g) is irreducible and unitary. Proof. Since the C-subalgebra of Cl(V, g) generated by Pin(V, g) is all of Cl(V, g), the irreducibility follows from the fact that μ in (2.14) is an isomorphism. For unitarity, it follows from (2.11) that μ(v) is self-adjoint for v ∈ V ; by (2.12), μ(v)2 = |v|2 I, so v ∈ V, |v| = 1 implies that ρ(v) is unitary, and unitarity of ρ on Pin(V, g) follows. The restriction of ρ to Spin(V, g) is not irreducible. In fact, set (3.14)

S+ (V, g, J) = Λeven C V,

S− (V, g, J) = Λodd C V.

Under ρ, the action of Spin(V, g) preserves both S+ and S− . In fact, we have (2.14) restricting to (3.15)

μ : Cl0 (V, g) −→ EndC S+ (V, g, J) ⊕ EndC S− (V, g, J) ,

this map being an isomorphism. On the other hand, (3.16)

z ∈ Cl1 (V, g) =⇒ μ(z) : S± −→ S∓ .

From (3.15) we get representations (3.17)

± : Spin(V, g) −→ Aut S± (V, g, J) , D1/2

which are irreducible and unitary.

3. Spinors

339

If V = R2k with its standard Euclidean metric, standard orthonormal basis e1 , . . . , e2k , we impose the complex structure Jei = ei+k , Jei+k = −ei , 1 ≤ i ≤ k, and set (3.18)

S(2k) = S(R2k , | |2 , J),

S± (2k) = S± (R2k , | |2 , J).

Then (3.17) defines representations ± : Spin(2k) −→ Aut S± (2k) . D1/2

(3.19)

We now consider the odd dimensional case. If V = R2k−1 , we use the isomorphism Cl(2k − 1) −→ Cl0 (2k)

(3.20) produced by the map (3.21)

v → ve2k ,

v ∈ R2k−1 .

Then the inclusion Spin(2k − 1) ⊂ Cl(2k − 1) composed with (3.20) gives an inclusion (3.22)

Spin(2k − 1) → Spin(2k).

+ from (3.19) gives a representation Composing with D1/2

(3.23)

+ D1/2 : Spin(2k − 1) −→ Aut S+ (2k).

− We also have a representation D1/2 of Spin(2k − 1) on S− (2k), but these two representations are equivalent. They are intertwined by the map

(3.24)

μ(e2k ) : S+ (2k) → S− (2k).

We now study spinor bundles on an oriented Riemannian manifold M , with metric tensor g. Over M lies the bundle of oriented orthonormal frames, (3.25)

P −→ M,

a principal SO(n)-bundle, n = dim M . A spin structure on M is a “lift,” (3.26)

P˜ −→ M,

a principal Spin(n)-bundle, such that P˜ is a double covering of P in such a way that the action of Spin(n) on the fibers of P˜ is compatible with the action of

340 10. Dirac Operators and Index Theory

SO(n) on the fibers of P , via the covering homomorphism τ : Spin(n) → SO(n). Endowed with such a spin structure, M is called a spin manifold. There are topological obstructions to the existence of a spin structure, which we will not discuss here (see [LM]). It turns out that there is a naturally defined element of H2 (M, Z2 ) whose vanishing guarantees the existence of a lift, and when such lifts exist, equivalence classes of such lifts are parameterized by elements of H1 (M, Z2 ). Given a spin structure as in (3.26), spinor bundles are constructed via the representations of Spin(n) described above. Two cases arise, depending on whether n = dim M is even or odd. If n = 2k, we form the bundle of spinors (3.27)

S(P˜ ) = P˜ ×ρ S(2k),

+ − ⊕ D1/2 is the sum of the representations in (3.19); this is a sum where ρ = D1/2 of the two vector bundles

(3.28)

S± (P˜ ) = P˜ ×D± S± (2k). 1/2

Recall that, as in §6 of Appendix C, on Connections and Curvature, the sections of S(P˜ ) are in natural correspondence with the functions f on P˜ , taking values in the vector space S(2k), which satisfy the compatibility conditions (3.29)

f (p · g) = ρ(g)−1 f (p),

p ∈ P˜ , g ∈ Spin(2k),

where we write the Spin(n)-action on P˜ as a right action. Recall that S(2k) is a Cl(2k)-module, via (2.13). This result extends to the bundle level. Proposition 3.4. The spinor bundle S(P˜ ) is a natural Cl(M )-module. Proof. Given a section u of Cl(M ) and a section ϕ of S(P˜ ), we need to define u · ϕ as a section of S(P˜ ). We regard u as a function on P˜ with values in Cl(n) and ϕ as a function on P˜ with values in S(n). Then u · ϕ is a function on P˜ with values in S(n); we need to verify the compatibility condition (3.29). Indeed, for p ∈ P˜ , g ∈ Spin(2k), u · ϕ(p · g −1 ) = τ (g)u(p) · ρ(g)ϕ(p) (3.30)

= gu(p)g # gϕ(p) = gu(p) · ϕ(p),

since gg # = 1 for g ∈ Spin(n). This completes the proof. Whenever (M, g) is an oriented Riemannian manifold, the Levi–Civita connection provides a connection on the principal SO(n)-bundle of frames P . If M has a spin structure, this choice of horizontal space for P lifts in a unique natural

3. Spinors

341

fashion to provide a connection on P˜ . Thus the spinor bundle constructed above has a natural connection, which we will call the Dirac–Levi–Civita connection. Proposition 3.5. The Dirac–Levi–Civita connection ∇ on S(P˜ ) is a Clifford connection. Proof. Clearly, ∇ is a metric connection, since the representation ρ of Spin(2k) on S(2k) is unitary. It remains to verify the compatibility condition (1.36), namely, (3.31)

∇X (v · ϕ) = (∇X v) · ϕ + v · ∇X ϕ,

for a vector field X, a 1-form v, and a section ϕ of S(P˜ ). To see this, we first note that as stated in (3.30), the bundle Cl(M ) can be obtained from P˜ → M as P˜ ×κ Cl(2k), where κ is the representation of Spin(2k) on Cl(2k) given by κ(g)w = gwg # . Furthermore, T ∗ M can be regarded as a subbundle of Cl(M ), obtained from P˜ ×κ R2k with the same formula for κ. The connection on T ∗ M obtained from that on P˜ is identical to the usual connection on T ∗ M defined via the Levi–Civita formula. Given this, (3.31) is a straightforward derivation identity. Using the prescription (1.31)–(1.33), we can define the Dirac operator on a Riemannian manifold of dimension 2k, with a spin structure: (3.32)

D : C ∞ (M, S(P˜ )) −→ C ∞ (M, S(P˜ )).

We see that Proposition 1.1 applies; D is symmetric. Note also the grading: (3.33)

D : C ∞ (M, S± (P˜ )) −→ C ∞ (M, S∓ (P˜ )).

In other words, this Dirac operator is of the form (1.3). On a Riemannian manifold of dimension 2k with a spin structure P˜ → M , let F → M be another vector bundle. Then the tensor product E = S(P˜ ) ⊗ F is a Cl(M )-module in a natural fashion. If F has a connection, then E gets a natural product connection. Then the construction (1.31)–(1.33) yields an operator DF of Dirac type on sections of E; in fact (3.34)

DF : C ∞ (M, E± ) −→ C ∞ (M, E∓ ),

E± = S± (P˜ ) ⊗ F.

If F has a metric connection, then E gets a Clifford connection. The operator DF is called a twisted Dirac operator. Sometimes it will be convenient to distinguish notationally the two pieces of DF ; we write (3.35)

DF+ : C ∞ (M, E+ ) −→ C ∞ (M, E− ),

DF− : C ∞ (M, E− ) −→ C ∞ (M, E+ ).

342 10. Dirac Operators and Index Theory

When dim M = 2k − 1 is odd, we use the representation (3.23) to form the bundle of spinors S+ (P˜ ) = P˜ ×D+ S+ (2k). 1/2

0

The inclusion Cl(2k − 1) → Cl (2k) defined by (3.20)–(3.21) makes S+ (2k) a Cl(2k − 1)-module, and analogues of Propositions 3.4 and 3.5 hold. Hence there arises a Dirac operator, D : C ∞ (M, S+ (P˜ )) → C ∞ (M, S+ (P˜ )). Twisted Dirac operators also arise; however, in place of (3.34), we have DF : C ∞ (M, E+ ) → C ∞ (M, E+ ), with E+ = S+ (P˜ ) ⊗ F .

Exercises 1. Verify that the map (3.15) is an isomorphism and that the representations (3.17) of Spin(V, g) are irreducible when dim V = 2k. 2. Let ν be as in Exercises 1–4 of §2, with n = 2k. Show that a) the center of Spin(V, g) consists of {1, −1, ν, −ν}, b) μ(ν) leaves S+ and S− invariant, c) μ(ν) commutes with the action of Cl0 (V, g) under μ, hence with the represen± of Spin(V, g), tations D1/2 d) μ(ν) acts as a pair of scalars on S+ and S− , respectively. These scalars are the two square roots of (−1)k . 3. Calculate μ(ν) · 1 directly, making use of the definition (2.11). Hence match the scalars in exercise 2d) to S+ and S− . (Hint: μ(ek+1 · · · e2k ) · 1 = (−i)k ek+1 ∧ · · · ∧ e2k in ΛkC V. Using ej+k = i ej in V, for 1 ≤ j ≤ k, we have μ(ν) · 1 = μ(e1 · · · ek )(e1 ∧ · · · ∧ ek ), and there are k interior products to compute.) 4. Show that Cl[2] (V, g), with the Lie algebra structure (2.19), is naturally isomorphic to the Lie algebra of Spin(V, g). In fact, if (ajk ) is a real, antisymmetric matrix, in the Lie algebra of SO(n), which is the same as that of Spin(n), show that there is the correspondence 1 ajk ej ek = κ(A). A = (ajk ) → 4 In particular, show that κ(A1 A2 − A2 A1 ) = κ(A1 )κ(A2 ) − κ(A2 )κ(A1 ). 5. If X is a spin manifold and M ⊂ X is an oriented submanifold of codimension 1, show that M has a spin structure. Deduce that an oriented hypersurface in Rn has a spin structure.

4. Weitzenbock formulas Let E → M be a Hermitian vector bundle with a metric connection ∇. Suppose E is also a Cl(M )-module and that ∇ is a Clifford connection. If we consider the Dirac-type operator D : C ∞ (M, E) → C ∞ (M, E) and the covariant derivative ∇ : C ∞ (M, E) → C ∞ (M, T ∗ ⊗ E), then D2 and ∇∗ ∇ are operators on

4. Weitzenbock formulas

343

C ∞ (M, E) with the same principal symbol. It is of interest to examine their difference, clearly a differential operator of order ≤ 1. In fact, the difference has order 0. This can be seen in principle from the following considerations. From Exercise 4 of §1, we have (4.1)

D2 (f ϕ) = f D2 ϕ − 2∇grad f ϕ − (Δf )ϕ

when ϕ ∈ C ∞ (M, E), f a scalar function. Similarly, we compute ∇∗ ∇(f ϕ). The derivation property of ∇ implies (4.2)

∇(f ϕ) = f ∇ϕ + df ⊗ ϕ.

To apply ∇∗ to this, first a short calculation gives (4.3)

∇∗ f (u ⊗ ϕ) = f ∇∗ (u ⊗ ϕ) − df, uϕ,

for u ∈ C ∞ (M, T ∗ ), ϕ ∈ C ∞ (M, E), and hence (4.4)

∇∗ (f ∇ϕ) = f ∇∗ ∇ϕ − ∇grad f ϕ.

This gives ∇∗ applied to the first term on the right side of (4.2). To apply ∇∗ to the other term, we can use the identity (see Appendix C, (1.35)) (4.5)

∇∗ (u ⊗ ϕ) = −∇U ϕ − (div U )ϕ,

where U is the vector field corresponding to u via the metric on M . Hence (4.6)

∇∗ (df ⊗ ϕ) = −∇grad f ϕ − (Δf )ϕ.

Then (4.6) and (4.4) applied to (4.2) gives (4.7)

∇∗ ∇(f ϕ) = f ∇∗ ∇ϕ − 2∇grad f ϕ − (Δf )ϕ.

Comparing (4.1) and (4.7), we have (4.8)

(D2 − ∇∗ ∇)(f ϕ) = f (D2 − ∇∗ ∇)ϕ,

which implies D2 − ∇∗ ∇ has order zero, hence is given by a bundle map on E. We now derive the Weitzenbock formula for what this difference is. Proposition 4.1. If E → M is a Cl(M )-module with Clifford connection and associated Dirac-type operator D, then, for ϕ ∈ C ∞ (M, E), (4.9)

D2 ϕ = ∇∗ ∇ϕ −

 j>k

vk vj K(ek , ej )ϕ,

344 10. Dirac Operators and Index Theory

where {ej } is a local orthonormal frame of vector fields, with dual frame field {vj }, and K is the curvature tensor of (E, ∇).  Proof. Starting with Dϕ = i vj ∇ej ϕ, we obtain D2 ϕ = − (4.10)

 j,k

=−



vk ∇ek vj ∇ej ϕ

vk vj ∇ek ∇ej ϕ + ∇ek vj ∇ej ϕ ,

j,k

using the compatibility condition (1.36). We replace ∇ek ∇ej by the Hessian, using the identity (4.11)

∇2ek ,ej ϕ = ∇ek ∇ej ϕ − ∇∇ek ej ϕ;

cf. (2.4) of Appendix C. We obtain D2 ϕ = − (4.12)

 j,k





vk vj ∇2ek ,ej ϕ

vk vj ∇∇ek ej ϕ + ∇ek vj ∇ej ϕ .

j,k

Let us look at each of the two double sums on the right. Using vj2 = 1 and the anticommutator property vk vj = −vj vk for k = j, we see that the first double sum becomes   ∇2ej ,ej ϕ − vk vj K(ek , ej )ϕ, (4.13) − j

j>k

since the antisymmetric part of the Hessian is the curvature. This is equal to the right side of (4.9), in light of the formula for ∇∗ ∇ established in Proposition 2.1 of Appendix C. As for the remaining double sum in (4.12), for any p ∈ M , we can choose a local orthonormal frame field {ej } such that ∇ej ek = 0 at p, and then this term vanishes at p. This proves (4.9). We denote the difference D2 − ∇∗ ∇ by K, so (4.14)

(D2 − ∇∗ ∇)ϕ = Kϕ,

K ∈ C ∞ (M, End E).

The formula for K in (4.9) can also be written as (4.15)

Kϕ = −

1 vk vj K(ek , ej )ϕ. 2 j,k

4. Weitzenbock formulas

345

Since a number of formulas that follow will involve multiple summation, we will use the summation convention. This general formula for K simplifies further in some important special cases. The first simple example of this will be useful for further calculations. Proposition 4.2. Let E = Λ∗ M , with Cl(M )-module structure and connection described in §1, so K ∈ C ∞ (M, End Λ∗ ). In this case, (4.16)

u ∈ Λ1 M =⇒ Ku = Ric(u).

Proof. The curvature of Λ∗ M is a sum of curvatures of each factor Λk M . In particular, if {ej , vj } is a local dual pair of frame fields, K(ei , ej )vk = −Rk ij v ,

(4.17)

where Rk ij are the components of the Riemann tensor, with respect to these frame fields, and we use the summation convention. In light of (4.15), the desired identity (4.16), will hold provided 1 vi vj v Rk ij = Ric(vk ), 2

(4.18)

so it remains to establish this identity. Since, if (i, j, ) are distinct, vi vj v = v vi vj = vj v vi , and since by Bianchi’s first identity Rk ij + Rk j i + Rk ij  = 0, it follows that in summing the left side of (4.18), the sum over (i, j, ) distinct vanishes. By antisymmetry of Rk ij , the terms with i = j vanish. Thus the only contributions arise from i =  = j and i =  = j. Therefore, the left side of (4.18) is equal to (4.19)

1 −vj Rk iij + vi Rk jij = vi Rk jij = Ric(vk ), 2

which completes the proof. We next derive Lichnerowicz’s calculation of K when E = S(P˜ ), the spinor bundle of a manifold M with spin structure. First we need an expression for the curvature of S(P˜ ). Lemma 4.3. The curvature tensor of the spinor bundle S(P˜ ) is given by (4.20)

K(ei , ej )ϕ =

1 k R ij vk v ϕ. 4

346 10. Dirac Operators and Index Theory

Proof. This follows from the relation between curvatures on vector bundles and on principal bundles established in Appendix C, §6, together with the identification of the Lie algebra of Spin(n) with Cl[2] (n) given in Exercise 4 of §3. Proposition 4.4. For the spin bundle S(P˜ ), K ∈ C ∞ (M, End S(P˜ )) is given by Kϕ =

(4.21)

1 Sϕ, 4

where S is the scalar curvature of M . Proof. Using (4.20), the general formula (4.15) yields (4.22)

1 1 Kϕ = − Rk ij vi vj vk v ϕ = vi vj v Rk ij vk ϕ, 8 8

the last identity holding by the anticommutation relations; note that only the sum over k =  counts. Now, by (4.18), this becomes 1 vi vk Rk jij ϕ 4 1 = Ricii ϕ (by symmetry) 4 1 = Sϕ, 4

Kϕ = (4.23)

completing the proof. We record the generalization of Proposition 4.4 to the case of twisted Dirac operators. We mention that one often sees a different sign before the sum, due to a different sign convention for Clifford algebras. Proposition 4.5. Let E → M have a metric connection ∇, with curvature RE . For the twisted Dirac operator on sections of F = S(P˜ ) ⊗ E, the section K of End F has the form (4.24)

Kϕ =

1 1 Sϕ − vi vj RE (ei , ej )ϕ. 4 2 i,j

Proof. Here RE (ei , ej ) is shorthand for I ⊗RE (ei , ej ) acting on S(P˜ )⊗E. This formula is a consequence of the general formula (4.15) and the argument proving Proposition 4.4, since the curvature of S(P˜ ) ⊗ E is K ⊗ I + I ⊗ RE , K being the curvature of S(P˜ ), given by (4.20). These Weitzenbock formulas will be of use in the following sections. Here we draw some interesting conclusions, due to Bochner and Lichnerowitz.

Exercises

347

Proposition 4.6. If M is compact and connected, and the section K in (4.14)– (4.15) has the property that K ≥ 0 on M and K > 0 at some point, then ker D = 0. Proof. This is immediate from (D2 ϕ, ϕ) = (Kϕ, ϕ) + ∇ϕ2L2 . Proposition 4.7. If M is a compact Riemannian manifold with positive Ricci tensor, then b1 (M ) = 0, that is, the deRham cohomology group H1 (M, R) = 0. Proof. Via Hodge theory, we want to show that if u ∈ Λ1 (M ) and du = d∗ u = 0, then u = 0. This hypothesis implies Du = 0, where D is the Dirac-type operator dealt with in Proposition 4.2. Consequently we have, for a 1-form u on M , Du2L2 = Ric(u), u + ∇u2L2 ,

(4.25) so the result follows.

Proposition 4.8. If M is a compact, connected Riemannian manifold with a spin structure whose scalar curvature is ≥ 0 on M and > 0 at some point, then M has no nonzero harmonic spinors, that is, ker D = 0 in C ∞ (M, S(P˜ )). Proof. In light of (4.21), this is a special case of Proposition 4.6.

Exercises 1. Let Δ be the Laplace operator on functions (0-forms) on a compact Riemannian manifold M, Δk the Hodge Laplacian on k-forms. If Spec(−Δ) consists of 0 = λ0 < λ1 ≤ λ2 ≤ · · · , show that λ1 ∈ Spec(−Δ1 ). 2. If Ric ≥ c0 I on M , show that λ1 ≥ c0 . 3. Recall the deformation tensor of a vector field u: Def u = Show that

1 1 Lu g = (∇u + ∇ut ), 2 2

Def : C ∞ (M, T ) → C ∞ (M, S 2 ).

Def∗ v = − div v,

where (div v)j = v jk ;k . Establish the Weitzenbock formula (4.26)

2 div Def u = −∇∗ ∇u + grad div u + Ric(u).

The operator div on the right is the usual divergence operator on vector fields. (This formula will appear again in Chap. 17, in the study of the Navier–Stokes equation.) 4. Suppose M is a compact, connected Riemannian manifold, whose Ricci tensor satisfies (4.27)

Ric(x) ≤ 0 on M,

Ric(x0 ) < 0, for some x0 ∈ M.

348 10. Dirac Operators and Index Theory Show that the operator Def is injective, so there are no nontrivial Killing fields on M , hence no nontrivial one-parameter groups of isometries. (Hint: From (4.26), we have   (4.28) 2 Def u 2L2 = ∇u 2L2 + div u 2 − Ric(u), u L2 .) 5. As shown in (3.39) of Chap. 2, the equation of a conformal Killing field on an n-dimensional Riemannian manifold M is (4.29)

Def X −

 1 div X g = 0. n

Note that the left side is the trace-free part of Def X ∈ C ∞ (M, S 2 T ∗ ). Denote it by DT F X. Show that   1 grad div X , (4.30) DT∗ F = − divS 2 T ∗ , DT∗ F DT F X = − div Def X + 0 n where S02 T ∗ is the trace-free part of S 2 T ∗ . Show that    1 1 1 1 2 2 (4.31) DT F X L2 = ∇X L2 + − div X 2L2 − Ric(X), X L2 . 2 2 n 2 Deduce that if M is compact and satisfies (4.27), then M has no nontrivial oneparameter group of conformal diffeomorphisms. 6. Show that if M is a compact Riemannian manifold which is Ricci flat (i.e., Ric = 0), then every conformal Killing field is a Killing field, and the dimension of the space of Killing fields is given by (4.32)

dimR ker Def = dim H1 (M, R).

(Hint: Combine (4.25) and (4.28).) 7. Suppose dim M = 2 and M is compact and connected. Show that, for u ∈ C ∞ (M, S02 T ∗ ),  1 DT∗ F u 2L2 = ∇u 2L2 + K|u|2 dV, 2 M

where K is the Gauss curvature. Deduce that if K ≥ 0 on M , and K(x0 ) > 0 for some x0 ∈ M , then Ker DT∗ F = 0. Compare with Exercises 6–8 of §10. 8. If u and v are vector fields on a Riemannian manifold M , show that   (4.33) div ∇u v = ∇u (div v) + Tr (∇u)(∇v) − Ric(u, v). Compare with formula (3.17) in Chap. 17, on the Euler equation. Relate this identity to the Weitzenbock formula for Δ on 1-forms (a special case of Proposition 4.2).

5. Index of Dirac operators If D : C ∞ (M, E0 ) → C ∞ (M, E1 ) is an elliptic, first-order differential operator between sections of vector bundles E0 and E1 over a compact manifold M , then, as we have seen, D : H k+1 (M, E0 ) → H k (M, E1 ) is Fredholm, for any real k. Furthermore, ker D is a finite-dimensional subspace of C ∞ (M, E0 ), independent

5. Index of Dirac operators

349

of k, and D∗ : H −k (M, E1 ) → H −k−1 (M, E0 ) has the same properties. A quantity of substantial importance is the index of D: (5.1)

Index D = dim ker D − dim ker D∗ .

In this section and the next we derive a formula for this index, due to Atiyah and Singer. Later sections will consider a few applications of this formula. One basic case for such index theorems is that of twisted Dirac operators. Thus, let F → M be a vector bundle with metric connection, over a compact Riemannian manifold M with a spin structure. Assume dim M = n = 2k is even. The twisted Dirac operator constructed in §3 in particular gives an elliptic operator (5.2)

DF : C ∞ (M, S+ (P˜ ) ⊗ F ) −→ C ∞ (M, S− (P˜ ) ⊗ F ).

The Atiyah–Singer formula for the index of this operator is given as follows. Theorem 5.1. If M is a compact Riemannian manifold of dimension n = 2k with spin structure and DF the twisted Dirac operator (5.2), then (5.3)

ˆ ) Ch(F ), [M ]. Index DF = A(M

ˆ What is meant by the right side of (5.3) is the following. A(M ) and Ch(F ) are certain characteristic classes; each is a sum of even-order differential forms on M , computed from the curvatures of S(P˜ ) and F , respectively. We will derive explicit formulas for what these are in the course of the proof of this theorem, in the next section, so we will not give the formulas here. The pairing with M indicated in (5.3) is the integration over M of the form of degree 2k = n arising ˆ in the product A(M )Ch(F ). ˆ The choice of notation in A(M ) and Ch(F ) indicates an independence of such particulars as the choice of Riemannian metric on M and of connection on F . This is part of the nature of characteristic classes, at least after integration is performed; for a discussion of this, see §7 of Appendix C. There is also a simple direct reason why Index DF does not depend on such choices. Namely, any two Riemannian metrics on M can be deformed to each other, and any two connections on F can be deformed to each other. The invariance of the index of DF is thus a special case of the following. Proposition 5.2. If Ds , 0 ≤ s ≤ 1, is a continuous family of elliptic differential operators Ds : C ∞ (M, E0 ) → C ∞ (M, E1 ) of first order, then Index Ds is independent of s. Proof. We have a norm-continuous family of Fredholm operators Ds : H 1 (M, E0 ) → L2 (M, E1 ); the constancy of the index of any continuous family of Fredholm operators is proved in Appendix A, Proposition 7.4. The proof of Theorem 5.1 will be via the heat-equation method, involving a comparison of the spectra of D∗ D and DD∗ , self-adjoint operators on L2 (M, E0 )

350 10. Dirac Operators and Index Theory

and L2 (M, E1 ), respectively. As we know, since D∗ D = L0 and DD∗ = L1 are both elliptic and self-adjoint, they have discrete spectra, with eigenspaces of finite dimension, contained in C ∞ (M, Ej ), say (5.4)

Eigen(Lj , λ) = {u ∈ C ∞ (M, Ej ) : Lj u = λu}.

We have the following result: Proposition 5.3. The spectra of L0 and L1 are discrete subsets of [0, ∞) which coincide, except perhaps at 0. All nonnero eigenvalues have the same finite multiplicity. Proof. It is easy to see that for each λ ∈ [0, ∞), D : Eigen(L0 , λ) → Eigen(L1 , λ) and D∗ : Eigen(L1 , λ) → Eigen(L0 , λ). For λ = 0, D and λ−1 D∗ are inverses of each other on these spaces. We know from the spectral theory of Chap. 8 that ϕ(L0 ) and ϕ(L1 ) are trace class for any ϕ ∈ S(R). We hence have the following. Proposition 5.4. For any ϕ ∈ S(R), with ϕ(0) = 1, (5.5)

Index D = Tr ϕ(D∗ D) − Tr ϕ(DD∗ ).

In particular, for any t > 0, (5.6)

Index D = Tr e−tD



D



− Tr e−tDD .

Now, whenever D is of Dirac type, so D∗ D = L0 and DD∗ = L1 have scalar principal symbol, results of Chap. 7 show that (5.7)

e−tLj u(x) =

 kj (t, x, y) u(y) dV (y), M

with (5.8)



kj (t, x, x) ∼ t−n/2 aj0 (x) + aj1 (x)t + · · · + aj (x)t + · · · ,

as t  0, with aj ∈ C ∞ (M, End Ej ), so (5.9)

  Tr e−tLj ∼ t−n/2 bj0 + bj1 t + · · · + bj t + · · · ,

with  (5.10)

bj =

Tr aj (x) dV (x). M

6. Proof of the local index formula

351

In light of (5.6), we have the following result: Proposition 5.5. If D is of Dirac type on M, of dimension n = 2k, then    (5.11) Index D = b0k − b1k = Tr a0k (x) − a1k (x) dV (x), M

where aj are the coefficients in (5.8). We remark that these calculations are valid for dim M = n odd. In that case, there is no coefficient of t0 in (5.8) or (5.9), so the identity (5.6) implies Index D = 0 for dim M odd. In fact, this holds for any elliptic differential operator, not necessarily of Dirac type. On the other hand, if dim M is odd, there exist elliptic pseudodifferential operators on M with nonzero index. We will establish the Atiyah–Singer formula (5.3) in the next section by showing that, for a twisted Dirac operator DF , the 2k-form part of the right side of ˆ the formula (5.3), with A(M ) and Ch(F ) given by curvatures in an appropriate fashion, is equal pointwise on M to the integrand in (5.11). Such an identity is called a local index formula.

6. Proof of the local index formula Let DF be a twisted Dirac operator on a compact spin manifold, as in (5.2). If L0 = DF∗ DF and L1 = DF DF∗ , we saw in §5 that, for all t > 0, 

Tr k0 (t, x, x) − Tr k1 (t, x, x) dV (x), (6.1) Index DF = M

where kj (t, x, y) are the Schwartz kernels of the operators e−tLj . In the index forˆ mula stated in (5.3), A(M ) and Ch(F ) are to be regarded as differential forms on M , arising in a fashion we will specify later in this section, from curvature forms given by the spin structure on M and a connection on F ; the product is the wedge product of forms. The following is the local index formula, which refines (5.3). Theorem 6.1. For the twisted Dirac operator DF , we have the pointwise identity  

 ˆ (6.2) lim Tr k0 (t, x, x) − Tr k1 (t, x, x) dV = A(M ) ∧ Ch(F ) n , t→0

where {β}n denotes the component of degree n = dim M of a differential form β, and dV denotes the volume form of the oriented manifold M . We first obtain a formula for the difference in the traces of k0 (t, x, x) and of k1 (t, x, x), which are elements of End((S± )x ⊗ Fx ). It is convenient to put these together, and consider   k0 0 (6.3) K= ∈ End(S ⊗ F ), 0 k1

352 10. Dirac Operators and Index Theory

where S = S+ ⊕S− , and we have dropped x and t. Using the isomorphism (2.14), μ : Cl(2k) → End S, we can write End(S ⊗ F ) = Cl(2k) ⊗ End(F ).

(6.4)

We will suppose dim M = n = 2k. In other words, we can think of an element of End(S ⊗ F ) as a combination of elements of the Clifford algebra, whose coefficients are linear transformations on F . Since (6.3) preserves S+ ⊗ F and S− ⊗ F , we have K ∈ Cl0 (2k) ⊗ End(F ).

(6.5)

For K of the form (6.3), the difference Tr k0 − Tr k1 is called the “supertrace” of K, written   1 0 (6.6) Str K = Tr(εK), with ε = . 0 −1 The first key step in establishing (6.2) is the following identity, which arose in the work of F. Berezin [Ber] and V. Patodi [Pt1]. Define the map τ : Cl(2k) −→ C

(6.7)

to be the evaluation of the coefficient of the “volume element” ν = e1 · · · e2k , introduced in Exercises 1–4 of §2. Similarly define (6.8)

τF : Cl(2k) ⊗ End(F ) −→ End F,

τ˜ : Cl(2k) ⊗ End(F ) −→ C

to be (6.9)

τF = τ ⊗ I,

τ˜ = Tr ◦ τF ,

where the last trace is Tr : End F → C. Lemma 6.2. The supertrace is given by Str K = (−2i)k τ˜(K),

(6.10) using the identification (6.4).

Proof. If this is established for the case F = C, the general case follows easily. We note that, with ν = e1 · · · e2k , S± = {x ∈ S : μ(ik ν)x = ±x}. Thus, for K ∈ Cl(2k),

6. Proof of the local index formula

353

Str K = Tr(ik νK).

(6.11) Thus (6.10) is equivalent to

Tr w = 2k w0 ,

(6.12)

for w ∈ Cl(2k) ≈ End S, where w0 is the scalar term in the expansion (2.2) for w. This in turn follows from Tr 1 = 2k

(6.13) and (6.14)

Tr ei11 · · · einn = 0

if i1 + · · · + in > 0,

iν = 0 or 1.

To verify these identities, note that 1 acts on S as the identity, so (6.13) holds by the computation of dim S. As for (6.14), using S ⊗ S  ≈ Cl(2k), we see that (6.14) is a multiple of the trace of ei11 · · · einn acting on Cl(2k) by Clifford multiplication, which is clearly zero. The proof is complete. Thus we want to analyze the Cl[2k] (2k) ⊗ End F component of K(t, x, x), the value on the diagonal of K(t, x, y), the Schwartz kernel of  −tL  2 e 0 0 −tDF = . e 0 e−tL1 We recall that a construction of K(t, x, y) was made in Chap. 7, §13. It was shown that, in local coordinates and with a local choice of trivializations of S(P˜ ) and of F , we could write, modulo a negligible error,  2 −tDF −n/2 u(x) = (2π) a(t, x, ξ)ˆ u(ξ)eix·ξ dξ, (6.15) e where the amplitude a(t, x, ξ) has an asymptotic expansion  aj (t, x, ξ). (6.16) a(t, x, ξ) ∼ j≥0

The terms aj (t, x, ξ) were defined recursively in the following manner. If, with such local coordinates and trivializations, (6.17)

DF2 = L(x, Dx ),

then, by the Leibniz formula, write

354 10. Dirac Operators and Index Theory

 i|α| L(α) (x, ξ) Dxα a(t, x, ξ) α! |α|≤2   2  ix·ξ =e B2− (x, ξ, Dx )a(t, x, ξ) , L2 (x, ξ)a(t, x, ξ) +

L(a eix·ξ ) = eix·ξ (6.18)

=1

where B2− (x, ξ, Dx ) is a differential operator (of order ) whose coefficients are polynomials in ξ, homogeneous of degree 2 −  in ξ. L2 (x, ξ) is the principal symbol of L = DF2 . Thus we want the amplitude a(t, x, ξ) in (6.15) to satisfy formally 2

(6.19)

 ∂a ∼ −L2 a − B2− (x, ξ, Dx )a. ∂t =1

If a is taken to have the form (6.16), we produce the following transport equations for aj : ∂a0 = −L2 (x, ξ)a0 (t, x, ξ) ∂t

(6.20) and, for j ≥ 1, (6.21)

∂aj = −L2 (x, ξ)aj + Ωj (t, x, ξ), ∂t

where (6.22)

Ωj (t, x, ξ) = −

2 

B2− (x, ξ, Dx )aj− (t, x, ξ).

=1

By convention, we set a−1 = 0. So that (6.15) reduces to Fourier inversion at t = 0, we set (6.23)

aj (0, x, ξ) = 0, for j ≥ 1.

a0 (0, x, ξ) = 1,

Then we have a0 (t, x, ξ) = e−tL2 (x,ξ) .

(6.24) The solution to (6.21) is

 (6.25)

aj (t, x, ξ) =

0

t

e(s−t)L2 (x,ξ) Ωj (s, x, ξ) ds.

Now, as shown in Chap. 7, we have

6. Proof of the local index formula

(6.26)

Tr e−tL ∼



355

 Tr

aj (t, x, ξ) dξ dx,

j≥0

with  (6.27)

aj (t, x, ξ) dξ = t(−n+j)/2 bj (x).

Furthermore, the integral (6.27) vanishes for j odd. Thus we have the expansion (6.28)



K(t, x, x) ∼ t−n/2 a0 (x) + a1 (x)t + · · · + a (x)t + · · · ,

with aj (x) = b2j (x). Our goal is to analyze the Cl[2k] ⊗ End F component of ak (x), with n = 2k. In fact, the way the local index formula (6.2) is stated, the claim is made that a (x) has zero component in this space, for  < k. The next lemma gives a more precise result. Its proof will also put us in a better position to evaluate the treasured Cl[2] ⊗ End F component of ak (x). Recall the filtration (2.20) of Cl0 (2k); complexification gives a similar filtration of Cl(2k). Lemma 6.3. In the expansion (6.28), we have (6.29)

aj (x) ∈ Cl(2j) (2k) ⊗ End F,

0 ≤ j ≤ k.

In order to prove this, we examine the expression for L = DF2 , in local coordinates, with respect to convenient local trivializations of S(P˜ ) and F . Fix x0 ∈ M . Use geodesic normal coordinates centered at x0 ; in these coordinates, x0 = 0. Let {eα } denote an orthonormal frame of tangent vectors, obtained by parallel translation along geodesics from x0 of an orthonormal basis of Tx0 M ; let {vα } denote the dual frame. The frame {eα } gives rise to a local trivialization of the spinor bundle S(P˜ ). Finally, choose an orthonormal frame {ϕμ } of F , obtained by parallel translation along geodesics from x0 of an orthonormal basis of Fx0 . The connection coefficients for the Levi–Civita connection will be denoted as Γk j for the coordinate frame, Γα βj for the frame {eα }; both sets of connection coefficients vanish at 0, their first derivatives at 0 being in terms of the Riemann given curvature tensor. Similarly, denote by θj = θμ νj the connection coefficients for F , with respect to the frame {ϕμ }. Denote by Φαβ the curvature of F , with respect to the frame {eα }. With respect to these choices, we write down a local coordinate expression for DF2 , using the Weitzenbock formula 1 1 DF2 = ∇∗ ∇ + S − vα vβ Φαβ , 4 2

356 10. Dirac Operators and Index Theory

together with the identity ∇∗ ∇ = −γ◦∇2 , proved in Proposition 2.1 of Appendix C. We obtain

(6.30)

   1 1 DF2 = −g j ∂j + Γβ αj vα vβ + θj ∂ + Γδ γ vγ vδ + θ 4 4   1 1 1 + g j Γi j ∂i + Γβ αi vα vβ + θi + S − Φαβ vα vβ . 4 4 2

This has scalar second-order part. The coefficients of ∂j are products of elements of Cl(2) (2k) with connection coefficients, which vanish at 0. Terms involving no derivatives include products of elements of Cl(2) (2k) with curvatures, which may not vanish, and products of elements of Cl(4) (2k) with coefficients that vanish to second order at 0. Hence we can say the following about the operators B2− (x, ξ, Dx ), which arise in (6.18) and which enter into the recursive formulas for aj (t, x, ξ). First, B0 (x, ξ, Dx ), a differential operator of order 2 that is homogeneous of degree 0 in ξ (thus actually independent of ξ), can be written as B0 (x, ξ, Dx ) =



B0α (x, ξ)Dxα ,

|α|≤2

where B00 (x, ξ) has coefficients in Cl(2) (2k), and also coefficients that are O(|x|2 ) in Cl(4) (2k); B0α (x, ξ) for |α| = 1 has some coefficients that are O(|x|) in Cl(2) (2k). Each B0α (x, ξ) for |α| = 2 is scalar. Note that B0 (x, ξ, Dx ) acts on aj−2 (t, x, ξ) in the recursive formula (6.21)–(6.22) for aj (t, x, ξ). The operator B1 (x, ξ, Dx ), a differential operator of order 1 that is homogeneous of degree 1 in ξ, can be written as B1 (x, ξ, Dx ) =



B1α (x, ξ)Dxα ,

|α|≤1

and among the coefficients are terms that are O(|x|) in Cl(2) (2k). The operator B1 (x, ξ, Dx ) acts on aj−1 (t, x, ξ) in (6.21)–(6.22). We see that while the coefficients in Cl[] (2k) in aj (t, x, ξ) give rise to coefficients in Cl[+2] (2k) in aj+1 (t, x, ξ) and in Cl[+4] (2k) in aj+2 (t, x, ξ), the degree of vanishing described above leads exactly to the sort of increase in “Clifford order” stated in Lemma 6.3, which is consequently proved. The proof of Lemma 6.3 gives more. Namely, the Cl[2j] -components of aj (x0 ), for 0 ≤ j ≤ k, are unchanged if we replace DF2 by the following: (6.31)

˜=− L

2 n   ∂ 1 1 − Ωj x − Φαβ vα vβ , ∂xj 8 2 j=1

where Ωj denotes the Riemann curvature tensor, acting on sections of S(P˜ ) as

6. Proof of the local index formula

(6.32)

357

Ωj = Rjαβ vα vβ .

In (6.31), summation over  is understood. At this point, we can exploit a key ob˜ x, y) of e−tL˜ can be evaluated servation of Getzler—that the Schwartz kernel K(t, in closed form at y = 0—by exploiting the similarity of (6.31) with the harmonic oscillator Hamiltonian, whose exponential is given by Mehler’s formula, provided ˜ in the following fashion. we modify L Namely, for the purpose of picking out the Cl[2j] -components of aj (x0 ), we ˜ act on sections of Cl(2k) ⊗ F rather than S(P˜ ) ⊗ F , and then might as well let L we use the linear isomorphism Cl(2k) ≈ Λ∗ Rn , and let the products involving vα and vβ in (6.31) and (6.32) be wedge products, which, after all, for the purpose of our calculation are the principal parts of the Clifford products. ˜ into two commuting parts. Let L ˜ 0 denote the sum over We can then separate L ˜ ˜ j in (6.31), and let K0 (t, x, y) be the Schwartz kernel of e−tL0 . We can evaluate ˜ 0 (t, x, 0) using Mehler’s formula, established in §6 of Chap. 8 (see particularly K Exercises 6 and 7 at the end of that section), which implies that whenever (Ωj ) is an antisymmetric matrix of imaginary numbers (hence a self-adjoint matrix), then (6.33)

˜ 0 (t, x, 0) = (4πt)−n/2 det K



Ωt/4 1/2 −(f (Ωt/4)x,x)/4t e , sinh(Ωt/4)

where f (s) = 2s coth 2s. Now it is straightforward to verify that this formula is also valid whenever Ω is a nilpotent element of any commutative ring (assumed to be an algebra over C), as in the case (6.32), where Ω is an End(Tx0 M )-valued 2-form. Evaluating (6.33) at x = 0 gives (6.34)

˜ 0 (t, 0, 0) = (4πt)−n/2 det K



Ωt/4 1/2 . sinh(Ωt/4)

When Ω is the curvature 2-form of M , with its Riemannian metric, this is to be interpreted in the same way as the characteristic classes discussed in §7 of ˆ Appendix C. The A-genus of M is defined to be this determinant, at t = 1/2πi: (6.35)

ˆ A(M ) = det



Ω/8πi 1/2 . sinh(Ω/8πi) ˜

The Cl[2k] -component of the t0 -coefficient in the expansion of e−tL is (−2i)−k times the 2k-form part of the product of (6.35) with Tr e−Φ/2πi , where Φ is the End F -valued curvature 2-form of the connection on F . This is also a characteristic class; we have the Chern character: (6.36)

Ch(F ) = Tr e−Φ/2πi .

This completes the proof of Theorem 6.1.

358 10. Dirac Operators and Index Theory

Exercises ˆ 1. Write out the first few terms in the expansion of the formula (6.35) for A(M ), such as forms of degree 0, 4, 8. 2. If M is a compact, oriented, four-dimensional manifold, show that  1 ˆ p1 (T M ), (6.37)

A(M ), [M ] = − 24 M

where p1 is the first Pontrjagin class, defined in §7 of Appendix C. ˆ ), [M ] = −1/8. Deduce that CP 2 has no spin struc3. If M = CP 2 , show that A(M ture. 4. If M is a spin manifold with positive scalar curvature, to which Proposition 4.8 applies, ˆ show that A(M ), [M ] = 0. What can you deduce about the right side of (5.3) in such a case? Consider particularly the case where dim M = 4. 5. Let Fj → M be complex vector bundles. Show that Ch(F1 ⊕ F2 ) = Ch(F1 ) + Ch(F2 ), Ch(F1 ⊗ F2 ) = Ch(F1 ) ∧ Ch(F2 ). 6. If F → M is a complex line bundle, relate Ch(F ) to the first Chern class c1 (F ), defined in §7 of Appendix C.

7. The Chern–Gauss–Bonnet theorem Here we deduce from the Atiyah–Singer formula (5.3) the generalized Gauss– Bonnet formula expressing as an integrated curvature the Euler characteristic χ(M ) of a compact, oriented Riemannian manifold M , of dimension n = 2k. As we know from Hodge theory, χ(M ) is the index of (7.1)

d + d∗ : Λeven M −→ Λodd M.

This is an operator of Dirac type, but it is not actually a twisted Dirac operator of the form (3.34), even when M has a spin structure. Rather, a further twist in the twisting procedure is required. Until near the end of this section, we assume that M has a spin structure. With V = R2k , we can identify CΛ∗ V , both as a linear space and as a Clifford module, with Cl(2k). Recall the isomorphism (2.14): (7.2)

μ : Cl(2k) −→ End S,

where S = S(2k) = S+ (2k) ⊕ S− (2k). This can be rewritten as Cl(2k) ≈ S ⊗ S  .

7. The Chern–Gauss–Bonnet theorem

359

Now if Cl(2k) acts on the left factor of this tensor product, then there is a twisted Dirac operator  (7.3)

DS− 0

0 DS+

 ,

produced from the grading S ⊗ S  = (S+ ⊗ S  ) ⊕ (S− ⊗ S  ), but this is not the operator (7.1). Rather, it is the signature operator. To produce (7.1), we use the identities CΛeven V = Cl0 (2k) and CΛodd V = Cl1 (2k). Recall the isomorphism (3.15): μ : Cl0 (2k) −→ End S+ ⊕ End S− .

(7.4) We rewrite this as

  ) ⊕ (S− ⊗ S− ). Cl0 (2k) ≈ (S+ ⊗ S+

(7.5)

Similarly, we have an isomorphism (7.6)

μ : Cl1 (2k) −→ Hom(S+ , S− ) ⊕ Hom(S− , S+ ),

which we rewrite as (7.7)

  ) ⊕ (S+ ⊗ S− ). Cl1 (2k) ≈ (S− ⊗ S+

It follows from this that the operator (7.1) is a “twisted” Dirac operator of the form   0 DS− ⊕ DS+ + − . (7.8) D= DS+ ⊕ DS− 0 +



In other words, the index χ(M ) of (7.1) is a difference: Index DS+ − Index DS+ , +



since Index DS− = − Index DS+ . Furthermore, this difference is respected in the − − local index formula, an observation that will be useful later when we remove the hypothesis that M have a spin structure. The Atiyah–Singer formula (5.3) thus yields (7.9)

    ˆ ) − Ch(S− )], [M ] . χ(M ) = A(M )[Ch(S+

360 10. Dirac Operators and Index Theory

The major step from here to the Chern–Gauss–Bonnet theorem is to produce a   ) − Ch(S− ) in purely differential geometric 2k-form on M expressing Ch(S+ terms, independent of a spin structure. If π± are the representations of Spin(2k) on S± , dπ± the derived representa˜ the spin(2k)-valued curvature form on P˜ , then tions of spin(2k), and Ω ˜

Ch(S± ) = Tr e−dπ± (Ω)/2πi ,

(7.10)

a sum of even-order forms formally related to the characters of π± , χ± (g) = Tr π± (g),

(7.11)

g ∈ Spin(2k).

Note that dim S+ = dim S− implies χ+ (e) − χ− (e) = 0. It is a fact of great significance that the difference χ+ (g) − χ− (g) vanishes to order k at the identity element e ∈ Spin(2k). More precisely, we have the following. Take X ∈ spin(2k) ≈ so(2k), identified with a real, skew-symmetric matrix, X = (Xij ); there is the exponential map Exp : spin(2k) → Spin(2k). The key formula is given as follows: Lemma 7.1. For X ∈ so(2k), (7.12)

  lim t−k χ+ (Exp tX) − χ− (Exp tX) = (−i)k Pf (X).

t→0

Here, Pf : so(2k) → R is the Pfaffian, defined as follows. Associate to X ∈ so(2k) the 2-form (7.13)

ξ = ξ(X) =

1 Xij ei ∧ ej , 2

e1 , . . . , e2k denoting an oriented orthonormal basis of R2k . Then (7.14)

k! (Pf X)e1 ∧ · · · ∧ e2k = ξ ∧ · · · ∧ ξ

(k factors).

It follows from this definition that if T : R2k → R2k is linear, then T ∗ ξ(X) = ξ(T t XT ), so (7.15)

Pf (T t XT ) = (det T )Pf (X).

Now any X ∈ so(n) can be written as X = T t AT , where T ∈ SO(n), and A is a sum of 2 × 2, skew-symmetric blocks, of the form  Aν =

 0 aν , −aν 0

aν ∈ R.

Thus ξ(A) = a1 e1 ∧ e2 + · · · + ak e2k−1 ∧ e2k , so

7. The Chern–Gauss–Bonnet theorem

361

Pf (X) = Pf (A) = a1 · · · ak .

(7.16) It follows that (7.17)

Pf (X)2 = det X.

We also note that, if one uses Clifford multiplication rather than exterior multiplication, on k factors of ξ(X), then the result has as its highest-order term k!(Pf X)e1 · · · e2k . In other words, in terms of the map τ : Cl(2k) → C of (7.7), (7.18)

k!(Pf X) = τ (ξ · · · ξ),

with k factors of ξ. To prove Lemma 7.1, note that the representation π = π+ ⊕ π− of Spin(2k) on S = S+ ⊕ S− is the restriction to Spin(2k) of the representation μ of Cl(2k) on S characterized by (2.11). Consequently, in view of Exercise 4 in §3, (7.19)

   Tr π+ (Exp tX) − Tr π− (Exp tX) = Str μ et Xij ei ej /4 ,

where Str stands for the supertrace, as in (6.6). This can be evaluated by  Berezin’s formula, (6.10), as (−2i)k times the coefficient of ν = e1 · · · e2k in et Xij ei ej /4 . Now the lowest power of t in the power-series expansion of this quantity, which has a multiple of ν as coefficient, is the kth power; the corresponding term is (7.20)

k tk 1 t k  Xij ei ej = k Pf X ν + · · · , k k! 4 2

by (7.18). Thus, by (6.10), the leading term in the expansion in powers of t of (7.19) is (−it)k (Pf X), which proves (7.12). We remark that the formula (7.12) plays a central role in the proof of the index formula for (twisted) Dirac operators, in the papers of Bismut [Bi] and of Berline– Vergne [BV]. In §8 of Appendix C, it is shown that the Pfaffian arises directly for the generalized Gauss–Bonnet formula for a hypersurface M ⊂ R2k+1 when one expresses the degree of the Gauss map M → S 2k as an integral of the Jacobian determinant of the Gauss map and evaluates this Jacobian determinant using the Weingarten formula and Gauss’ Theorema Egregium. From (7.12) it follows that (7.21)

  ) − Ch(S− ) = (2π)−k Pf (Ω). Ch(S+

This is defined independently of any spin structure on M . Since locally any manifold has spin structures, the local index formula of §6 provides us with the following conclusion.

362 10. Dirac Operators and Index Theory

Theorem 7.2. If M is a compact, oriented Riemannian manifold of dimension n = 2k, then the Euler characteristic χ(M ) satisfies the identity χ(M ) = (2π)−k

(7.22)

 Pf (Ω). M

Proof. It remains only to note that in the formula ˆ (2π)−k A(M )Pf (Ω), [M ] = χ(M ), since the factor Pf(Ω) is a pure form of degree 2k = n, only the leading term 1 in ˆ A(M ) contributes to this product.

Exercises 1. Verify that when dim M = 2, the formula (7.22) coincides with the classical Gauss– Bonnet formula:  K dV = 2πχ(M ). (7.23) M

2. Work out “more explicitly” the formula (7.22) when dim M = 4. Show that 

1 |R|2 − 4|Ric|2 + S 2 dV, (7.24) χ(M ) = 2 8π M

where R is the Riemann curvature tensor, Ric the Ricci tensor, and S the scalar curvature. For some applications, see [An]. 3. Evaluate (7.19); show that (7.25)

sinh tX/2 1/2  Pf X. Str μ et Xij ei ej /4 = (−it)k det tX/2

(Hint: Reduce to the case where X is a sum of 2 × 2 blocks.) + , using 4. Apply Theorem 6.1 to give a formula for the index of the signature operator DH + − the representation (7.3) of DH ⊕ DH as a twisted Dirac operator. Justify the formula when M has no spin structure. Show that, if M is a compact, oriented 4-manifold, then (7.26)

+ ˆ = −8 A(M ), [M ]. Index DH

(Hint: Take a peek in [Roe].)

8. Spinc manifolds Here we consider a structure that arises more frequently than a spin structure, namely a spinc structure. Let M be an oriented Riemannian manifold of dimen-

8. Spinc manifolds

363

sion n, P → M the principal SO(n)-bundle of oriented orthonormal frames. A spinc structure on M is a principal bundle Q → M with structure group (8.1)

Spinc (n) = Spin(n) × S 1 /{(1, 1), (−1, −1)} = G.

Note that {−1, 1} ⊂ Spin(n) is the pre-image of the identity element of SO(n). For this principal bundle Q, we require that there be a bundle map ρ : Q → P , commuting with the natural Spin(n) actions on Q and P . There is a natural injection Spin(n) → Spinc (n), as a normal subgroup. Note that taking the quotient R = Q/ Spin(n) produces a principal S 1 -bundle, over which Q projects. We display the various principal bundles:

(8.2)

Q −−−−→ ⏐ ⏐ 

R ⏐ ⏐ 

P −−−−→ M There is a topological obstruction to the existence of a spinc structure on M , though it is weaker than the obstruction to the existence of a spin structure. We refer to [LM] for these topological considerations; we will give some examples of spinc -manifolds later in this section. The standard representation of S 1 on C produces a complex line bundle (8.3)

L −→ M.

+ − ⊕ D1/2 of Spin(n) on S(2k) Suppose n = 2k. Recall the representation D1/2 from (3.19). If we take the product with the standard representation of S 1 on C, this is trivial on the factor group appearing in (8.1), so we get a representation of + − ⊕ D1/2 . This representaSpinc (n) on S(2k), which we continue to denote D1/2 tion produces a vector bundle over M , which we continue to call a spinor bundle:

(8.4)

S(Q) = S+ (Q) ⊕ S− (Q);

S± (Q) = Q ×D± S± (2k). 1/2

In case n is odd, we have instead the bundle of spinors constructed from the representation (3.24) of Spin(n), via the same sort of procedure. As in §3, we will be able to define a Dirac operator on C ∞ (M, S(Q)) in terms of a connection on Q, which we now construct. The Levi–Civita connection on M defines an so(n)-valued form θ0 on P , which pulls back to an so(n)-valued form θ0 on Q. Endow the bundle R = Q/ Spin(n) → M with a connection θ1 , so L → M gets a metric connection. Then θ1 pulls back to an iR-valued form θ1 on Q, and (8.5)

θ = θ0 + θ1

364 10. Dirac Operators and Index Theory

defines a spinc (n)-valued form on Q, which gives rise to a connection on Q. This leads to a connection on the spinor bundle S(Q) → M , and the analogues of Propositions 3.4 and 3.5 hold. Thus we produce the Dirac operator (8.6)

D = i m ◦ ∇ : C ∞ (M, S) −→ C ∞ (M, S).

More generally, if E → M is a vector bundle with a metric connection, one gets a Clifford connection on S(Q) ⊗ E and hence a twisted Dirac operator (8.7)

DE : C ∞ (M, S ⊗ E) −→ C ∞ (M, S ⊗ E).

If dim M is even, DE maps sections of S± ⊗ E to sections of S∓ ⊗ E. We consider some ways in which spinc structures arise. First, a spin structure gives rise to a spinc structure. Indeed, if the frame bundle P → M lifts to a principal Spin(n)-bundle P˜ → M , then Q can be taken to be the quotient of the product bundle P˜ ×S 1 → M by the natural Z2 -action on the fibers. The canonical flat connection on S 1 × M → M is used, to provide Q → M with a connection, and then the Dirac operator (8.6) defined by Q → M coincides with that defined by P˜ → M . Another family of examples of spinc structures of considerable importance arises as follows. Suppose M is a manifold of dimension n = 2k with an almost complex structure, J : Tx M → Tx M, J 2 = −I. Endow M with a Riemannian metric such that J is an isometry. T M , which is (T M, J) regarded as a complex vector bundle of fiber dimension k, then acquires a natural Hermitian metric, as in (2.9). The associated frame bundle F → M is a principal U (k) bundle. Note that U(k) ≈ SU(k) × S 1 /Γ,

(8.8)

where Γ = {(I, 1), (−I, −1)}. Since SU(k) is simply connected, the inclusion U(k) → SO(n) yields a uniquely defined homomorphism SU(k) −→ Spin(n),

(8.9) and hence a homomorphism (8.10)

U(k) ≈ SU(k) × S 1 /Γ −→ Spinc (n).

From the bundle F → M , this gives rise to a principal Spinc (n) bundle Q → M . In this case, the map U(k) → Spinc (n) → S 1 is given by the determinant, det : U(k) → S 1 . The principal S 1 -bundle R → M is obtained by taking the quotient of the principal U(k)-bundle F by the action of SU(k). The associated line bundle L → M is seen to be (8.11)

L = ΛkC T .

8. Spinc manifolds

365

Other geometrical structures give rise to spinc structures; we refer to [LM] for more on this. We mention the following: namely, any oriented hypersurface in a spinc manifold inherits a natural spinc structure. In this fashion the sphere bundle S ∗ M over a Riemannian manifold gets a spinc structure, as a hypersurface of T ∗ M , which can be given an almost complex structure. Though a spinc structure is more general than a spin structure, it is a very significant fact that a spinc structure in turn gives rise to a spin structure, in the following circumstance. Namely, suppose the principal S 1 -bundle R → M lifts to a double cover (8.12)

˜ −→ M, R

corresponding to the natural two-to-one surjective homomorphism sq : S 1 → S 1 . This is equivalent to the hypothesis that the line bundle L → M possess a “square root” λ → M : (8.13)

λ ⊗ λ = L.

˜ → M by the natural action of S 1 on each In such a case, the quotient of Q × R factor gives a lift of Q to a principal Spin(n) × S 1 -bundle (8.14)

˜ −→ M. Q

Then the quotient (8.15)

˜ 1 −→ M P˜ = Q/S

defines a spin structure on M . The vector bundles S(Q) and S(P˜ ) are related by (8.16)

S(Q) = S(P˜ ) ⊗ λ.

Furthermore, the connection on S(Q) defined above coincides with the product connection on S(P˜ ) ⊗ λ arising from the natural connections on each factor. 0 are respectively the twisted Dirac operator associated Therefore, if DE and DE with a vector bundle E → M (given a metric connection) via the spinc and spin structures described above, then (8.17)

0 . DE = Dλ⊗E

This holds, we recall, provided L has a square root λ. One consequence of this is the following extension of the Weitzenbock formula (4.24). Namely, if DE is the twisted Dirac operator on S(Q) ⊗ E described there, then applying (4.24) to the right side of (8.17) gives (8.18)

2 = ∇∗ ∇ + K, DE

366 10. Dirac Operators and Index Theory

with (8.19)

Kϕ =

1 1 1 Sϕ − vi vj ω λ (ei , ej )ϕ − vi vj RE (ei , ej )ϕ, 4 2 i,j 2 i,j

or equivalently (8.20)

Kϕ =

1 1 1 Sϕ − vi vj ω L (ei , ej )ϕ − vi vj RE (ei , ej )ϕ, 4 4 i,j 2 i,j

where, as before, {ej } is a local orthonormal frame of vector fields on M , with dual frame field {vj }. Here ω λ is the curvature form of the line bundle λ and ω L that of L. Now locally there is no topological obstruction to the existence of the lift (8.12). Consequently, the identity (8.20) holds regardless of whether L possesses a global square root. Therefore, the proof of the local index formula given in §6 extends to this case. Furthermore, we have the pointwise identity of forms: (8.21)

Ch(λ ⊗ E) = ec1 (λ) Ch(E),

c1 (λ) =

1 c1 (L), 2

where c1 is the first Chern class, defined in §7 of Appendix C. Therefore, we have the following extension of Theorem 5.1: Theorem 8.1. If M is a compact Riemannian manifold of dimension n = 2k with spinc structure and DE : C ∞ (M, S+ ⊗ E) → C ∞ (M, S− ⊗ E) is a twisted Dirac operator, then (8.22)

  ˆ Index DE = ec1 (L)/2 Ch(E)A(M ), [M ] ,

where L is the line bundle (8.3), and c1 (L) is its first Chern class. The index formula for twisted Dirac operators on spinc manifolds furnishes a tool with which one can evaluate the index of general elliptic pseudodifferential operators. Indeed, let P be any elliptic pseudodifferential operator (of order m), (8.23)

P : C ∞ (M, E0 ) −→ C ∞ (M, E1 ),

Ej → M being vector bundles. Then, as seen in Chap. 7, we have the principal symbol (8.24)

˜0 , E ˜1 )), σP ∈ C ∞ (S ∗ M, Hom(E

˜j → S ∗ M being the pull-backs of Ej → M . The ellipticity of P is equivalent E to σP being an isomorphism at each point of S ∗ M . Now, we can construct a new

9. The Riemann–Roch theorem

367

 , the double of the ball bundle B ∗ M , as follows. We vector bundle E over BM ˜ let Ej also denote the pull-back of Ej to B ∗ M , and, when the two copies of  , we also glue together E ˜0 and B ∗ M are glued together along S ∗ M to form BM ˜1 , over S ∗ M , using the isomorphism (8.24). The construction of E → BM  by E  can be given a this process is known as the “clutching construction.” Now BM Riemannian metric, and also a spinc structure, arising from the almost complex structure on B ∗ M . If E is endowed with a connection, one obtains a twisted Dirac  . The following result, together with the formula for Index operator DE on BM DE given by Theorem 8.1, provides the general Atiyah–Singer index formula. Theorem 8.2. If P is an elliptic pseudodifferential operator, giving rise to a twisted Dirac operator DE by the clutching construction described above, then Index P = Index DE .

(8.25)

The proof of this result will not be given here; it involves use of the Bott periodicity theorem. Related approaches, computing Index P from a knowledge of the index of twisted signature operators, are discussed in [Pal] and [ABP]. A refinement of (8.25), involving an identity in K-homology is established in [BDT].

Exercises 1. Consider the following zero-order pseudodifferential operator on L2 (S 1 ): Q = Mf P + Mg (I − P ), where P is the projection P

∞ 

inθ

cn e

−∞

=

∞ 

cn einθ .

0

We assume f and g are smooth, complex-valued functions; Mf u = f u. If f and g are nowhere vanishing on S 1 , Q is elliptic. A formula for its index is produced in Exercises 1–5 of Chap. 4, §3. Construct the associated twisted Dirac operator DE , acting on sections of a vector

1 ≈ T2 . Evaluate the index of DE using Theorem 8.1, bundle over the manifold BS and verify the identity (8.25) in this case.

9. The Riemann–Roch theorem In this section we will show how the index formula (8.22) implies the classical Riemann–Roch formula on compact Riemann surfaces, and we also discuss some of the implications of that formula. For implications of generalizations of the Riemann–Roch formula to higher-dimensional, compact, complex manifolds, which also follows from (8.22), see [Har] and [Hir].

368 10. Dirac Operators and Index Theory

Let M be a compact two-dimensional manifold, with a complex structure, defined by J : Tx M → Tx M, J 2 = −I. As shown in Chap. 5, §10, this a priori “almost complex” structure automatically gives rise to holomorphic charts on M in this dimension. We can put a Riemannian metric and an orientation on M such that J is an isometry on each tangent space, counterclockwise rotation by 90◦ . Then T M gets the structure of a complex line bundle, which we denote T M , with a Hermitian metric. We have the dual line bundle T  M . Note that the Hermitian metric on T M yields a Hermitian metric on T  M and also produces a conjugate linear bundle isomorphism of T M with T  M . We also define the complex line bundle T M to be the tangent bundle T M with complex structure given  by −J and T M to be its dual. A function u ∈ C ∞ (M ) is holomorphic if ∂u/∂z = 0 in any local holomorphic coordinate system and is antiholomorphic if ∂u/∂z = 0. We denote the space of holomorphic functions on an open set U ⊂ M by OU , and antiholomorphic functions by OU . There are invariantly defined operators (9.1)

∂ : C ∞ (M ) −→ C ∞ (M, T  ),



∂ : C ∞ (M ) −→ C ∞ (M, T ),

given as follows. If X is a real vector field, namely, a section of T M , set (9.2)

∂X u =

1 Xu − i(JX)u , 2

∂X u =

1 Xu + i(JX)u . 2

Note that (9.3)

∂JX u = i∂X u,

∂ JX u = −i∂ X u,

which justifies (9.1). In addition to holomorphic functions, we also have the notion of a holomorphic line bundle over M. Given a complex line bundle L → M, let {Uj } be a covering of M by geodesically convex sets. A holomorphic structure on L is a choice of nowhere-vanishing sections sj of L over Uj such that sj = σjk sk on Ujk = Uj ∩ Uk , with σjk holomorphic complex-valued functions. Similarly, a choice of nowhere-vanishing sections tj of L over Uj such that tj = τjk tk on Ujk , τjk antiholomorphic, gives L the structure of an antiholomorphic line bundle. The bundle T M has a natural structure of a holomorphic line bundle; in a local holomorphic coordinate system {Uj }, let sj = ∂/∂x. T  is a holomorphic line bundle with sj = dx. To see this, note that if ψ : U → V is a holomorphic map relating two local coordinate charts on M, ψ = u+iv, then (Dψ)(∂/∂x) is equal to  ∂u ∂v ∂ ∂u ∂ ∂v ∂ ∂v  ∂ ∂ψ ∂ ∂u ∂ + = + J = +i = . ∂x ∂x ∂x ∂y ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x

9. The Riemann–Roch theorem

369

Here, the first two quantities are regarded as local sections of T M , the last two  as local sections of T M . Similarly, T and T have natural structures as antiholomorphic line bundles, using the same choices of local sections as above. It is also common to identify T M and T M with complementary subbundles of the complexified tangent bundle TC M = C ⊗ T M , a complex vector bundle whose fibers are two-dimensional complex vector spaces. Namely, the local section ∂/∂x of T M is identified with (1/2)(∂/∂x − i∂/∂y) = ∂/∂z to yield T M → TC M and it is identified with (1/2)(∂/∂x + i∂/∂y) = ∂/∂z to yield T M → TC M . More generally, these two maps are given respectively by X → (1/2)(X − iJX) and X → (1/2)(X + iJX). Identifying T M and T M with their images in TC M , we have TC M = T M ⊕ T M. Similarly, we have the complexified cotangent bundle TC∗ M = C ⊗ T ∗ M , and  natural injections T  M → TC∗ M, T M → TC∗ M , so that 

TC∗ M = T  M ⊕ T M. In this case, dx is mapped respectively to (dx + idy)/2 = dz/2 and to (dx − idy)/2 = dz/2. We use the following common notation for these line bundles equipped with these extra structures: (9.4)

T = κ−1 ,

T  = κ,

T = κ−1 ,



T = κ.

We can rewrite (9.1) as ∂ : C ∞ (M ) −→ C ∞ (M, κ),

∂ : C ∞ (M ) −→ C ∞ (M, κ).

We note that κ−1 and κ are isomorphic as C ∞ -line bundles; κ is called the canonical bundle. More generally, if L → M is any holomorphic line bundle, we have a naturally defined operator (9.5)

∂ : C ∞ (M, L) −→ C ∞ (M, L ⊗ κ),

defined as follows. Pick any local (nowhere-vanishing) holomorphic section S of L, for example, S = sj on Uj , used in the definition above of holomorphic structure. Then an arbitrary section u is of the form u = vS, v complex-valued, and we set (9.6)

∂u =

∂v S ⊗ dz. ∂z

370 10. Dirac Operators and Index Theory

It is easy to see that this is independent of the choice of holomorphic section S or of local holomorphic coordinate system. Sometimes, to emphasize the dependence of (9.5) on L, we denote this operator by ∂ L . The operator (9.5) is a first-order, elliptic differential operator, and the Riemann–Roch formula is a formula for its index. The kernel of ∂ L in (9.5) consists of holomorphic sections of L; namely, sections u such that, with respect to the defining sections sj on Uj , u = vj sj with vj holomorphic. We denote this space of holomorphic sections by O(L) = ker ∂ L .

(9.7)

The significance of the Riemann–Roch formula lies largely in its use as a tool for understanding as much as possible about the spaces (9.7). The cokernel of ∂ L in (9.5) can be interpreted as follows. The Hermitian metric on T gives rise to a trivialization of κ ⊗ κ and to a duality of L2 (M, L ⊗ κ) with L2 (M, L−1 ⊗ κ). With respect to this duality, the adjoint of ∂ L is (9.8)

−∂ : C ∞ (M, L−1 ⊗ κ) −→ C ∞ (M, L−1 ⊗ κ ⊗ κ).

Consequently, (9.9)

Index ∂ L = dim O(L) − dim O(L−1 ⊗ κ).

The Riemann–Roch theorem will produce a formula for (9.9) in terms of topological information, specifically, in terms of c1 (L) and c1 (κ). Recall that M has a natural spinc structure, arising from its complex structure. We will produce a twisted Dirac operator on M whose index is the same as that of ∂ L . In fact, when the construction of the spinor bundle made in §8 is specialized to the case at hand, we get (9.10)

S+ = 1,

S− = T ≈ κ,

where 1 denotes the trivial line bundle over M . Furthermore, the line bundle denoted as L in (8.11) is T ≈ κ−1 . If L is a (holomorphic) line bundle over M , we give L a Hermitian metric and metric connection ∇. Then the twisted Dirac operator (9.11)

DL : C ∞ (M, L) −→ C ∞ (M, L ⊗ κ)

is given by (9.12)

DL u, X =

1 ∇X u + i∇JX u , 2

for X a section of T M , identified with T M ≈ κ , noting that (9.13)

DL u, JX = −DL u, X.

9. The Riemann–Roch theorem

371

It is easy to see that ∂ L and DL are differential operators with the same principal symbol. Disregarding the question of whether one can pick a connection on L making these operators equal, we clearly have (9.14)

Index ∂ L = Index DL .

Now applying the index formula (8.22) to the right side of (9.14) gives (9.15)

  ˆ ), [M ] . Index DL = e−c1 (κ)/2 Ch(L)A(M

ˆ Since A(M ) is 1 plus a formal sum of forms of degree 4, 8, . . . , we obtain (9.16)

1 Index DL = c1 (L)[M ] − c1 (κ)[M ]. 2

Putting together (9.9), (9.14), and (9.16) gives the Riemann–Roch formula: Theorem 9.1. If L is a holomorphic line bundle over a compact Riemann surface M , with canonical bundle κ, then (9.17)

1 dim O(L) − dim O(L−1 ⊗ κ) = c1 (L)[M ] − c1 (κ)[M ]. 2

According to the characterization of the Chern classes given in §7 of Appendix C, if L has a connection with curvature 2-form ωL , then (9.18)

1 c1 (L)[M ] = − 2πi

 ωL . M

In particular, c1 (κ)[M ] is given by the Gauss–Bonnet formula: (9.19)

c1 (κ)[M ] = −χ(M ) = 2g − 2,

where χ(M ) is the Euler characteristic and g is the genus of M . We begin to draw some conclusions from the Riemann–Roch formula (9.17). First, for the trivial line bundle 1 we clearly have (9.20)

dim O(1) = 1,

assuming M is connected, since holomorphic functions on M must be constant. If we apply (9.17) to L = κ, using κ−1 ⊗ κ = 1 and the formula (9.19), we obtain (9.21)

dim O(κ) = g.

372 10. Dirac Operators and Index Theory

The space O(κ) is called the space of holomorphic 1-forms, or “Abelian differentials.” We claim there is a decomposition (9.22)

H1 (M ) = O(κ) ⊕ O(κ),

of the space H1 (M ) of (complex) harmonic 1-forms on M into a direct sum of O(κ) and the space O(κ) of antiholomorphic sections of κ. In fact, the Hodge star operator ∗ : Λ1 M → Λ1 M , extended to be C-linear on C ⊗ Λ1 M , acts on H1 (M ), with ∗∗ = −1, and O(κ) and O(κ) are easily seen to be the i and −i eigenspaces of ∗ in H1 (M ). Furthermore, there is a conjugate linear isomorphism C : O(κ) −→ O(κ)

(9.23)

given in local holomorphic coordinates by C u(z) dz = u(z) dz. Now (9.22) and (9.23) imply (9.24)

dim O(κ) =

1 1 dim H1 (M ) = dim H1 (M, C), 2 2

where H1 (M, C) is a deRham cohomology group, and the last identity is by Hodge theory. Granted that dim H1 (M, C) = 2g, this gives an alternative derivation of (9.21), not using the Riemann–Roch theorem. Compare the derivation of (10.28) in Chapter 5. The Hodge theory used to get the last identity in (9.24) is contained in Proposition 8.3 of Chap. 5. Actually, in §8 of Chap. 5, H1 denoted the space of real harmonic 1-forms, which was shown to be isomorphic to the real deRham cohomology group H1 (M, R), which in turn was denoted H1 (M ) there. Just for fun, we note the following. Suppose that instead of (9.17) one had in hand the weaker result (9.25)

dim O(L) − dim O(L−1 ⊗ κ) = Ac1 (L)[M ] + Bc1 (κ)[M ],

with constants A and B that had not been calculated. Then using the results (9.19) and (9.21), one can determine A and B. Indeed, substituting L = 1 into (9.25) gives 1 − g = B(2g − 2), while substituting L = κ in (9.25) gives g − 1 = (A + B)(2g − 2). As long as g = 1, this forces A = 1, B = −1/2. The g = 1 case would also follow if one knew that (9.25) held with constants independent of M . Before continuing to develop implications of the Riemann–Roch formula, we note that, in addition to O(L), it is also of interest to study M(L), the space of meromorphic sections of a holomorphic line bundle. The following is a fundamental existence result.

9. The Riemann–Roch theorem

373

Proposition 9.2. If L → M is a holomorphic line bundle, there exist nontrivial elements of M(L). Proof. The operator (9.5) extends to (9.26)

∂ : H s+1 (M, L) −→ H s (M, L ⊗ κ),

which is Fredholm. There are elements v1 , . . . , vK ∈ C ∞ (M, L−1 ⊗κ) such that, for all s ∈ R, if f ∈ H s (M, L ⊗ κ) and f, vj  = 0 for j = 1, . . . , K, then there exists u ∈ H s+1 (M, L) such that ∂u = f . Now, for s < −1, there is a finite linear combination of “deltafunctions,” in H s (M, L ⊗ κ), orthogonal to these vj . Denote such an f by f = aj δpj . Then let u ∈ H s+1 (M, L) satisfy ∂u = f . In particular, ∂u = 0 on the complement of a finite set of points. Near each p ∈ supp f, u looks like the Cauchy kernel, so u is a nontrivial meromorphic section of L. Such an existence result need not hold for O(L); in Corollary 9.4 we will see a condition that guarantees O(L) = 0. Such a result should not be regarded in a negative light; indeed knowing that O(L) = 0 for some line bundles can give important information on O(L1 ) for certain other line bundles, as we will see. Any nontrivial u ∈ M(L) will have a finite number of zeros and poles. If p is a zero of u, let νu (p) be the order of the zero; if p is a pole of u, let −νu (p) be the order of the pole. We define the “divisor” of u ∈ M(L) to be the formal finite sum (9.27)

ϑ(u) =



νu (p) · p

p

over the set of zeros and poles of u. It is a simple exercise in complex analysis that ifu is a nontrivial meromorphic function on M (i.e., an element of M(1)), then p νu (p) = 0. The following is a significant generalization of that. Proposition 9.3. If L → M is a holomorphic line bundle and u ∈ M(L) is nontrivial, then  νu (p). (9.28) c1 (L)[M ] = p

Proof. The left side of (9.28) is given by (9.18), where ωL is the curvature 2-form associated to any connection on L. We will use the formula (9.29)

1 − 2πi

 ωL = Index X, M

for any X ∈ C ∞ (M, L) with nondegenerate zeros, proved in Appendix C, Proposition 5.4, as a variant of the Gauss–Bonnet theorem. The section X will be

374 10. Dirac Operators and Index Theory

constructed from u ∈ M(L) as follows. Except on the union of small neighborhoods of the poles of u, we take X = u. Near the poles of u, write u = vS, S a nonvanishing holomorphic section of L defined on a neighborhood of such poles, v meromorphic. Pick R > 0 sufficiently large, and replace u by (R2 /v)S, where |v| ≥ R. Smooth out X near the loci |v| = R. Then the formula (9.29) for X is equivalent to the desired formula, (9.28). The following is an immediate consequence. Corollary 9.4. If L → M is a holomorphic line bundle with c1 (L)[M ] < 0, then every nontrivial u ∈ M(L) has poles; hence O(L) = 0. Note that if c1 (L)[M ] = 0 and O(L) = 0, by (9.28) we have that any u ∈ O(L) not identically zero is nowhere vanishing. Thus we have (9.30)

c1 (L)[M ] = 0, O(L) = 0 =⇒ L is trivial holomorphic line bundle.

To relate Corollary 10.4 to the Riemann–Roch formula (9.17), we note that since dim O(L−1 ⊗ κ) ≥ 0, (9.17) yields Riemann’s inequality: dim O(L) ≥ c1 (L)[M ] − g + 1.

(9.31) In view of the identities (9.32)

c1 (L1 ⊗ L2 )[M ] = c1 (L1 )[M ] + c1 (L2 )[M ], c1 (L−1 )[M ] = −c1 (L)[M ],

we see that (9.33)

c1 (L)[M ] > 2g − 2 =⇒ O(L−1 ⊗ κ) = 0.

Thus we have the following sharpening of Riemann’s inequality: Proposition 9.5. If M has genus g and c1 (L)[M ] > 2g − 2, then (9.34)

dim O(L) = c1 (L)[M ] − g + 1.

Generalizing (9.27), we say a divisor on M is a finite formal sum (9.35)

ϑ=



ν(p) · p,

p

ν(p) taking values in Z. One defines −ϑ and the sum of two divisors in the obvious fashion. To any divisor ϑ we can associate a holomorphic line bundle, denoted Eϑ ; one calls Eϑ a divisor bundle. To construct Eϑ , it is most convenient to use the method of transition functions. Cover M with holomorphic coordinate sets Uj ,

9. The Riemann–Roch theorem

375

pick ψj ∈ MUj , having a pole of order exactly |ν(p)| at p, if ν(p) < 0, a zero of order exactly ν(p) if ν(p) > 0 (provided p ∈ Uj ), and no other poles or zeros. The transition functions ϕjk = ψk−1 ψj

(9.36)

define a holomorphic line bundle Eϑ . The collection {ψj , Uj } defines a meromorphic section ψ ∈ M(Eϑ )

(9.37) and

−ϑ(ψ) = ϑ.

(9.38) Thus Proposition 9.3 implies (9.39)

c1 (Eϑ ) = −



ν(p) = ϑ,

p

where the last identity defines ϑ. Divisor bundles help one study meromorphic sections of one line bundle in terms of holomorphic sections of another. A basic question in Riemann surface theory is when can one construct a meromorphic function on M (more generally, a meromorphic section of L) with prescribed poles and zeros. A closely related question is the following. Given a divisor ϑ on M , describe the space (9.40)

M(L, ϑ) = {u ∈ M(L) : ϑ(u) ≥ ϑ},

 μ(p) · p where ϑ1 ≥ ϑ means ϑ1 − ϑ ≥ 0, that is, all integers μ(p) in ϑ1 − ϑ = are ≥ 0. When L = 1, we simply write M(ϑ) for the space (9.40). A straightforward consequence of the construction of Eϑ is the following: Proposition 9.6. There is a natural isomorphism (9.41)

M(L, ϑ) ≈ O(L ⊗ Eϑ ).

Proof. The isomorphism takes u ∈ M(L, ϑ) to uψ, where ψ is described by (9.36)–(9.37). We can hence draw some conclusions about the dimension of M(L, ϑ). From the identity (9.34) we have (9.42) c1 (L)[M ] + ϑ > 2g − 2 =⇒ dim M(L, ϑ) = c1 (L)[M ] + ϑ − g + 1, and, in particular,

376 10. Dirac Operators and Index Theory

(9.43)

ϑ > 2g − 2 =⇒ dim M(ϑ) = ϑ − g + 1.

Also one has general inequalities, as a consequence of (9.31). Now Corollary 9.4 and Proposition 9.5 specify precisely dim O(L) provided either c1 (L)[M ] < 0 or c1 (L)[M ] > 2g − 2, but (9.31) gives weaker information if 0 ≤ c1 (L)[M ] ≤ 2g − 2; in fact, for c1 (L)[M ] ≤ g − 1, it gives no information at all. In this range the lower bound (9.31) can be complemented by an upper bound. For example, (9.30) implies (9.44)

c1 (L)[M ] = 0 =⇒ dim O(L) = 0 or 1.

We will show later that both possibilities can occur. We now establish the following generalization of (9.44). Proposition 9.7. Let k = 0, 1, . . . , g − 1. Then, for a holomorphic line bundle L → M, (9.45)

c1 (L)[M ] = g − 1 − k =⇒ 0 ≤ dim O(L) ≤ g − k

and (9.46)

c1 (L)[M ] = g − 1 + k =⇒ k ≤ dim O(L) ≤ g.

Proof. First we establish (9.46). The lower estimate follows from (9.31). For the upper estimate, pick any divisor ϑ ≤ 0 with ϑ = k. Then dim O(L) ≤ dim M(L, ϑ) = dim O(L ⊗ Eϑ ), which is equal to g since c1 (L ⊗ Eϑ )[M ] = 2g − 1 and Proposition 9.5 applies. The upper estimate in (9.45) follows by interchanging L and L−1 ⊗ κ in the Riemann–Roch identity. To illustrate (9.46), we note the following complement to (9.44): (9.47)

c1 (L)[M ] = 2g − 2 =⇒ dim O(L) = g − 1 or g.

On the other hand, the closer c1 (L)[M ] gets to g − 1, the greater the uncertainty in dim O(L), except of course when g = 0; then Corollary 9.4 and Proposition 9.5 cover all possibilities. It turns out that, for “typical” L, the minimum value of dim O(L) in (9.45)–(9.46) is achieved; see [Gu]. We now use some of the results derived above to obtain strong results on the structure of compact Riemann surfaces of genus g = 0 and 1. Proposition 9.8. If M is a compact Riemann surface of genus g = 0, then M is holomorphically diffeomorphic to the Riemann sphere S 2 . Proof. Pick p ∈ M ; with ϑ = −p, so ϑ = 1, (9.43) implies dim M(ϑ) = 2. Of course, the constants form a one-dimensional subspace of M(ϑ); thus we know that there is a nonconstant u ∈ M(ϑ); u cannot be holomorphic, so it must have a simple pole at p. The proof thus follows from the next result.

9. The Riemann–Roch theorem

377

Proposition 9.9. If there exists a meromorphic function u on a compact Riemann surface M , regular except at a single point, where it has a simple pole, then M is holomorphically diffeomorphic to S 2 . Proof. By the simple argument mentioned above (9.28), u must have precisely one zero, a simple zero. By the same reasoning, for any λ ∈ C, u − λ must have precisely one simple zero, so u : M → C ∪ {∞} = S 2 is a holomorphic diffeomorphism. Proposition 9.10. If M is a compact Riemann surface of genus g = 1, then there exists a lattice Γ ⊂ C such that M is holomorphically diffeomorphic to C/Γ. Proof. By (9.21), or alternatively  by (9.24), dim O(κ) = 1 in this case. Pick a nontrivial section ξ. By (9.28), νξ (p) = 2g − 2 = 0. Since ξ has no poles, it also has no zeros, that is, κ is holomorphically trivial if g = 1. (Compare with (9.30).) We use a topological fact. Namely, since dim H1 (M, C) = 2 if g = 1, by deRham’s theorem there exist closed curves γ1 , γ2 in M such that, for any closed curve γ in M , there are integers m1 , m2 such that 

 v = m1 γ

 v + m2

γ1

v, γ2

for any closed 1-form v on M . Granted this, it follows that if we pick p0 ∈ M , the map  (9.48)

M  z →

z

ξ p0

defines a holomorphic map (9.49)

Φ : M −→ C/Γ ,

 where Γ is the lattice in C generated by ζj = γj ξ, j = 0, 1. Since ξ is nowhere vanishing, the map (9.49) is a covering map. It follows that there is a holomorphic covering map Ψ : C → M , and the covering transformations form a group of translations of C (a subgroup of Γ , call it Γ). This gives the holomorphic diffeomorphism M ≈ C/Γ. We remark that, with a little extra argument, one can verify that (9.49) is already a diffeomorphism. Propositions 9.8 and 9.10 are special cases of the uniformization theorem for compact Riemann surfaces. The g ≥ 2 case will be established in Chap. 14 as a consequence of solving a certain nonlinear PDE. Also in that chapter, an alternative proof of Proposition 9.10 will be presented; in that case the PDE becomes linear. Also in Chap. 14 we present a linear PDE proof that treats the case g = 0.

378 10. Dirac Operators and Index Theory

We note that in the treatment of the g = 1 case given above, the Riemann–Roch theorem is not essential; the analysis giving (9.22)–(9.24) suffices. We return to the study of dim O(L), for L = Eϑ . We illustrate how the first possibility can occur in (9.44). In fact, pick distinct points p, q ∈ M , and consider ϑ = p−q. Clearly, c1 (Ep−q )[M ] = 0. Now O(Ep−q ) ≈ M(p−q), and it follows from Proposition 9.9 that if there is a nontrivial member of M(p − q), then M must be the sphere S 2 . We thus have (9.50)

O(Ep−q ) = 0

if p = q ∈ M, of genus g ≥ 1.

On the other hand, if p, q, r ∈ M are distinct, then c1 (E−p−q+r )[M ] = 1, and (9.34) applies for g = 1; hence (9.51)

g = 1 =⇒ dim M(−p − q + r) = 1.

By the discussion above, a nontrivial u ∈ M(−p − q + r) cannot have just a simple pole; it must have poles at p and q. This proves the next result: Proposition 9.11. If p, q, and r are distinct points in M , of genus 1, there is a meromorphic function on M with simple poles at p and q, and a zero at r, unique up to a multiplicative constant. Similarly, if p = q = r ∈ M , one has a meromorphic u with a double pole at p, and a zero at r. Given that M = C/Γ, these meromorphic functions are the elliptic functions of Weierstrass, and they can be constructed explicitly. The uniqueness statement can also be established on elementary grounds. Note that, with p, q, and r as in Proposition 9.11, the corresponding elliptic function u vanishes at one other uniquely determined point s (or perhaps has a double zero at r, so s = r). In other words, if we set ϑ = −p − q + r + s, for M of genus 1, the line bundle Eϑ is trivial for a unique s ∈ M , given p, q, r ∈ M, r different from p or q. Actually, this last qualification can be dispensed with; r = p forces s = q. It is a basic general question in Riemann surface theory to specify conditions on a divisor ϑ (in addition to ϑ = 0) necessary and sufficient for Eϑ to be a trivial holomorphic line bundle over M . The question of whether Eϑ is trivial is equivalent to the question of whether there exists a nontrivial meromorphic function on M , with poles at p of order exactly |ν(p)|, where ν(p) < 0, in the representation (9.35) for ϑ, and zeros of order exactly ν(p), where ν(p) > 0. This question is answered by a theorem of Abel; see [Gu] for a discussion. The answer is essentially equivalent to a classification of holomorphic line bundles over M .

Exercises 1. Show that the conjugate linear map C in (9.23) is indeed well defined, independently of a choice of local holomorphic coordinates. 2. Show that if M is a compact Riemann surface, then the complex line bundle κ has a square root, i.e., a line bundle λ such that κ ≈ λ ⊗ λ. Show that λ can even be

Exercises

3. 4. 5. 6. 7.

379

taken to be a holomorphic square root. Thus M actually has a spin structure. (Note also Exercise 5 of §3.) Deduce the index formula (9.15), which leads to the Riemann–Roch formula, directly from Theorem 5.1, for twisted Dirac operators on spin manifolds. Is it possible to choose a connection on L such that the operators ∂ L and DL in (9.14) are actually equal? Sections of the line bundle κ ⊗ κ are called quadratic differentials. Compute the dimension of O(κ ⊗ κ). Given a divisor ϑ ≤ 0, compute dim M(κ ⊗ κ, ϑ). Show that (9.41)–(9.42) provide an alternative proof of the existence result, Proposition 9.2. Deduce from Proposition 9.2 that every holomorphic line bundle L over a Riemann surface is isomorphic to a divisor bundle Eϑ .

A nonconstant meromorphic function f : M → C ∪ {∞} can be regarded as a holomorphic map f : M → S 2 , which is onto. It is called a branched covering of S 2 by the Riemann surface M . A branch point of M is a point p ∈ M such that df (p) = 0. The order o(p) is the order to which df (p) vanishes at p. 8. If f : M → S 2 is a holomorphic map with branch points pj , show that  o(pj ) = 2 deg(f ) + 2g − 2. (9.52) j

(Hint: Reduce to the case where all poles of f are simple, so (counting multiplicity) # poles of f  = 2 × # poles of f, while the left side of (9.52) is equal to # zeros of f  . Think of f  as a meromorphic section of κ.) 9. Give another derivation of (9.52) by triangulating S 2 so that the points qj = f (pj ) are among the vertices, pulling this triangulation back to M , and comparing the numbers of vertices, edges, and faces. The formula (9.52) is called Hurwitz’ formula. 10. Let X be a “real” vector field on a compact Riemann surface M . Assume M is given a Riemannian metric compatible with its complex structure, so that J : Tx M → Tx M is an isometry. Picture X as a section of the complex line bundle T = κ−1 . Show that X generates a group of conformal diffeomorphisms of M if and only if it is a holomorphic section of κ−1 . If g is the genus of M , show that g ≥ 2 =⇒ O(κ−1 ) = 0, g = 1 =⇒ dimC O(κ−1 ) = 1, g = 0 =⇒ dimC O(κ−1 ) = 3. Deduce the dimension of Lie groups of conformal diffeomorphisms in these cases. Compare the conclusion in case g ≥ 2 with that of Exercise 5 of §4, given (see Chap. 14, §2) that one could choose a Riemannian metric of curvature −1. Compare the g = 1 case with Exercise 6 of §4. 11. Considering M(κ, p) = {u ∈ O(κ) : u(p) = 0} ≈ O(κ ⊗ Ep ), show that g ≥ 1 =⇒ dim M(κ, p) = g − 1.

380 10. Dirac Operators and Index Theory Deduce that, for each p ∈ M , there exists u ∈ O(κ) such that u(p) = 0, provided g ≥ 1. Hint. Use (9.17) to get dim O(κ ⊗ Ep ) − dim O(Ep−1 ) = g − 2. Then show that dim O(Ep−1 ) = dim M(−p) = 1 if g ≥ 1. (Cf. Proposition 9.9) 12. Consider ∂ κ : H s (M, κ) → H s−1 (M, κ ⊗ κ) ≈ H s−1 (M ). Show that the range of ∂ κ has codimension 1. Hint. As in (9.8), the adjoint is −∂ : H 1−s (M ) → H −s (M, κ). 13. Let uj be meromorphic 1-forms on neighborhoods Oj of pj (1 ≤ k ≤ K), with poles holomorphic on M \ {pj }, at pj . Use Exercise 12 to show there exists u ∈ M(κ),  such that u − uj |Oj is pole free for each j, if and only if K j=1 Respj uj = 0. 14. Let E → M be a holomorphic vector bundle over a compact Riemann surface, of rank k. That is, each fiber Ep has complex dimension k. Modify the proof of Theorem 9.1 to show that dim O(E) − dim O(E  ⊗ κ) = c1 (E)[M ] −

k c1 (κ)[M ]. 2

Here E  is the dual bundle of E. (Hint. Obtain an analogue of (9.15) and use Ch(E) = Tr e−Φ/2πi , as in (6.36), where Φ is the End(E)-valued curvature form of a connection on E, to get k e−c1 (κ)/2 Ch(E) = c1 (e) − c1 (κ).) 2 15. Use Theorem 8.1 to formulate a version of the Riemann-Roch theorem for a compact, complex manifold of higher dimension.

10. Direct attack in 2-D Here we produce a direct analysis of the index formula for a first-order, elliptic operator (10.1)

D : C ∞ (M, E0 ) −→ C ∞ (M, E1 )

of Dirac type when dim M = 2. In view of (5.11), if kj (t, x, y) are the integral ∗ ∗ kernels of e−tD D and e−tDD , j = 0, 1, then (10.2)

kj (t, x, x) ∼ aj0 (x)t−1 + aj1 (x) + aj2 (x)t + · · · ,

as t  0, and (10.3)

 Index D =

  a01 (x) − a11 (x) dV (x).

M

As shown in Chap. 7, §14, we can produce explicit formulas for aj1 (x) via calculations using the Weyl calculus. Thus, pick local frame fields for E0 and E1 so that, in a local coordinate chart, D = A(X, D), with

10. Direct attack in 2-D

A(x, ξ) =

(10.4)



381

Aj (x)ξj + C(x),

a K × K matrix-valued symbol. Assume that D∗ D = g(X, D) + 0 (X, D) + B0 (x), DD∗ = g(X, D) + 1 (X, D) + B1 (x),

(10.5)

where g(x, ξ) defines a metric tensor, while j (x, ξ) and Bj (x) are K ×K matrixvalued, and  (ν) j (x)ξj . (10.6) ν (x, ξ) = j

By (14.86) of Chap. 7, we have the following: Proposition 10.1. If D is an operator of Dirac type satisfying the hypotheses above and dim M = 2, then Index D is equal to

(10.7)

1 4π

⎧ ⎫  ⎨  ⎬  (0) 2    (1) j (x) − j (x)2 + Tr B1 (x) − B0 (x) Tr dV. ⎩ ⎭ M

j

Of course, the individual terms in the integrand in (10.7) are not generally globally well defined on M ; only the total is. We want to express these terms directly in terms of the symbol of D. Assuming the adjoint is computed using L2 (U, dx), we have D∗ D = L0 (X, D) and DD∗ = L1 (X, D), with

(10.8)

i L0 (x, ξ) = A(x, ξ)∗ A(x, ξ) + {A∗ , A}, 2 i L1 (x, ξ) = A(x, ξ)A(x, ξ)∗ + {A, A∗ }. 2

Hence i 0 (x, ξ) = A1 (x, ξ)∗ C(x) + C(x)∗ A1 (x, ξ) + {A∗1 , A1 }, 2 (10.9) i 1 (x, ξ) = A1 (x, ξ)C(x)∗ + C(x)A1 (x, ξ)∗ + {A1 , A∗1 }, 2  where A1 (x, ξ) = Aj (x)ξj , and

(10.10)

i i B0 (x) = C(x)∗ C(x) + {C ∗ , A1 } + {A∗1 , C}, 2 2 i i B1 (x) = C(x)C(x)∗ + {C, A∗1 } + {A1 , C ∗ }. 2 2

382 10. Dirac Operators and Index Theory

Suppose that, for a given point x0 ∈ M , we arrange C(x0 ) = 0. Then i ∗ i  ∂A∗1 ∂A1 ∂A∗1 ∂A1  , {A1 , A1 } = − 2 2 j ∂ξj ∂xj ∂xj ∂ξj i i  ∂A1 ∂A∗1 ∂A1 ∂A∗1  1 (x0 , ξ) = {A1 , A∗1 } = , − 2 2 j ∂ξj ∂xj ∂xj ∂ξj

0 (x0 , ξ) = (10.11)

and i ∗ i {C , A1 } + {A∗1 , C} 2 2 ∂A∗1 ∂C  i  ∂C ∗ ∂A1 − , + = 2 j ∂xj ∂ξj ∂ξj ∂xj

B0 (x0 ) =

(10.12)

i i {C, A∗1 } + {A1 , C ∗ } 2 2 ∂A1 ∂C ∗  i  ∂C ∂A∗1 − . + = 2 j ∂xj ∂ξj ∂ξj ∂xj

B1 (x0 ) =

Note that if A1 (x, ξ) is scalar, then 0 (x0 , ξ) = −1 (x0 , ξ) (granted that C(x0 ) = 0). Hence their contributions to the integrand in (10.7) cancel. Also, if A1 (x, ξ) is scalar, then B1 (x0 ) = −B0 (x0 ). Thus, at x0 , the integrand in (10.7) is equal to    ∂C ∂C ∗ Aj − Aj (10.13) 2 Tr B1 (x0 ) = − Tr ∂xj ∂xj j in this case. This situation arises for elliptic differential operators on sections of complex line bundles. In such a case, C(x) is also scalar, and we can rewrite (10.13) as (10.14)

−2 Im

 j

Aj

∂C . ∂xj

Let’s take a look at the operator DL : C ∞ (M, L) → C ∞ (M, L ⊗ κ), where M is a Riemann surface, L → M is a complex line bundle, with a Hermitian metric and a metric connection ∇, and, for a vector field X, (10.15)

DL u, X = ∇X u + i∇JX u.

This is the same as (9.11)–(9.12), up to a factor of 2. Here J is the complex structure on T M . We can assume M has a Riemannian metric with respect to which J is rotation by 90◦ . Pick x0 ∈ M . Use a geodesic normal coordinate system centered at x0 , so the metric tensor gjk satisfies (10.16)

∇gjk (x0 ) = 0.

10. Direct attack in 2-D

383

Let X(x0 ) = ∂/∂x1 , and define X by parallel transport radially from x0 (along geodesics). Then (10.17)

X(x) = a11 (x)

∂ ∂ + a21 (x) , ∂x1 ∂x2

with (10.18)

a11 (x0 ) = 1,

a21 (x0 ) = 0,

∇aj1 (x0 ) = 0.

Furthermore, (10.19)

JX(x) = a12 (x)

∂ ∂ + a22 (x) , ∂x1 ∂x2

with (10.20)

a12 (x0 ) = 0,

a22 (x0 ) = 1,

∇aj2 (x0 ) = 0.

Next, let ϕ be a local section of L such that ϕ(x0 ) has norm 1, and ϕ(x) is obtained from ϕ(x0 ) by radial parallel translation. Thus (10.21)

u = vϕ =⇒ ∇∂j u = (∂j v + iθj v)ϕ,

where the connection coefficients satisfy θj (x0 ) = 0.

(10.22)

In such a coordinate system, and with respect to such choices, the operator DL takes the form (10.23)

DL (vϕ) =

∂v 1  Aj − Aj θj v ϕ ⊗ ϑ, i ∂xj

where (10.24)

Aj = i aj1 + iaj2

and where ϑ ∈ C ∞ (U, κ) satisfies X, ϑ = 1,

JX, ϑ = i.

∗ : C ∞ (M, L ⊗ κ) → C ∞ (M, L) is given by Then DL ∗ (10.25) DL (w ϕ ⊗ ϑ) =

∂ 1  −1/2 g + (∂j Aj + Aj θj ) g 1/2 w ϕ. Aj i ∂xj

384 10. Dirac Operators and Index Theory

√ Now we want to take adjoints using L2 (U, dx) rather than L2 (U, gdx), so we conjugate by g 1/4 , and replace DL by

 ∂ −1/4 ˜L = 1 g 1/4 Aj g v − Aj θj v . (10.26) D i ∂xj Thus we are in the situation of considering an operator of the form (10.4), with Aj given by (10.24) and '  & i ∂Aj 1 −1 ∂g (10.27) C(x) = − Aj θj − g Aj . 2 ∂xj 4 ∂xj Thus C(x0 ) = 0, by (10.18)–(10.22), while ' & i 1 (10.28) ∂k C(x0 ) = −Aj (∂k θj ) + ∂k ∂j Aj − Aj (∂k ∂j g) . 2 4 j Now ∂k θj (x0 ) is given by the curvature of ∇ on L: ∂θj 1 (x0 ) = Fjk (x0 ). ∂xk 2

(10.29)

Meanwhile, as shown in §3 of Appendix C, ∂k ∂j Aj can be expressed in terms of the Riemannian curvature: (10.30)

1 1 ∂j ∂k am (x0 ) = − Rjmk − Rkmj , 6 6

and of course so can ∂k ∂j g(x0 ). Consequently, at x0 , the formula (10.14) for the integrand in (10.7) becomes 1 2 − F12 + S(x0 ). i 2

(10.31)

Note that S/2 = K, the Gauss curvature. Thus the formula (10.7) becomes    2 − F12 + K dV i M   1 1 ωL + K dV, =− 2πi 4π

Index DL = (10.32)

1 4π

M

M

where ωL is the curvature form of L. We have the identities   1 1 1 (10.33) − ωL = c1 (L)[M ], K dV = χ(M ), 2πi 4π 2 M

M

10. Direct attack in 2-D

385

the latter being the Gauss–Bonnet theorem. Now, if L → M is a holomorphic line bundle, then (1/2)DL has the same principal symbol, hence the same index, as (10.34)

∂ L : C ∞ (M, L) −→ C ∞ (M, L ⊗ κ).

Hence we obtain the Riemann–Roch formula: (10.35)

1 Index ∂ L = c1 (L)[M ] + χ(M ), 2

in agreement with (9.17). We finish with a further comment on the Gauss–Bonnet formula; χ(M ) is the index of (10.36)

d + δ : Λ0 M ⊕ Λ2 M −→ Λ1 M

if dim M = 2. If M is oriented, both Λ1 M and (Λ0 ⊕ Λ2 )M get structures of complex line bundles via the Hodge ∗ operator; use (10.37)

J = ∗ on Λ1 ,

J = −∗ : Λ0 → Λ2 ,

J = ∗ : Λ2 → Λ 0 .

It follows easily that (d + δ)J = J(d + δ), so we get a C-linear differential operator (10.38)

ϑ : Λe M −→ Λo M,

where Λe = Λ0 ⊕ Λ2 , Λo = Λ1 , regarded as complex line bundles, so Index ϑ =

1 Index(d + δ). 2

Ker ϑ is a one-dimensional complex vector space: Ker ϑ = span(1) = span(∗1). The cokernel of d + δ in (10.36) consists of the space H1 of (real) harmonic 1forms on M . This is invariant under ∗, so it becomes a complex vector space:

(10.39)

dimC H1 =

1 dimR H1 = g. 2

Thus (10.40)

Index ϑ =

1 (2 − 2g) = 1 − g. 2

386 10. Dirac Operators and Index Theory

When one applies an analysis parallel to that above, leading to (10.32), one gets (10.41)

Index ϑ =

1 4π

 K dV. M

Putting together (10.40) and (10.41), we again obtain the Gauss–Bonnet formula, for a compact, oriented surface.

Exercises 1. Use (10.36)–(10.39) to give another proof of (9.24), that is, dim O(κ) =

1 dim H1 (M, C) = g. 2

In Exercises 2–4, suppose Ej → M are complex line bundles over M , a compact manifold of dimension 2, and suppose D : C ∞ (M, E0 ) −→ C ∞ (M, E1 ) is a first-order, elliptic differential operator. 2. Show that the symbol of D induces an R-linear isomorphism (10.42)

σD (x) : Tx∗ −→ L(E0x , E1x ).

Hence M has a complex structure, making this C-linear. This gives M an orientation; reversing the orientation makes (10.42) conjugate linear. 3. If M is oriented so that (10.42) is conjugate linear, show that D has a principal symbol homotopic to that of DL , given by (10.15), with L = E0 , L ⊗ κ ≈ E1 . Deduce that (10.43)

Index D =

1 1 c1 (E0 )[M ] + c1 (E1 )[M ]. 2 2

4. What happens to the formula for Index D∗ ? In Exercises 5–8, S02 T ∗ denotes the bundle of symmetric second-order tensors with trace zero on a Riemannian manifold M , and S01,1 denotes the bundle of symmetric tensors of type (1, 1) with trace 0. The metric tensor provides an isomorphism of these two bundles. 5. If M is a compact, oriented 2-fold, with associated complex structure J : Tx M → 1,1 ⊂ Hom Tx by Tx M , show that a complex structure is defined on S0x (10.44)

J(A) =

1 [J, A] = JA. 2

Thus S01,1 and S02 T ∗ become complex line bundles. 6. Recall the first-order operator considered in (4.29)–(4.31): (10.45) DT F : C ∞ (M, T ) −→ C ∞ (M, S02 T ∗ ),

DT F X = Def X −

1 (div X)g, 2

11. Index of operators of harmonic oscillator type

387

in case n = dim M = 2. If T and S02 T ∗ are regarded as complex line bundles, show that DT F is C-linear. 7. Recall that ker DT F consists of vector fields that generate conformal diffeomorphisms of M , hence of holomorphic sections of T = κ−1 . Show that there is an isomorphism S02 T ∗ ≈ κ−1 ⊗ κ transforming (10.45) to (10.46)

∂ : C ∞ (M, κ−1 ) −→ C ∞ (M, κ−1 ⊗ κ).

Note that Index ∂ = −(3g − 3) in this case, if g is the genus of M . 8. In view of (4.30), the orthogonal complement of the range of DT F is the finite dimensional space (10.47)

V = {u ∈ C ∞ (M, S02 T ∗ ) : div u = 0}.

Comparing (10.45) and (10.46), show that V ≈ O(κ ⊗ κ). If M has genus g ≥ 2, ∂ in (10.46) is injective (by Exercise 12, §9). Deduce that (10.48)

dimR V = 6g − 6, if g ≥ 2.

Compare Exercise 5 of §9. For g = 0, compare Exercise 7 of §4. For connections with the dimension of Teichmuller space, see [Tro].

11. Index of operators of harmonic oscillator type In this section we study elliptic operators of harmonic oscillator type, introduced in §15 of Chap. 7. We recall that a symbol p(x, ξ) belongs to S1m (Rn ) if it is smooth in (x, ξ) ∈ Rn × Rn and satisfies estimates (11.1)

|Dxβ Dξα p(x, ξ)| ≤ Cαβ (1 + |x| + |ξ|)m−|α|−|β| .

The associated operator P = p(X, D) ∈ OP S1m (Rn ) is defined using the Weyl calculus. The operator is elliptic provided that, for |x|2 + |ξ|2 large enough, (11.2)

|p(x, ξ)−1 | ≤ C(1 + |x| + |ξ|)−m .

In such a case, P has a parametrix Q ∈ OP S1−m (Rn ), such that P Q−I and QP − I belong to OP S1−∞ (Rn ). The class S m (Rn ) of classical symbols is defined to consist of elements p(x, ξ) ∈ S1m (Rn ) such that (11.3)

p(x, ξ) ∼



pj (x, ξ),

j≥0

where pj (x, ξ) ∈ S1m−2j (Rn ) is homogeneous of degree m − 2j in (x, ξ) for |x|2 + |ξ|2 ≥ 1. If such a symbol satisfies the ellipticity condition (11.2), then P = p(X, D) has parametrix Q ∈ OP S −m (Rn ). A paradigm example of such an operator is the harmonic oscillator

388 10. Dirac Operators and Index Theory

(11.4)

H = −Δ + |x|2 ,

which is elliptic in OP S 2 (Rn ), with symbol |x|2 + |ξ|2 . It is a positive definite operator, and, as shown in Chap. 7, (11.5)

H s ∈ OP S 2s (Rn ),

∀ s ∈ R.

There are Sobolev-type spaces Hs (Rn ), s ∈ R, such that, for s = k ∈ Z+ , (11.6)

Hk (Rn ) = {u ∈ L2 (Rn ) : xα Dxβ u ∈ L2 (Rn ), ∀ |α| + |β| ≤ k}.

As shown in Chap. 7, if P ∈ OP S m (Rn ), then, for all s ∈ R, (11.7)

P : Hs (Rn ) −→ Hs−m (Rn ),

and if P is elliptic, this map is Fredholm. We want to study its index. For simplicity, we stick to operators with symbols of classical type. If P = p(X, D) is an elliptic operator (k × k matrix valued), with symbol expansion of the form, we call p0 (x, ξ) the principal symbol. Recall we assume p0 (x, ξ) is homogeneous of order m for |x|2 + |ξ|2 ≥ 1. We then have the symbol map (11.8)

σP : S 2n−1 −→ G(k, C), σP (x, ξ) = p0 (x, ξ),

|x|2 + |ξ|2 = 1.

Note that P ∈ OP S m (Rn ) and P H μ ∈ OP S m+2μ (Rn ) have the same symbol map, and they have the same index, one on Hs (Rn ) → Hs−m (Rn ) and the other on Hs (Rn ) → Hs−m−2μ (Rn ). Basic Fredholm theory gives the following. Proposition 11.1. Given elliptic k × k systems Pj ∈ OP S mj (Rn ), if σP1 and σP2 are homotopic maps from S 2n−1 to G(k, C), then Index P1 = Index P2 . Let us take n = 1 and k = 1 and look for specific index formulas. In this case, given elliptic scalar P ∈ OP S m (R), we have (11.9)

σP : S 1 −→ G(1, C) = C \ 0.

Such a map is specified up to homotopy by the winding number (11.10)

1 ind σP = 2πi

 S1

σP (ζ) dζ, σP (ζ)

where ζ = x + iξ. If P1 and P2 are two such elliptic operators, we have (11.11)

Index P1 P2 = Index P1 + Index P2 ,

11. Index of operators of harmonic oscillator type

389

and (11.12)

ind σP1 P2 = ind σP1 + ind σP2 .

Let us consider the operator D1 =

(11.13)

∂ + x1 , ∂x1

acting on functions of x1 ∈ R. Its symbol is x1 + iξ1 , so ind σD1 = 1.

(11.14) Note that D1∗ = −∂1 + x1 , and (11.15)

D!∗ D1 = −∂12 + x21 − 1,

D1 D1∗ = −∂12 + x21 + 1.

We have 2

(11.16)

Ker D1 = Span{e−x1 /2 }, and D1 D1∗ ≥ 2I =⇒ Ker D1∗ = 0,

hence (11.17)

Index D1 = 1.

Putting together (11.9)–(11.17) and Proposition 11.1, we have the following. Proposition 11.2. If P ∈ OP S m (R) is a scalar elliptic operator, then (11.18)

Index P = ind σP .

We next keep n = 1 and let P ∈ OP S m (R) be an elliptic k × k system, so (11.19)

σP : S 1 −→ G(k, C).

We want to classify these maps, up to homotopy. To do this, we bring in the following topological fact about (11.20)

S(k, C) = {A ∈ G(k, C) : det A = 1},

namely (11.21)

S(k, C) is simply connected.

Using this fact, we prove the following.

390 10. Dirac Operators and Index Theory

Proposition 11.3. Given a symbol map (11.19), define σ ˜P : S 1 → G(k, C) by  (11.22)

σ ˜P (x, ξ) =

 0 , I

det σP (x, ξ) 0

˜P are where I denotes the (k − 1) × (k − 1) identity matrix. Then σP and σ homotopic. Proof. Given (11.19) and (11.22), we set (11.23)

˜P σP−1 : S 1 −→ S(k, C). γ1 = σ

Using (11.21), we can deform γ1 to γ0 ≡ I, through γτ : S 1 → S(k, C), 0 ≤ ˜P is then given by σPτ (x, ξ) = γτ (x, ξ)σP (x, ξ), τ ≤ 1. A homotopy from σP to σ 0 ≤ τ ≤ 1. We have a scalar operator P( ∈ OP S km (R), defined uniquely mod OP S km−2 (R) by the condition (11.24)

σP = det σP .

Then (11.25)

Index P = Index

 P(

 I

,

which by Proposition 11.2 is given by ind det σP . We have proved the following. Proposition 11.4. If P ∈ OP S m (R) is an elliptic k × k system, (11.26)

Index P = ind det σP .

Returning to (11.21), we note that it is equivalent to the result (11.27)

SU (k) is simply connected.

To see this, we use the polar decomposition (11.28) where

A ∈ G(k, C) =⇒ A = U (A)Π(A), Π(A) = (A∗ A)1/2 is positive definite, U (A) = A(A∗ A)−1/2 ∈ U (k).

With this, we can define a 1-parameter family of maps

11. Index of operators of harmonic oscillator type

(11.29)

ϑτ : G(k, C) −→ G(k, C),

391

τ ∈ [0, 1],

by ϑτ (A) = U (A)Π(A)τ .

(11.30) We have

ϑ0 (A) = U (A),

(11.31)

ϑ1 (A) = A.

This makes U (k) a deformation retract of G(k, C). As a consequence, each continuous map σ : S 2n−1 → G(k, C) is homotopic to the map ϑ0 ◦ σ : S 2n−1 → U (k). Note that (11.32)

det Π(A) = | det A|,

det U (A) =

det A , | det A|

so ϑ0 : S(k, C) −→ SU (k),

(11.33)

and ϑτ makes SU (k) a deformation retract of S(k, C). This establishes the equivalence of (11.21) and (11.27). In case k = 2, we have ) (11.34)

SU (2) =

a −b b a



* : a, b ∈ C, |a|2 + |b|2 = 1 ≈ S 3 ,

which is clearly simply connected. For k > 2, (11.27) is a special case of (11.56) below. Let us now take n ≥ 2 and consider a k × k elliptic system P ∈ OP S m (Rn ), giving a symbol map (11.8). Making use of the deformation retract (11.29)– (11.31), we see that σP is homotopic to a symbol map (11.35)

σP # : S 2n−1 −→ U (k),

for an operator P # ∈ OP S m (Rn ), uniquely defined mod OP S m−2 (Rn ), and Index P = Index P # . For k = 1, we have the following topological result. Lemma 11.5. If n ≥ 2, every continuous map σ : S 2n−1 → U (1) = S 1 is homotopic to a constant map. Proof. Indeed, since S 2n−1 is simply connected for n ≥ 2, σ lifts to a continuous map σ ˜ : S 2n−1 → R, which is clearly homotopic to a constant map. In light of this, if we have (11.35) and set (as in (11.22))

392 10. Dirac Operators and Index Theory

 (11.36)

σ ˜ (x, ξ) =

 det σP # (x, ξ) 0 , 0 I

σ ˜ : S 2n−1 → U (k),

then, for n ≥ 2, σ ˜ is homotopic to a constant. Hence σP # and (11.37)

σ b : S 2n−1 → SU (k),

σ b (x, ξ) = σ ˜ (x, ξ)−1 σP # (x, ξ),

are homotopic. Given μ ∈ R, this is the symbol map of an operator P ∈ OP S μ (Rn ), uniquely determined up to a lower order operator. We have the following result. Proposition 11.6. For n ≥ 2, if P ∈ OP S m (Rn ) is an elliptic k × k system, there exists for each μ ∈ R an elliptic k × k system P ∈ OP S μ (Rn ) whose symbol map σP : S 2n−1 −→ SU (k)

(11.38)

is homotopic to σP , as maps S 2n−1 → G(k, C). Hence Index P = Index P.

(11.39)

Let us now specialize to n = 2. By Lemma 11.5, every scalar elliptic P ∈ OP S m (R2 ) must have index 0. We construct an elliptic 2×2 system with nonzero index as follows. With D1 as in (11.13), set  D2 = 

(11.40) =

∂2 − x2 −∂1 + x1  ∗ −L2 , D1∗

∂1 + x1 ∂2 + x2 D1 L2



where L2 = ∂2 + x2 ,

(11.41)

L∗2 = −∂2 + x2 .

Note that  (11.42)

σ D2 =

x1 + iξ1 x2 + iξ2

 −x2 + iξ2 , x1 − iξ2

so σD2 : S 3 → SU (2) ≈ S 3

is essentially the identity map. A computation gives (11.43) and

D2∗ D2 =



D1∗ D1 + L∗2 L2

 D1 D1∗ + L2 L∗2

,

11. Index of operators of harmonic oscillator type

(11.44)

D2 D2∗ =



D1 D1∗ + L∗2 L2

393

 D1∗ D1 + L2 L∗2

.

We recall the formulas for D1∗ D1 and D1 D1∗ in (11.15). Similarly, (11.45)

L∗2 L2 = −∂22 + x22 − 1,

L2 L∗2 = −∂22 + x22 + 1.

2

Hence Ker L2 = Span{e−x2 /2 } and L2 L∗2 ≥ 2I, and we have for the four diagonal elements of (11.43)–(11.44) that dim Ker(D1∗ D1 + L∗2 L2 ) = 1, D1 D1∗ + L2 L∗2 ≥ 4I,

(11.46)

D1 D1∗ + L∗2 L2 ≥ 2I, D1∗ D1 + L2 L∗2 ≥ 2I.

Hence dim Ker D2 = 1,

(11.47)

dim Ker D2∗ = 0,

so Index D2 = 1.

(11.48)

Now consider an arbitrary 2×2 elliptic system P ∈ OP S m (R2 ). As in (11.38), we have an adjusted operator P, with the same index as P , and σP : S 3 −→ SU (2) ≈ S 3 .

(11.49)

The homotopy class of this map is an element of π3 (S 3 ). Results on this homotopy group, which we will discuss in more detail below, imply the following. Proposition 11.7. Let P ∈ OP S m (R2 ) be a 2 × 2 elliptic system. For σP as in (11.49), there is a unique integer  such that either  > 0 and σP is homotopic to σD2 , (11.50)

 = 0 and σP is homotopic to a constant map,  < 0 and σP is homotopic to σ(D2∗ )|| .

We denote this  by (11.51) Then

 = ind σP .

394 10. Dirac Operators and Index Theory

(11.52)

Index P = Index P = ind σP .

To see that (11.52) follows from (11.50), note that in the first case Index P = Index D2 =  and in the third case Index P = Index(D2∗ )|| = −|| = . We now discuss some homotopy theory behind (11.50). It is convenient to place this in a more general setting. If M is a smooth, connected manifold and j ∈ N, πj (M ) denotes the set of homotopy classes of continuous maps ϕ : S j → M (which is equivalent to the set of homotopy classes of smooth maps). This can be given a group structure as follows. Fix p0 ∈ S j , q0 ∈ M . Given maps ϕ, ψ : S j → M , one can produce maps homotopic to these that take p0 to q0 , so assume ϕ and ψ have this property. Now take S j and collapse its “equator,” which is homeomorphic to S j−1 , to a point. You obtain two copies of S j , joined at a point, which we identify with p0 . Then map the top sphere to M by ψ and the bottom sphere to M by ϕ, and compose with the collapse map, to get a map σ : S j → M , whose homotopy class [σ] = [ϕ] · [ψ]. In case G is a connected Lie group, there is another way to define a product on πj (G). Namely, if ϕ, ψ : S j → G, consider the map ϕ · ψ : S j → G given by (ϕ · ψ)(x) = ϕ(x)ψ(x), using the product on G. If ϕ and ϕ˜ are homotopic ˜ we have ϕ · ψ ∼ ϕ˜ · ψ, ˜ so this gives a product (write ϕ ∼ ϕ) ˜ and also ψ ∼ ψ, on πj (G). It is a basic fact that this product on πj (G) agrees with the previously defined one; cf. [Spa], Chap. 1. What makes (11.50) work is the j = 3 case of the following fundamental result of H. Hopf. Proposition 11.8. For each j ∈ N, (11.53)

πj (S j ) ≈ Z,

and (the homotopy class of) the identity map Id : S j → S j is a generator. In fact, if ϕ, ψ : S j → S j are smooth, they have degrees, defined in Chap. 1, §19, and the Hopf theorem says they are homotopic if and only if they have the same degree. Cf. [Spa], p. 398. Under the identification (11.34) of SU (2) with (a, b) ∈ S 3 , σD2 : S 3 → S 3 is the identity map, and σD2 ∈ π3 (S 3 ) is an -fold product, hence corresponds to  ∈ Z under this isomorphism, while σD2∗ = −1 ∈ π3 (S 3 ), and σ(D2∗ )|| = −|| ∈ π3 (S 3 ). Let us next consider a k × k elliptic system P ∈ OP S m (R2 ), giving rise to P # as in (11.35) and P as in (11.38), all having the same index. The following result is useful. Proposition 11.9. For each k ∈ N, the natural inclusion SU (k) → U (k) induces an isomorphism (11.54)



→ πj (U ((k)), πj (SU (k)) −

if j > 1.

11. Index of operators of harmonic oscillator type

395

Furthermore, the inclusions U (k) → U (k + ) and SU (k) → SU (k + ), given by   A (11.55) A → , I where I denotes the  ×  identity matrix, induce isomorphisms (11.56) ≈ → πj (U (k + )), πj (U (k)) −



πj (SU (k)) − → πj (SU (k + )),

if j ≤ 2k − 1.

We mention that a proof of (11.54) requires just a few arguments beyond the proof of Proposition 11.6. The proof of (11.56), with  = 1, which then proceeds inductively, follows by applying the “homotopy exact sequence” to (11.57)

U (k + 1)/U (k) ≈ S 2k+1 ,

SU (k + 1)/SU (k) ≈ S 2k+1 .

See (11.83) below. According to Proposition 11.9, when j = 3, (11.56) holds for k ≥ 2. Taking (11.53) into account, we have π3 (SU (k)) ≈ π3 (U (k)) ≈ Z,

(11.58)

∀ k ≥ 2.

We can now augment Proposition 11.7 as follows. Proposition 11.10. Let P ∈ OP S m (R2 ) be a k × k elliptic system, k > 2. For σP : S 3 → SU (k) as in (11.49), there is a unique integer  such that, with I denoting the (k − 2) × (k − 2) identity matrix, either   > 0 and σP is homotopic to (11.59)



σD2 I

,

 = 0 and σP is homotopic to a constant map,   σ(D2∗ )||  < 0 and σP is homotopic to . I

We denote this  by (11.60)

 = ind σP .

Then (11.61)

Index P = Index P = ind σP .

We now turn to higher dimensions. Our next task is to construct, for each j an elliptic system Dj ∈ OP S 1 (Rj ) (actually a system of differential operators) of

396 10. Dirac Operators and Index Theory

index 1. The construction is inductive. Assume we have such an elliptic system Dn−1 , with the properties dim Ker Dn−1 = 1,

(11.62) and

∗ ≥ 2I. Dn−1 Dn−1

(11.63)

By (11.15)–(11.16) we have this for n − 1 = 1, and by (11.43)–(11.47) we have this for n − 1 = 2. We then set     Dn−1 ∂n − xn Dn−1 −L∗n (11.64) Dn = = , ∗ ∗ ∂n + xn Dn−1 Ln Dn−1 where Ln = ∂n + xn ,

(11.65)

L∗n = −∂n + xn .

Parallel to (11.43)–(11.44), a computation gives  ∗ Dn−1 Dn−1 + L∗n Ln ∗ (11.66) Dn Dn =

 ,

∗ Dn−1 Dn−1 + Ln L∗n

and (11.67)

Dn Dn∗

=

 ∗ Dn−1 Dn−1 + L∗n Ln

 ∗ Dn−1 Dn−1 + Ln L∗n

.

Parallel to (11.45), we have (11.68)

L∗n Ln = −∂n2 + x2n − 1,

Ln L∗n = −∂n2 + x2n + 1.

2

We see that Ln annihlates e−xn /2 and Ln L∗n ≥ 2I. Hence, parallel to (11.46), we have

(11.69)

∗ Dn−1 + L∗n Ln ) = 1, dim Ker(Dn−1 ∗ Dn−1 Dn−1 + Ln L∗n ≥ 4I, ∗ Dn−1 Dn−1 + L∗n Ln ≥ 2I, ∗ Dn−1 Dn−1 + Ln L∗n ≥ 2I.

Consequently, we have (11.70)

dim Ker Dn = 1,

11. Index of operators of harmonic oscillator type

397

and Dn Dn∗ ≥ 2I,

(11.71) hence

Index Dn = 1.

(11.72)

This completes the inductive construction. Note that the matrix doubles in size at each iteration, so Dn is a 2n−1 × 2n−1 matrix of differential operators. We can extend Proposition 11.10, using the following fundamental result of R. Bott. Cf. [Mil], §23. Proposition 11.11. For n ∈ N, (11.73)

π2n−1 (U (k)) ≈ π2n−1 (SU (k)) ≈ Z,

if k ≥ n.

Note that (11.58) is the case n = 2 of this result. Given this proposition, the calculation (11.72) implies the following. Proposition 11.12. For n ∈ N, the map σDn : S 2n−1 −→ U (2n−1 )

(11.74)

defines a generator of π2n−1 (U (2n−1 )). Note: The calculation (11.66) implies σDn∗ Dn (x, ξ) = σDn (x, ξ)∗ σDn (x, ξ) = I, for |x|2 + |ξ|2 = 1. From here, we have the following extension of Proposition 11.10. Proposition 11.13. Let P ∈ OP S m (Rn ) be a k × k elliptic system, with associated symbol map σP : S 2n−1 → SU (k). If k = 2n−1 , there is a unique integer  such that either  > 0 and σP is homotopic to σDn , (11.75)

 = 0 and σP is homotopic to a constant map,  < 0 and σP is homotopic to σ(Dn∗ )|| .

we denote this  by (11.76) Then

 = ind σP .

398 10. Dirac Operators and Index Theory

(11.77)

Index P = Index P = ind σP .

If k < 2n−1 , then Index P = Index P = Index P(, where  (11.78)

σP =



σP I

,

I being the (2n−1 −k)×(2n−1 −k) identity matrix, and the considerations above apply to give Index P(, hence Index P . If k > 2n−1 , then there is a unique integer  such that either   > 0 and σP is homotopic to (11.79)



σDn I

,

 = 0 and σP is homotopic to a constant map,   σ(Dn∗ )||  < 0 and σP is homotopic to , I

I being the (k − 2n−1 ) × (k − 2n−1 ) identity matrix, and analogues of (11.76)– (11.77) hold. Remark: An integral formula for Index P is given in [Fed]; see also [Ho]. Also of use in index theory is the following complement to Proposition 11.11. Proposition 11.14. Given k ≥ 1, (11.80)

j∈ / {1, 3, . . . , 2k − 1} =⇒ πj (U (k)) is finite.

Thanks to Shrawan Kumar for mentioning this and for explaining the proof, which we now sketch. One ingredient is the result that (11.81)

πj (S 2k−1 ) is finite for all j = 2k − 1.

See [Spa], p. 515. The proof of (11.80) goes by induction on k. The case k = 1 is clear. The case k = 2 follows from (11.54), which reduces (11.80) with k = 2 to the assertion that πj (SU (2)) = πj (S 3 ) is finite for j = 3. To do the inductive step, we assume that (11.82)

j = {1, 3, . . . , 2k − 3} =⇒ πj (U (k − 1)) is finite,

and aim to deduce (11.80). Another ingredient for this is the homotopy exact sequence for U (k)/U (k − 1) = S 2k−1 , which includes the segment

11. Index of operators of harmonic oscillator type

(11.83)

399

πj+1 (S 2k−1 ) → πj (U (k − 1)) → πj (U (k)) → πj (S 2k−1 ),

cf. [Mil], p. 128. We tensor with Q, denoting πj (X) ⊗ Q by πjQ (X). (11.84)

Q πj+1 (S 2k−1 ) → πjQ (U (k − 1)) → πjQ (U (k)) → πjQ (S 2k−1 ).

By (11.81), πjQ (S 2k−1 ) = 0 if j = 2k − 1.

(11.85) Thus (11.86)

j∈ / {2k − 2, 2k − 1} =⇒ πjQ (U (k)) ≈ πjQ (U (k − 1)).

With this, (11.82) leads to / {1, 3, . . . , 2k − 3} and j ∈ / {2k − 2, 2k − 1}. (11.87) πjQ (U (k)) = 0 if j ∈ On the other hand, setting j = 2k − 2 in (11.84) gives (11.88)

Q Q (U (k − 1)) → π2k−2 (U (k)) → 0, Q → π2k−2

so (11.89)

Q Q (U (k − 1)) = 0 =⇒ π2k−2 (U (k)) = 0, π2k−2

giving (11.80). See the exercises for an application of Proposition 11.14. Remark: S. Kumar has also shown the author how further arguments yield, for k ≥ 2, (11.90)

π2k+1 (U (k)) = 0 Z/(2)

if k is odd, if k is even.

In case k = 2, one has (11.91) See [Spa], p. 520.

π5 (U (2)) = π5 (SU (2)) = π5 (S 3 ) = Z/(2).

400 10. Dirac Operators and Index Theory

Exercises 1. Give a Clifford algebra description of the operator Dn in (11.64). 2. Show that if k ≥ n, there exists for each ∈ Z a k × k elliptic system P ∈ OP S m (Rn ) such that Index P = . 3. Suppose you know that π2n−1 (U (k)) is a finite group. (By (11.73) this would require k < n.) Show that if P ∈ OP S m (Rn ) is a k ×k elliptic system, Index P = 0. (Hint. Index P j = j Index P .) 4. Using Exercise 3 and Proposition 11.14, show that if P ∈ OP S m (Rn ) is a k × k elliptic system, k < n =⇒ Index P = 0.

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402 10. Dirac Operators and Index Theory [MS] H. McKean and I. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom. 1(1967), 43–69. [Mel] R. Melrose, Index Theory on Manifolds with Corners, A. K. Peters, Boston, 1994. [Mil] J. Milnor, Morse Theory, Princeton University Press, Princeton, NJ, 1963. [MiS] J. Milnor and J. Stasheff, Characteristic Classes, Princeton University Press, Princeton, NJ, 1974. [Mor] J. Morgan, The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-Manifolds, Princeton University Press, Princeton, NJ, 1996. [Pal] R. Palais, ed., Seminar on the Atiyah–Singer Index Theorem, Princeton University Press, Princeton, NJ, 1963. [Pt1] V. Patodi, Curvature and the eigenforms of the Laplace operator, J. Diff. Geom. 5(1971), 233–249. [Pt2] V. Patodi, An analytic proof of the Riemann–Roch–Hirzebruch theorem for Kahler manifolds, J. Diff. Geom. 5(1971), 251–283. [Poo] W. Poor, Differential Geometric Structures, McGraw-Hill, New York, 1981. [Roe] J. Roe, Elliptic Operators, Topology, and Asymptotic Methods, Longman, New York, 1988. [Rog] A. Rogers, A superspace path integral proof of the Gauss–Bonnet–Chern theorem, J. Geom. Phys. 4(1987), 417–437. [Spa] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. [Spi] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1–5, Publish or Perish, Berkeley, 1979. [Stb] S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, 1964. [T] M. Taylor, Pseudodifferential operators and K-homology, Proc. Symp. Pure Math. 51, Part 1 (1990), 561–583. [Tro] A. Tromba, A classical variational approach to Teichmuller theory, pp.155–185 in LNM #1365, Springer, New York, 1989. [Wit] E. Witten, Constraints on supersymmetry breaking, Nucl. Phys. B202(1982), 253.

11 Brownian Motion and Potential Theory

Introduction Diffusion can be understood on several levels. The study of diffusion on a macroscopic level, of a substance such as heat, involves the notion of the flux of the quantity. If u(t, x) measures the intensity of the quantity that is diffusing, the flux J across the boundary of a region O in x-space satisfies the identity   ∂ u(t, x) dV (x) = − ν · J dS(x), (0.1) ∂t O

∂O

as long as the substance is being neither created nor destroyed. By the divergence theorem, this implies (0.2)

∂u = − div J. ∂t

The mechanism of diffusion creates a flux in the direction from greater concentration to lesser concentration. In the simplest model, the quantitative relation specified is that the flux is proportional to the x-gradient of u: (0.3)

J = −D grad u,

with D > 0. Applying (0.2), we obtain for u the PDE (0.4)

∂u = D Δu, ∂t

in case D is constant. In such a case we can make D = 1, by rescaling, and this PDE is the one usually called “the heat equation.” Many real diffusions result from jitterings of microscopic or submicroscopic particles, in a fashion that appears random. This motivates a probabilistic attack on diffusion, including creating probabilistic tools to analyze the heat equation. This is the topic of the present chapter. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  M. E. Taylor, Partial Differential Equations II, Applied Mathematical Sciences 116, https://doi.org/10.1007/978-3-031-33700-0 11

403

404 11. Brownian Motion and Potential Theory

In §1 we give a construction of Wiener measure on the space of paths in Rn , governed by the hypothesis that a particle located at x ∈ Rn at time t1 will have the probability P (t, x, U ) of being in an open set U ⊂ Rn at time t1 + t, where  (0.5)

p(t, x, y) dy,

P (t, x, U ) = U

and p(t, x, y) is the fundamental solution to the heat equation. We prove that, with respect to Wiener measure, almost every path is continuous, and we establish a modulus of continuity. Our choice of etΔ rather than etΔ/2 to define such probabilities differs from the most popular convention and leads to minor differences in various formulas. Of course, translation between the two conventions is quite easy. In §2 we establish the Feynman–Kac formula, for the solution to (0.6)

∂u = Δu + V (x)u, ∂t

in terms of an integral over path space. A limiting argument made in §3 gives us formulas for the solution to (0.4) on a bounded domain Ω, with Dirichlet boundary conditions. This also leads to formulas for solutions to (0.7)

Δu = f on Ω,

u = 0 on ∂Ω,

Δu = 0 on Ω,

u = g on ∂Ω.

and (0.8)

A different, and more natural, formula for the solution to (0.8) is derived in §5, after the development in §4 of a tool known as the “strong Markov property.” In §6 we present a study of the Newtonian capacity of a compact set K ⊂ Rn , in the case n ≥ 3, which is related to the probability that a Brownian path starting outside K will hit K. We give Wiener’s criterion for a point y in ∂Ω to be regular for the Dirichlet problem (0.8), in terms of the capacity of Kr = {z ∈ ∂Ω : |z − y| ≤ r}, as r → 0, which has a natural probabilistic proof. In §7 we introduce the notion of the stochastic integral, such as  (0.9)

t

  f s, ω(s) dω(s),

0

which is not straightforward since almost all Brownian paths fail to have locally bounded variation. We show how the solution to (0.10)

∂u = Δu + Xu ∂t

11

Brownian Motion and Potential Theory

405

can be given in terms of an integral over path space, whose integrand involves a stochastic integral, in case X is a first-order differential operator. The derivation of this formula, like the derivation of the Feynman–Kac formula in §2, uses a tool from functional analysis known as the Trotter product formula, which we establish in Appendix A at the end of this chapter. In §8 we consider a more general sort of stochastic integral, needed to solve stochastic differential equations: (0.11)

dX = b(t, X) dt + σ(t, X) dω,

which we study in §9. Via Ito’s formulas, stochastic differential equations can be used to treat diffusion equations of the form (0.12)

 ∂u  = Ajk (x) ∂j ∂k u + bj (x) ∂j u + V (x)u, ∂t

in terms of path space integrals. We look at this in §10. Results there, specialized to (0.10), yield a formula with a different appearance than that derived in §7. The identity of these two formulas leads to a formula of Cameron-Martin-Girsanov, representing the “Jacobian determinant” of a certain nonlinear transformation of path space. In §11 we consider diffusion on Riemannian manifolds. We produce Wiener measure on path space, by a process parallel to that used in §1 for the case M = Rn , centered about the formula (0.5). A crucial difference is that the heat kernel p(t, x, y) has a simple explicit formula for Rn . For more general Riemannian manifolds, we make use of various results established in Chapters 6 and 8. We assume M is a complete Riemannian manifold, and add other hypotheses as desirable, to yield needed heat kernel estimates. We establish “stochastic continuity,” as an alternative to path continuity. This easier result still allows a derivation of the Feynman-Kac formula. We also treat diffusion with drift, involving semigroups generated by Δ+X, where X is a vector field on M , satisfying convenient hypotheses. A final topic we take up in §11 is relativistic diffusion on Minkowski space, which we relate to diffusion on hyperbolic space. A number of sources use stochastic ODE as an essential tool in the study of diffusion on manifolds, but we do not take this up here. The reader who gets through this chapter will be in a good position to appreciate such treatments, which can be found in [Stk3], [Hsu], and [Em]. Another important topic that we do not treat here is Malliavin’s stochastic calculus of variations, introduced in [Mal], which has had numerous interesting applications to PDE. We refer the reader to [Stk2] and [B] for material on this, and further references.

406 11. Brownian Motion and Potential Theory

1. Brownian motion and Wiener measure One way to state the probabilistic connection with the heat equation ∂u = Δu ∂t

(1.1)

is in terms of the heat kernel, p(t, x, y), satisfying  etΔ f (x) =

(1.2)

p(t, x, y)f (y) dV (y).

If Δ in (1.1) is the Friedrichs extension of the Laplacian on any Riemannian manifold M , the maximum principle implies p(t, x, y) ≥ 0.

(1.3)

In many cases, including all compact M and M = Rn , we also have  p(t, x, y) dV (y) = 1.

(1.4)

Consequently, for each x ∈ M, p(t, x, y) dV (y) defines a probability distribution, which we can interpret as giving the probability that a particle starting at the point x at time 0 will be in a given region in M at time t. Restricting our attention to the case M = Rn , we proceed to construct a probability measure, known as “Wiener measure,” on the set of paths ω : [0, ∞) → Rn , undergoing a random motion, sometimes called Brownian motion, described as follows. Given t1 < t2 and that ω(t1 ) = x1 , the probability density for the location of ω(t2 ) is (1.5)

etΔ δx1 (x) = p(t, x − x1 ) = (4πt)−n/2 e−|x−x1 |

2

/4t

,

t = t2 − t 1 .

The motion of a random path for t1 ≤ t ≤ t2 is supposed to be independent of its past history. Thus, given 0 < t1 < t2 < · · · < tk , and given Borel sets Ej ⊂ Rn , the probability that a path, starting at x = 0 at t = 0, lies in Ej at time tj for each j ∈ [1, k] is 

 ···

(1.6) E1

p(tk − tk−1 , xk − xk−1 ) · · · p(t1 , x1 ) dxk · · · dx1 .

Ek

It is not obvious that there is a countably additive measure characterized by these properties, and Wiener’s result was a great achievement. The construction we give here is a slight modification of one in Appendix A of [Nel2].

1. Brownian motion and Wiener measure

407

Anticipating that Wiener measure is supported on the set of continuous paths, we will take a path to be characterized by its locations at all positive rational t. Thus, we consider the set of “paths” 

P=

(1.7)

R˙ n .

t∈Q+

˙ n is the one-point compactification of Rn (i.e., R˙ n = Rn ∪ {∞}). Thus Here, R P is a compact, metrizable space. We construct Wiener measure W as a positive Borel measure on P. By the Riesz theorem, it suffices to construct a positive linear functional E : C(P) → R, on the space C(P) of real-valued, continuous functions on P, satisfying E(1) = 1. We first define E on the subspace C # , consisting of continuous functions that depend on only finitely many of the factors in (1.7); that is, functions on P of the form (1.8)

  ϕ(ω) = F ω(t1 ), . . . , ω(tk ) ,

t 1 < · · · < tk ,

k where F is continuous on 1 R˙ n , and tj ∈ Q+ . To be consistent with (1.6), we take   E(ϕ) = · · · p(t1 , x1 )p(t2 − t1 , x2 − x1 ) (1.9)

· · · p(tk − tk−1 , xk − xk−1 ) F (x1 , . . . , xk ) dxk · · · dx1 .

If ϕ(ω) in (1.8) actually depends only on ω(tν ) for some proper subset {tν } of {t1 , . . . , tk }, there arises a formula for E(ϕ) with a different appearance from (1.9). The fact that these two expressions are equal follows from the semigroup property of etΔ . From this it follows that E : C # → R is well defined. It is also a positive linear functional, satisfying E(1) = 1. Now, by the Stone-Weierstrass theorem, C # is dense in C(P). Since E : C # → R is a positive linear functional and E(1) = 1, it follows that E has a unique continuous extension to C(P), possessing these properties. Thus there is a unique probability measure W on P such that  (1.10)

ϕ(ω) dW (ω).

E(ϕ) = P

This is the Wiener measure. Proposition 1.1. The set P0 of paths from Q+ to Rn , which are uniformly continuous on bounded subsets of Q+ (and which thus extend uniquely to continuous paths from [0, ∞) to Rn ), is a Borel subset of P with Wiener measure 1.

408 11. Brownian Motion and Potential Theory

For a set S, let oscS (ω) denote sups,t∈S |ω(s) − ω(t)|. Set (1.11)

 E(a, b, ε) = ω ∈ P : osc[a,b] (ω) > 2ε ;

here [a, b] denotes {s ∈ Q+ : a ≤ s ≤ b}. Its complement is (1.12)



E c (a, b, ε) =

ω ∈ P : |ω(s) − ω(t)| ≤ 2ε ,

t,s∈[a,b]

which is closed in P. Below we will demonstrate the following estimate on the Wiener measure of E(a, b, ε): (1.13)

  ε  W E(a, b, ε) ≤ 2ρ , |b − a| , 2

where  (1.14)

ρ(ε, δ) = sup t≤δ

p(t, x) dx,

|x|>ε

with p(t, x) = etΔ δ(x), as in (1.5). In fact, the sup is assumed at t = δ, so  (1.15)

ρ(ε, δ) = √ |y|>ε/ δ

 ε  p(1, y) dy = ψn √ , δ

where (1.16)

−n/2



ψn (r) = (4π)

e−|y|

2

/4

dy ≤ αn rn−1 e−r

2

/4

,

|y|>r

as r → ∞. The relevance of the analysis of E(a, b, ε) is that if we set  (1.17) F (k, ε, δ) = ω ∈ P : ∃ J ⊂ [0, k] ∩ Q+ , (J) ≤ δ, oscJ (ω) > 4ε , where (J) is the length of the interval J, then (1.18)

F (k, ε, δ) =



E(a, b, 2ε) : [a, b] ⊂ [0, k], |b − a| ≤ δ

is an open set, and, via (1.13), we have (1.19)

  ρ(ε, δ) . W F (k, ε, δ) ≤ 2k δ



1. Brownian motion and Wiener measure

409

Furthermore, with F c (k, ε, δ) = P \ F (k, ε, δ),

(1.20)

 P0 = ω : ∀k < ∞, ∀ε > 0, ∃δ > 0 such that ω ∈ F c (k, ε, δ)

F c (k, ε, δ) = k ε=1/ν δ=1/μ

is a Borel set (in fact, an Fσδ set), and we can conclude that W (P0 ) = 1 from (1.19), given the observation that, for any ε > 0, ρ(ε, δ) −→ 0, as δ → 0, δ

(1.21)

which follows immediately from (1.15) and (1.16). Thus, to complete the proof of Proposition 1.1, it remains to establish the estimate (1.13). Lemma 1.2. Given ε, δ > 0, take ν numbers tj ∈ Q+ , 0 ≤ t1 < · · · < tν , such that tν − t1 ≤ δ. Let (1.22)

 A = ω ∈ P : |ω(t1 ) − ω(tj )| > ε, for some j = 1, . . . , ν .

Then ε  W (A) ≤ 2ρ , δ . 2

(1.23) Proof. Let

 ε , B = ω : |ω(t1 ) − ω(tν )| > 2  ε Cj = ω : |ω(tj ) − ω(tν )| > , 2 Dj = {ω : |ω(t1 ) − ω(tj )| > ε and

(1.24)

|ω(t1 ) − ω(tk )| ≤ ε, ∀ k ≤ j − 1}. Then A ⊂ B ∪

ν 

 Cj ∩ Dj , so

j=1

(1.25)

W (A) ≤ W (B) +

ν 

  W Cj ∩ Dj .

j=1

Clearly, W (B) ≤ ρ(ε/2, δ). Furthermore, via (1.8)–(1.9), if we set   D ω(t1 ), . . . , ω(tj ) = 1, if ω ∈ Dj , 0 otherwise,   C ω(tj ), ω(tν ) = 1, if ω ∈ Cj , 0 otherwise,

410 11. Brownian Motion and Potential Theory

we have C(xj , xν ) = C1 (xj − xν ) and (1.26) W (Cj ∩ Dj )   = · · · D(x1 , . . . , xj )C(xj , xν )p(t1 , x1 )p(t2 − t1 , x2 − x1 ) · · · p(tj − tj−1 , xj − xj−1 )p(tν − tj , xν − xj ) dxν dxj · · · dx1   ε  · · · D(x1 , . . . , xj )p(t1 , x1 ) · · · p(tj − tj−1 , xj − xj−1 ) ≤ ρ ,δ 2 · dxj · · · dx1 ε  ≤ ρ , δ W (Dj ), 2 so (1.27)

 j

  ε W Cj ∩ Dj ≤ ρ( , δ), 2

since the Dj are mutually disjoint. This proves (1.23). Let us note an intuitive approach to (1.26). Since Dj describes properties of ω(t) for t ∈ [t1 , tj ] and Cj describes a property of ω(tν ) − ω(tj ), these sets describe independent events, so W (Cj ∩ Dj ) = W (Cj )W (Dj ); meanwhile W (Cj ) ≤ ρ(ε/2, δ). We continue the demonstration of (1.13). Now, given such tj as in the statement of Lemma 1.2, if we set (1.28)

 E = ω : |ω(tj ) − ω(tk )| > 2ε, for some j, k ∈ [1, ν] ,

it follows that (1.29)

ε  W (E) ≤ 2ρ , δ , 2

since E is a subset of A, given by (1.22). Now, E(a, b, ε), given by (1.11), is a countable increasing union of sets of the form (1.28), obtained, say, by letting {t1 , . . . , tν } consist of all t ∈ [a, b] that are rational with denominator ≤ K, and taking K +∞. Thus we have (1.13), and the proof of Proposition 1.1 is complete. We make the natural identification of paths ω ∈ P0 with continuous paths ω : [0, ∞) → Rn . Note that a function ϕ on P0 of the form (1.8), with tj ∈ R+ , not necessarily rational, is a pointwise limit on P0 of functions in C # , as long as k F is continuous on 1 R˙ n , and consequently such ϕ is measurable. Furthermore, (1.9) continues to hold, by the dominated convergence theorem. An alternative approach to the construction of W would be to replace (1.7) 

=  R˙ n : t ∈ R+ . With the product topology, this is compact but not by P

1. Brownian motion and Wiener measure

411

but not a Baire set, metrizable. The set of continuous paths is a Borel subset of P, so some extra measure-theoretic considerations arise if one takes this route. Looking more closely at the estimate (1.19) of the measure of the set F (k, ε, δ), defined by (1.17), we note that you can take ε = K δ log 1/δ, in which case (1.30)

   1 n/2−1 K 2 /4 1 ≤ Cn log δ . ρ(ε, δ) = ψn K log δ δ

Then we obtain the following refinement of Proposition 1.1. Proposition 1.3. For almost all ω ∈ P, we have the modulus of continuity  8 δ log 1/δ, that is, given 0 ≤ s, t ≤ k < ∞,  (1.31)

lim sup |s−t|=δ→0

    ω(s) − ω(t) − 8 δ log 1 ≤ 0. δ

In fact, (1.30) gives W (Sk ) = 1, where Sk is the set of paths satisfying (1.31), with 8 replaced by 8+1/k, and then k Sk is precisely the set of paths satisfying (1.31). This result  is not quite sharp; P. Levy showed that, for almost all ω ∈ P, with μ(δ) = 2 δ log 1/δ, 0 ≤ s, t ≤ k < ∞, (1.32)

lim sup |s−t|→0

|ω(s) − ω(t)| = 1. μ(|s − t|)

See [McK] for a proof. We also refer to [McK] for a proof of the result, due to Wiener, that almost all paths ω are nowhere differentiable. By comparison with (1.31), note that if we define functions Xt on P, taking values in Rn , by Xt (ω) = ω(t),

(1.33)

then a simple application of (1.8)–(1.10) yields (1.34)

Xt 2L2 (P)

 =

|x|2 p(t, x) dx = 2nt,

and more generally (1.35)

Xt − Xs L2 (P) =



2n |s − t|1/2 .

Note that (1.35) depends on n, while (1.32) does not. Via a simple translation of coordinates, we have a similar construction for the set of Brownian paths ω starting at a general point x ∈ R , yielding the positive functional Ex : C(P) → R, and Wiener measure Wx , such that

412 11. Brownian Motion and Potential Theory

 (1.36)

Ex (ϕ) =

ϕ(ω) dWx (ω). P

When ϕ(ω) is given by (1.8), Ex (ϕ) has the form (1.9), with the function p(t1 , x1 ) replaced by p(t1 , x1 − x). To put it another way, Ex (ϕ) has the form (1.9) with F (x1 , . . . , xk ) replaced by F (x1 + x, . . . , xk + x). We will often use such notation as   Ex f (ω(t)      instead of P f Xt (ω) dWx (ω) or Ex f (Xt (ω)) . The following simple observation is useful. Proposition 1.4. If ϕ ∈ C(P), then Ex (ϕ) is continuous in x. Proof. Continuity for ϕ ∈ C # , the set of functions of the form (1.8), is clear from (1.9) and its extension to x = 0 discussed above. Since C # is dense in C(P), the result follows easily.

Exercises 1. Given a > 0, define a transformation Da : P0 → P0 by (Da ω)(t) = aω(a−2 t). Show that Da preserves the Wiener measure W . This transformation is called Brownian scaling.  0 ) = 1.  0 = {ω ∈ P0 : lims→∞ s−1 ω(s) = 0}. Show that W (P 2. Let P  0 → P0 by Define a transformation ρ : P (ρω)(t) = tω(t−1 ), for t > 0. Show that ρ preserves the Wiener measure W . 3. Given a > 0, define a transformation Ra : P0 → P0 by (Ra ω)(t) = ω(t), 2ω(a) − ω(t),

for 0 ≤ t ≤ a, for t ≥ a.

Show that Ra preserves the Wiener measure W . 4. Show that Lp (P0 , dW0 ) is separable, for 1 ≤ p < ∞. (Hint: P is a compact metric space. Show that C(P) is separable.) 5. If 0 ≤ a1 < b1 ≤ a2 < b2 , show that Xb1 − Xa1 is orthogonal to Xb2 − Xa2 in L2 (P, dWx , Rn ), where Xt (ω) = ω(t), as in (1.33). 6. Verify the following identities (when n = 1):   2 (1.37) Ex eλ(ω(t)−ω(s)) = e|t−s|λ ,  2k  (2k)! = Ex ω(t) − ω(s) (1.38) |t − s|k , k!

2. The Feynman–Kac formula

413

  E ω(s)ω(t) = 2 min(s, t).

(1.39) 2

7. Show that eλ|ω(t)| ∈ L2 (P0 , dW0 ) if and only if λ < 1/8t.

2. The Feynman–Kac formula To illustrate the application of Wiener measure to PDE, we now derive a formula, known as the Feynman–Kac formula, for the solution operator et(Δ−V ) to ∂u = Δu − V u, ∂t

(2.1)

u(0) = f,

given f in an appropriate Banach space, such as Lp (Rn ), 1 ≤ p < ∞, or f ∈ Co (Rn ), the space of continuous functions on Rn vanishing at infinity. To start, we will assume V is bounded and continuous on Rn . Following [Nel2], we will use the Trotter product formula  k f. et(Δ−V ) f = lim e(t/k)Δ e−(t/k)V

(2.2)

k→∞

 For any k,

e(t/k)Δ e−(t/k)V

k f is expressed as a k-fold integral:

k e(t/k)Δ e−(t/k)V f (x)    t = · · · f (xk )e−(t/k)V (xk ) p , xk − xk−1 e(t/k)V (xk−1 ) · · · k  t −(t/k)V (x1 ) ·e p , x − x1 dx1 · · · dxk . k

 (2.3)

Comparison with (1.36) gives  (2.4)

e(t/k)Δ e−(t/k)V

k f (x) = Ex (ϕk ),

where (2.5)

  ϕk (ω) = f ω(t) e−Sk (ω) ,

Sk (ω) =

k t    jt  . V ω k j=1 k

We are ready to prove the Feynman–Kac formula. Proposition 2.1. If V is bounded and continuous on Rn , and f ∈ C(Rn ) vanishes at infinity, then, for all x ∈ Rn ,

414 11. Brownian Motion and Potential Theory

(2.6)

    t et(Δ−V ) f (x) = Ex f ω(t) e− 0 V (ω(τ )) dτ .

Proof. We know that et(Δ−V ) f is equal to the limit of (2.4) as k → ∞, in the sup norm. Meanwhile, since almost all ω ∈ P are continuous paths, Sk (ω) → t V (ω(τ ))dτ boundedly and a.e. on P. Hence, for each x ∈ Rn , the right side 0 of (2.4) converges to the right side of (2.6). This finishes the proof. Note that if V is real-valued and in L∞ (Rn ), then et(Δ−V ) is defined on L (Rn ), by duality from its action on L1 (Rn ), and ∞

(2.7)

fν ∈ C0∞ (Rn ), fν 1 =⇒ et(Δ−V ) fν et(Δ−V ) 1.

Thus, if V is real-valued, bounded, and continuous, then, for all x ∈ Rn ,  t  (2.8) et(Δ−V ) 1(x) = Ex e− 0 V (ω(τ ))dτ . We can extend these identities to some larger classes of V . First we consider the nature of the right side of (2.6) for more general V . Lemma 2.2. Fix t ∈ [0, ∞). If V ∈ L∞ (Rn ), then  (2.9)

IV (ω) =

t

V (ω(τ )) dτ 0

is well defined in L∞ (P). If Vν is a bounded sequence in L∞ (Rn ) and Vν → V in measure, then IVν → IV boundedly and in measure on P. This is true for each measure Wx , x ∈ Rn . Proof. Here, L∞ is the set of equivalence classes (mod a.e. equality) of bounded measurable functions, that is, elements of L∞ (Rn ). Suppose W ∈ L∞ (Rn ) is a t pre-image of V . Then 0 W (ω(τ )) dτ = ιW (ω) is defined and measurable, and ιW L∞ (P) ≤ W L∞ (Rn ) t. If W # is also a pre-image of V , then W = W # almost everywhere on Rn . Look at U , defined on P × R+ by U (ω, s) = W (ω(s)) − W # (ω(s)). This is measurable. Let K ⊂ Rn be the set where W (x) = W # (x); this has measure 0. Now, for fixed s, the set of ω ∈ P such that ω(s) ∈ K has Wiener + measure 0. By Fubini’s theorem it follows that U = 0 a.e.  t on #P × R , and + hence, for almost all ω ∈ P, U (ω, ·) = 0 a.e. on R . Thus 0 W (ω(τ )) dτ = t W (ω(τ )) dτ for a.e. ω ∈ P, so IV is well defined in L∞ (P) for each V ∈ 0 ∞ L (Rn ). Clearly, IV L∞ ≤ V L∞ t. If Vν → V boundedly and in measure, in view of the previous argument we can assume without loss of generality that, upon passing to a subsequence, Vν (x) → V (x) for all x. Consider

2. The Feynman–Kac formula

415

Uν (ω, s) = V (ω(s)) − Vν (ω(s)), which is bounded in L∞ (P×R+ ). This converges to 0 for each (ω, s) ∈ P×R+ , t so by Fubini’s theorem again, 0 Uν (ω, s) ds → 0 for a.e. ω. This completes the proof. A similar argument yields the following. Lemma 2.3. If V ∈ L1loc (Rn ) is bounded from below, then (2.10)

eV (ω) = e−

t 0

V (ω(τ )) dτ

is well defined in L∞ (P). If Vν ∈ L1loc (Rn ) are uniformly bounded below and Vν → V in L1loc , then eVν → eV boundedly and in measure on P. Thus, if V ∈ L1loc (Rn ), V ≥ −K > −∞, take bounded, continuous Vν such that Vν ≥ −K and Vν → V in L1loc . We have et(Δ−Vν )  ≤ eKt for all ν, where  ·  can be the operator norm on Lp (Rn ) or on Co (Rn ). Now, if we replace V by Vν in (2.6), then Lemma 2.3 implies that, for any f ∈ C0∞ (Rn ), the right side converges, for each x, namely,     t (2.11) Ex f ω(t) e− 0 Vν (ω(τ )) dτ −→ P (t)f (x), as ν → ∞. Clearly |P (t)f (x)| ≤ eKt Ex (|f |) ≤ eKt f L∞ . Consequently, for each x ∈ Rn , if f ∈ C0∞ (Rn ),     t (2.12) et(Δ−Vν ) f (x) −→ P (t)f (x) = Ex f ω(t) e− 0 V (ω(τ )) dτ . It follows that P (t) : C0∞ (Rn ) → L∞ (Rn ). Since  t(Δ−V )  ν e (2.13) f (x) ≤ eKt etΔ |f |(x), we also have P (t) : C0∞ (Rn ) → L1 (Rn ). Furthermore, we can pass to the limit in the PDE ∂uν /∂t = Δuν − Vν uν for uν = et(Δ−Vν ) f , to obtain for u(t) = P (t)f the PDE (2.14)

∂u = Δu − V u, ∂t

u(0) = f.

If Δ − V , with domain D = D(Δ) ∩ D(V ), is self-adjoint, or has self-adjoint closure A, the uniqueness result of Proposition 9.11 in Appendix A, Functional Analysis, guarantees that P (t)f = etA f . For examples of such self-adjointness results on Δ − V , see Chap. 8, §2, and the exercises following that section. Thus the identity (2.6) extends to such V , for example, to V ∈ L∞ (Rn ); so does the identity (2.8). We can derive a similar formula for the solution operator S(t, 0) to

416 11. Brownian Motion and Potential Theory

(2.15)

∂u = Δu − V (t, x)u, ∂t

u(0) = f,

using the time-dependent Trotter product formula, Proposition A.5, and its consequence, Proposition A.6. Thus, we obtain (2.16)

    t S(t, 0)f (x) = Ex f ω(t) e− 0 V (τ,ω(τ )) dτ

  when V (t) ∈ C [0, ∞), BC(Rn ) , BC(Rn ) denoting the space of bounded continuous functions on Rn . By arguments such as those used above, we can extend this identity to larger classes of functions V (t).

Exercises 1. Given ε > 0, λ ∈ R, compute the integral operator giving 2

et(∂x −εx

(2.17)

2

−λx)

f (x).

(Hint: Use εx2 + λx = ε(x + λ/2ε)2 − λ2 /4ε to reduce this to the problem of computing the integral operator giving 2

2

et(∂x −εx ) g(x).

(2.18)

For this, see the material on the harmonic oscillator in §6 of Chap. 8, in particular, Mehler’s formula.) 2. Obtain a formula for   t t 2 2 2 (2.19) Ex e−ε 0 ω(s) ds−λ 0 ω(s) ds = et(∂x −εx −λx) 1(x), in the case of one-dimensional Brownian motion. (Hint: Use the formula 2

(2.20)

2

2

et(∂x −εx ) 1(x) = a(t)e−b(t)x ,  √ √ −1/2 1√ , b(t) = ε tanh 2 εt, a(t) = cosh 2 εt 2

which follows from the formula for (2.18). Alternatively, verify (2.20) directly, examining the system of ODE a (t) = −2a(t)b(t),

b (t) = ε − 4b(t)2 .)

3. Pass to the limit ε  0 in (2.19), to evaluate   t (2.21) Ex e−λ 0 ω(s)ds . Note that the monotone convergence theorem applies. Exercises 4 and 5 will investigate

a (2.22) ψ(ε) = W0 ω∈P: ω(s)2 ds < ε =P 0

a 0

ω(s)2 ds < ε .

3. The Dirichlet problem and diffusion on domains with boundary

417

4. Using Exercise 2, show that, for all λ > 0, ∞   a 2 ψ  (s)e−λs ds = E0 e−λ 0 ω(s) ds 0 (2.23) √  √ −1/2 √ −a√λ   −1/2 1 + e−4a λ = 2e . = cosh 2a λ Other derivations of (2.23) can be found in [CM] and [Lev]. 5. The subordination identity, given as (5.22) in Chap. 3, implies ∞ √ √ 2 a ϕa (s)e−λs ds = 2e−a λ if ϕa (s) = √ s−3/2 e−a /4s . 2π 0 Deduce that ψ  (s) = ϕa (s) − hence that

(2.24)

d P dε



a

1 3 ϕ5a (s) + ϕ9a (s) − · · · , 2 8

ω(s)2 ds < ε

0

  2 2 2 a 1 3 = √ ε−3/2 e−a /4ε − · 5e−25a /4ε + · 9e−81a /4ε − · · · . 2 8 2π

Show that the terms in this alternating series have progressively decreasing magnitude provided ε/a2 ≤ 1/2. (Hint: Use the power series 1 3 (1 + y)−1/2 = 1 − y + y 2 − · · · 2 8 √

with y = e−4a λ .) 6. Suppose now that ω(t) is Brownian motion in Rn . Show that    a √ −n/2 2 E0 e−λ 0 |ω(s)| ds = cosh 2a λ . Deduce that in the case n = 2,    2 2 2 d  a 2a |ω(s)|2 ds < ε = √ ε−3/2 e−a /ε − 3e−9a /ε + 5e−25a /ε − · · · . P dε π 0 Show that the terms in this alternating series have progressively decreasing magnitude provided ε ≤ 2a2 .

3. The Dirichlet problem and diffusion on domains with boundary We can use results of §2 to provide connections between Brownian motion and the Dirichlet boundary problem for the Laplace operator. We begin by extending Lemma 2.3 to situations where Vν V , with V (x) possibly equal to +∞ on a big set. We have the following analogue of Lemma 2.3. Lemma 3.1. Let Vν ∈ L1loc (Rn ), −K ≤ Vν V , with possibly V (x) = +∞ on a set of positive measure. Then eV (ω), given by (2.10), is well defined in L∞ (P),

418 11. Brownian Motion and Potential Theory

provided we set e−∞ = 0, and eVν → eV boundedly and in measure on Ω, for each t. Proof. This follows from the monotone convergence theorem. Thus we again have convergence with bounds in (2.11)–(2.13). We will look at a special class of such sequences. Let Ω ⊂ Rn be open, with smooth boundary (in fact, Lipschitz boundary will more than suffice), and set E = Rn \ Ω. Let Vν ≥ 0 be continuous and bounded on Rn and satisfy Vν = 0 on Ω,

(3.1)

Vν ≥ ν on Eν ,

Vν ,

where Eν is the set of points of distance ≥ 1/ν from Ω. Given f ∈ L2 (Rn ), g ∈ L2 (Ω), set PΩ f = f |Ω ∈ L2 (Ω), and define EΩ g ∈ L2 (Rn ) to be g(x) for x ∈ Ω, 0 for x ∈ E = Rn \ Ω. Proposition 3.2. Under the hypotheses above, if f ∈ L2 (Rn ), then   (3.2) et(Δ−Vν ) f −→ EΩ etΔΩ PΩ f , as ν → ∞, where ΔΩ is the Laplace operator with Dirichlet boundary condition on Ω. Proof. We will first show that, for any λ > 0, (3.3)



λ − Δ + Vν

−1

 −1 f → EΩ λ − ΔΩ PΩ f.

Indeed, denote the left side of (3.3) by uν , so (λ − Δ + Vν )uν = f . Taking the inner product with uν , we have (3.4) λuν 2L2 +∇uν 2L2 +



Vν |uν |2 dx = (f, uν ) ≤

λ 1 uν 2L2 + f 2L2 , 2 2λ

so (3.5)

λ uν 2L2 + ∇uν 2L2 + 2



Vν |uν |2 dx ≤

1 f 2L2 . 2λ

fixed λ > 0, {uν : ν ∈ Z+ } is bounded in H 1 (Rn ), while Thus, for 2 |u | dx ≤ C/ν. Thus {uν } has a weak limit point u ∈ H 1 (Rn ), and ν Eν u = 0 on ∪Eν . The regularity hypothesized for ∂Ω implies u ∈ H01 (Ω). Clearly, (λ − Δ)u = f on Ω, so (3.3) follows, with weak convergence in H 1 (Rn ). But note that, parallel to (3.4), λu2L2 + ∇u2L2 = (f, u) = lim (f, uν ), ν→∞

so

3. The Dirichlet problem and diffusion on domains with boundary

(3.6)

419

λu2L2 + ∇u2L2 ≥ lim sup λuν 2L2 + ∇uν 2L2 . ν→∞

Hence, in fact, we have H 1 -norm convergence in (3.3), and a fortiori L2 -norm convergence. Now consider the set F of real-valued ϕ ∈ Co ([0, ∞)) such that, for all f ∈ L2 (Rn ), (3.7)

ϕ(−Δ + Vν )f −→ EΩ ϕ(−ΔΩ )PΩ f, in L2 (Rn )-norm,

where ϕ(H) is defined via the spectral theorem for a self-adjoint operator H. (Material on this functional calculus can be found in §1 of Chap. 8.) The analysis above shows that, for each λ > 0, rλ (s) = (λ + s)−1 belongs to F. Since PΩ EΩ is the identity on L2 (Ω), it is clear that F is an algebra; it is also easily seen to be a closed subset of Co ([0, ∞)). Since it contains rλ for λ > 0, it separates points, so by the Stone-Weierstrass theorem all real-valued ϕ ∈ Co ([0, ∞)) belong to F. This proves (3.2). The version of (2.12) we have this time is the following. Proposition 3.3. Let Ω ⊂ Rn be open, with smooth boundary, or more generally with the property that {u ∈ H 1 (Rn ) : supp u ⊂ Ω} = H01 (Ω). Let F ∈ C0∞ (Rn ), f = F |Ω . Then, for all x ∈ Ω, t ≥ 0,     t etΔ f (x) = Ex f ω(t) e− 0 Ω (ω(τ )) dτ .

(3.8)

On the left, etΔ is the solution operator to the heat equation on R+ × Ω with Dirichlet boundary condition on ∂Ω, and in the expression on the right ◦

Ω (x) = 0 on Ω, + ∞ on Rn \ Ω = E.

(3.9)

Note that, for ω continuous, (3.10)

e−

t 0

Ω (ω(τ )) dτ

= ψΩ (ω, t) = 1 0

if ω([0, t]) ⊂ Ω, otherwise.

The second identity defines ψΩ (ω, t). Of course, for ω continuous, ω([0, t]) ⊂ Ω if and only if ω([0, t] ∩ Q) ⊂ Ω. We now extend Proposition 3.3 to the case where Ω ⊂ Rn is open, with no regularity hypothesis on ∂Ω. Choose a sequence Ωj of open regions with smooth boundary, such that Ωj ⊂⊂ Ωj+1 ⊂⊂ · · · , j Ωj = Ω. Let Δj denote the

420 11. Brownian Motion and Potential Theory

Laplace operator on Ωj , with Dirichlet boundary condition, and let Δ denote that of Ω, also with Dirichlet boundary condition. Lemma 3.4. Given f ∈ L2 (Ω), t ≥ 0, etΔ f = lim Ej etΔj Pj f,

(3.11)

j→∞

where Pj f = f |Ωj and, for g ∈ L2 (Ωj ), Ej g(x) = g(x) for x ∈ Ωj , 0 for x ∈ Ω \ Ωj . Proof. Methods of Chap. 5, §5, show that, for λ > 0, (3.12)

Ej (λ − Δj )−1 Pj f → (λ − Δ)−1 f

in L2 -norm, and then (3.11) follows from this, by reasoning used in the proof of Proposition 3.2. Suppose f ∈ C0∞ (ΩL ). Then, for j ≥ L, Ej etΔj f → etΔ f in L2 -norm, as we have just seen. Furthermore, local regularity implies (3.13)

Ej etΔj f −→ etΔ f

locally uniformly on Ω.

Thus, given such f , and any x ∈ Ω (hence x ∈ Ωj for j large), (3.14)

    etΔ f (x) = lim Ex f ω(t) ψΩj (ω, t) . j→∞

Now, as j → ∞, ψΩj (ω, t) ψΩ (ω, t),

(3.15) where we define (3.16)

ψΩ (ω, t) = 1 0

if ω([0, t]) ⊂ Ω, otherwise.

This yields the following: Proposition 3.5. For any open Ω ⊂ Rn , given f ∈ C0∞ (Ω), x ∈ Ω, (3.17)

    etΔ f (x) = Ex f ω(t) ψΩ (ω, t) .

In particular, if Ω has smooth boundary, one can use either ψΩ (ω, t) or ψΩ (ω, t) in the formula for etΔ f (x). However, if ∂Ω is not smooth, it is ψΩ (ω, t) that one must use.

3. The Dirichlet problem and diffusion on domains with boundary

421

It is useful to extend this result to more general f . Suppose fj ∈ C0∞ (Ω), f ∈ tΔ tΔ L (Ω), and fj (x)  f (x) for each x ∈ Ω. Then,  for any t > 0, e fj → e f 2 ∞ in L (Ω) ∩ C (Ω), while, for each x ∈ Ω, Ex fj (ω(t))ψΩ (ω, t) converges  to the right side of (3.17), by the monotone convergence theorem. Hence (3.17) holds for all such f ; denote this class by L(Ω). Clearly, the characteristic function χK ∈ L(Ω) for each compact K ⊂ Ω. By the same reasoning, the class of functions in L2 (Ω) for which (3.17) holds is closed under forming monotone limits, either fj f or fj  f , of sequences bounded in L2 (Ω). An argument used in Lemma 2.2 shows that modifying f ∈ L2 (Ω) on a set of measure zero does not change the right side of (3.17). If S ⊂ Ω is measurable, then χS (x) = lim χKj (x), a.e., 2

j→∞

for an increasing sequence of compact sets Kj ⊂ S, so (3.17) holds for f = χS . Thus it holds for finite linear combinations of such characteristic functions, and an easy limiting argument gives the following: Proposition 3.6. The identity (3.17) holds for all f ∈ L2 (Ω) when t > 0, x ∈ Ω. Suppose now that Ω is bounded. Then, for f ∈ Lp (Ω), 1 ≤ p ≤ ∞, −Δ−1 f =

(3.18)





etΔ f dt,

0

the integral being absolutely convergent in Lp -norm. If f ∈ C0∞ (Ω), we hence have, for each x ∈ Ω,   ∞   f ω(t) ψΩ (ω, t) dt . (3.19) −Δ−1 f (x) = Ex 0

Furthermore, by an argument such as used to prove Proposition 3.6, this identity holds for almost every x ∈ Ω, given f ∈ L2 (Ω), and for every x if fj ∈ C0∞ (Ω) and fj (x) f (x) for all x. In particular, for Ω bounded, (3.20)

  −Δ−1 1(x) = Ex ϑΩ (ω) ,

x ∈ Ω,

where, if ω is a continuous path starting inside Ω, we define  (3.21)

ϑΩ (ω) =

0



ψΩ (ω, t)dt = sup {t : ω([0, t]) ⊂ Ω} = min {t : ω(t) ∈ ∂Ω}.

In other words, ϑΩ (ω) is the first time ω(t) hits ∂Ω; it is called the “first exit time.” Since Δ−1 1 ∈ C ∞ (Ω), it is clear that the first exit time for a path starting at any x ∈ Ω is finite for Wx -almost every ω when Ω is bounded. (If ω starts at a point in ∂Ω or in Rn \ Ω, set ϑΩ (ω) = 0.) Note that we can write

422 11. Brownian Motion and Potential Theory

−Δ−1 f (x) = Ex

(3.22)



ϑΩ (ω)

   f ω(t) dt .

0

If ∂Ω is smooth enough for Proposition 3.3 to hold, we have the formula (3.19), with ψΩ (ω, t) replaced by ψΩ (ω, t), valid for all x ∈ Ω. In particular, for smooth bounded Ω,   −Δ−1 1(x) = Ex ϑΩ (ω) ,

(3.23)

x ∈ Ω,

where we define (3.24)

    ϑΩ (ω) = inf t : ω(t) ∈ Rn \ Ω = max t : ω [0, t] ⊂ Ω .

(If ω(0) ∈ Rn \ Ω, set ϑΩ (ω) = 0.) Comparing this with (3.20), noting that ϑΩ (ω) ≥ ϑΩ (ω), we have the next result. Proposition 3.7. If Ω is bounded and ∂Ω is smooth enough for Proposition 3.3 to hold, then (3.25)

x ∈ Ω =⇒ ϑΩ (ω) = ϑΩ (ω), for Wx - almost every ω,

and (3.26)

x ∈ ∂Ω =⇒ ϑΩ (ω) = 0, for Wx - almost every ω.

The probabilistic interpretation of this result is that, for any x ∈ Ω, once a Brownian path ω starting at x hits ∂Ω, it penetrates into the interior of Rn \ Ω within an arbitrarily short time, for Wx -almost all ω. From here one can show that, given x ∈ ∂Ω, Wx -a.e. path ω spends a positive amount of time in both Ω and Rn \ Ω, on any time interval [0, s0 ], for any s0 > 0, however small. This is one manifestation of how wiggly Brownian paths are. Note that taking f = 1 in (3.17) gives, for all x ∈ Ω, any open set in Rn , (3.27)

  etΔ 1(x) = Wx {ω : ϑΩ (ω) > t} ,

x ∈ Ω,

the right side being the probability that a path starting in Ω at x has first exit time > t. Meanwhile, if ∂Ω is regular enough for Proposition 3.3 to hold, then (3.28)

etΔ 1(x) = Wx ({ω : ϑΩ (ω) > t}) .

Comparing these identities extends Proposition 3.7 to unbounded Ω. The following is an interesting consequence of (3.28). Proposition 3.8. For one-dimensional Brownian motion, starting at the origin, given t > 0, λ > 0,

3. The Dirichlet problem and diffusion on domains with boundary

(3.29)

423

    W {ω : sup ω(s) ≥ λ} = 2W {ω : ω(t) ≥ λ} . 0≤s≤t

∞ 2 2 Proof. The right side is λ p(t, x) dx, with p(t, x) = etd /dx δ(x) = 2 −x /4t (4πt)−1/2 , the n = 1 case  of (1.5). The left side of (3.29) is the same  e as W {ω : ϑ(−∞,λ) (ω) < t} , which by (3.28) is equal to 1 − etL 1(0) if L = d2 /dx2 on (−∞, λ), with Dirichlet boundary condition at x = λ. By the method of images we have, for x < λ,  p(t, y)H(λ − x + y) dy,

etL 1(x) =

where H(s) = 1 for s > 0, −1 for s < 0. From this, the identity (3.29) readily follows. We next derive an expression for the Poisson integral formula, for the solution PI f = u to (3.30)

Δu = 0 on Ω,

u|∂Ω = f.

This can be expressed in terms of the integral kernel G(x, y) of Δ−1 if ∂Ω is smooth. In fact, an application of Green’s formula gives  (3.31)

f (y)

PI f (x) =

∂ G(x, y) dS(y), ∂νy

∂Ω

where νy is the outward normal to ∂Ω at y. A closely related result is the following. Let f be defined and continuous on a neighborhood of ∂Ω. Given small δ > 0, set (3.32)

Sδ = {x ∈ Ω : dist(x, ∂Ω) < δ},

and define uδ by (3.33)

Δuδ = δ −2 fδ on Ω, uδ = 0 on ∂Ω, fδ = f on Sδ , 0 on Ω \ Sδ .

Lemma 3.9. If ∂Ω is smooth, then, locally uniformly on Ω, (3.34)

1 lim uδ = − PI f. 2

δ→0

Proof. If ν is the outward normal, we have

424 11. Brownian Motion and Potential Theory

uδ (x) = δ −2

  0

∂Ω

= −δ −2

(3.35)

1 =− 2



δ

G(x, y − sν)f (y) ds dS(y) + o(1)

 f (y)

∂Ω

f (y)

 δ  ∂ G(x, y) s ds dS(y) + o(1) ∂νy 0

∂ G(x, y) dS(y) + o(1), ∂νy

∂Ω

so the result follows from (3.31). Comparing this with (3.22), we conclude that when ∂Ω is smooth, (3.36)

2 PI f (x) = lim 2 Ex δ0 δ



ϑΩ (ω)

0

  f ω(t) ιSδ (ω, t) dt , 

where Sδ is as in (3.32), and, for S ⊂ Ω, (3.37)

ιS (ω, t) = 1 0

if ω(t) ∈ S, otherwise.

We will discuss further formulas for PI f in §5.

Exercises 1. Looking at the definitions, check that ψΩ (ω, t) and ϑΩ (ω) are measurable when Ω ⊂ Rn is open with smooth boundary and that ψΩ (ω, t) and ϑΩ (ω) are measurable, for general open Ω ⊂ Rn . 2. Show that if x ∈ O, then  {ω ∈ P0 : ω(s) ∈ Rn \ O}. (3.38) {ω ∈ P0 : ϑO (ω) < t0 } = s∈[0,t0 )∩Q

3. For any finite set S = {s1 , . . . , sK } ⊂ Q+ , N ∈ Z+ , set   FN,S (ω) = ΦN,S ω(s1 ) . . . ω(sK ) ,   ΦN,S (x1 , . . . , xK ) = min N, min{sν : xν ∈ Rn \ O} . Show that, for any continuous path ω, (3.39)

ϑO (ω) = sup inf FN,S (ω). N

S

Note that the collection of such sets S is countable. 4. If PO,N = {ω ∈ P0 : ϑO (ω) ≤ N } and O is bounded, show that (3.40)

  Wx P0 \ PO,N ≤ CN −1 .

Exercises

425

(Hint: Use (3.23).) 5. If ω ∈ PO,N , show that (3.41)

ϑO (ω) = lim ϑν,N (ω), ν→∞

   /O . ϑν,N (ω) = min N, inf s ∈ 2−ν Z+ : ω(s) ∈   Write ϑν,N (ω) = Φν,N ω(s1 ), . . . , ω(sL ) , where Φν,N has a form similar to ΦN,S in Exercise 3. 6. For one-dimensional Brownian motion, establish the following, known as Kolmogorov’s inequality:   2t (3.42) W {ω : sup |ω(s)| ≥ ε} ≤ 2 , ε > 0. ε 0≤s≤t where

  (Hint: Write the left side of (3.42) as W {ω : ϑ(−ε,ε) (ω) < t} , and relate this to the heat equation on Ω = [−ε, ε], with Dirichlet boundary condition, in a fashion parallel to the proof of Proposition 3.8.) Note that this estimate is nontrivial only for t < ε2 /2. By Brownian scaling, it suffices to consider the case ε = 1. Compare the estimate   ∞ ≤4 p(t, x) dx, W ω : sup |ω(s)| ≥ ε 0≤s≤t

ε

which follows from (3.29). 7. Given Ω ⊂ Rn open, with complement K, and Δ with Dirichlet boundary condition on ∂Ω, show that, for x ∈ Ω,   (3.43) Wx {ω : ϑΩ (ω) = ∞} = HK (x), where (3.44)

HK (t, x) = etΔ 1(x)  HK (x), as t ∞.

 K (x) ∈ C(Ω), 8. Suppose that K = Rn \ Ω is compact, and suppose there exists H   harmonic on Ω, such that HK = 0 on ∂K and HK (x) → 1, as |x| → ∞. Show that  K (x), for all t < ∞. HK (t, x) ≥ H (Hint: Show that ΔHK (t, x) ≤ 0 and that HK (t, x) → 1 as |x| → ∞, and use the maximum principle.)    K (x) exists, then Wx {ω : ϑΩ (ω) = ∞} > 0. Deduce that if such H  K exists, then in fact 9. In the context of Exercise 8, show that if such H (3.45)

 K (x), for all x ∈ Ω. HK (x) = H

(Hint: Show that HK must be harmonic in Ω and that lim sup|x|→∞ HK (x) ≤ 1.) By explicit construction, produce such a function on Rn \ B when B is a ball of radius a > 0, provided n ≥ 3. 10. Using Exercises 7–9, show that when n ≥ 3,

426 11. Brownian Motion and Potential Theory (3.46)

  Wx {ω : |ω(t)| → ∞ as t → ∞} = 1.

(Hint: Given R > 0, the probability that |ω(t)| ≥ R for some t is 1. If R >> a, and |ω(t0 )| ≥ R, show that the probability that |ω(t0 + s)| ≤ a for some s > 0 is small, using (3.43) for K = Ba = {x : |x| ≤ a}.) To restate (3.46), one says that Brownian motion in Rn is “non-recurrent,” for n ≥ 3. 11. If n ≤ 2 and K = Ba , show that HK (t, x) = 0 in (3.44), and hence the probability defined in (3.46) is zero. Deduce that if n ≤ 2 and U ⊂ Rn is a nonempty open set, almost every Brownian path ω visits U at an infinite sequence of times tν → ∞. One says that Brownian motion in Rn is “recurrent,” for n ≤ 2. 12. Relate the formula (3.34) for PI f to representations of PI f by double-layer potentials, discussed in §11 of Chap. 7. Where is the second layer coming from? 13. If Ω is a bounded domain with smooth boundary, show that (3.36) remains true with Sδ replaced by Sδ = {x ∈ Rn \ Ω : dist(x, ∂Ω) < δ} and with ϑΩ (ω) replaced by ϑΩδ (ω), where Ωδ = Ω ∪ Sδ . (Hint: Start by showing that u δ (x) → −(1/2)PI f (x), for x ∈ Ω, where, in place of (3.33), Δ uδ = δ −2 fδ on Ωδ ,

u δ = 0 on ∂Ωδ ,

with fδ = f on Sδ , 0 on Ω.

4. Martingales, stopping times, and the strong Markov property Given t ∈ [0, ∞), let Bt be the σ-field of subsets of P0 generated by sets of the form (4.1)

{ω ∈ P0 : ω(s) ∈ E},

where s ∈ [0, t] and E is a Borel subset of Rn . One easily sees that each element of Bt is a Borel set in P. As t increases, Bt is an increasing family of σ-fields, n each  of sets which are Wx -measurable, for all x ∈ R . Set B∞ =  consisting σ t 0, f ∈ C(R˙ n ), (4.5)

     Ex f (ω(t + s))Bs = Eω(s) f (ω(t)) , for Wx -almost all ω.

Proof. The right side of (4.5) is Bs -measurable, so the identity is equivalent to the statement that     f (ω(t + s)) dWx (ω) = ω ) dWx (ω), (4.6) f ( ω (t)) dWω(s) ( S

S

for all S ∈ Bs . It suffices to verify (4.6) for all S of the form S = {ω ∈ P0 : ω(t1 ) ∈ E1 , . . . , ω(tK ) ∈ EK }, given tj ∈ [0, s], Ej Borel sets in Rn . For such S, (4.6) follows directly from the characterization of the Wiener integral given in §1, that is, from (1.6)–(1.9) in the case x = 0, together with the identity    (4.7) f ( ω (t)) dWy ( ω ) = E f (y + ω(t))

428 11. Brownian Motion and Potential Theory

used to define (1.36). We can easily extend (4.5) to      (4.8) Ex F (ω(s + t1 ), . . . , ω(s + tk ))Bs = Eω(s) F (ω(t1 ), . . . , ω(tk )) , k for Wx -almost all ω, given t1 , . . . , tk > 0, and F continuous on 1 R˙ n , as in (1.8). Also, standard limiting arguments allow us to enlarge the class of functions F for which this works. We then get the following more definitive statement of the Markov property. Proposition 4.2. For s > 0, define the map σs : P0 −→ P0 ,

(4.9)

(σs ω)(t) = ω(t + s).

Then, given ϕ bounded and B∞ -measurable, we have    (4.10) Ex ϕ ◦ σs Bs = Eω(s) (ϕ), for Wx -almost all ω. The following is a useful restatement of Proposition 4.2. Corollary 4.3. For s > 0, define the map ϑs : P0 → P0 ,

(4.11)

(ϑs ω)(t) = ω(t + s) − ω(s).

Then, given ϕ ∈ L1 (P0 , dW0 ), we have    Ex ϕ ◦ ϑs Bs = E0 (ϕ).

(4.12) In particular,

      Ex f ϑs ω(t) Bs = E0 f (ω(t)) .

(4.13)

Note that (4.12) implies ϑs is measure preserving, in the sense that   Wx ϑ−1 s (S) = W0 (S),

(4.14)

for W0 -measurable sets S. The map ϑs is not one-to-one, of course, but it is onto the set of paths in P0 satisfying ω(0) = 0. The Markov property also implies certain independence properties. A function ϕ ∈ L1 (P0 , dWx ) is said to be independent of the σ-algebra Bt provided that, for all continuous F ,    F ϕ(ω) dWx (ω) = Wx (S)Ex (F ◦ ϕ), ∀ S ∈ Bt . (4.15) S

An equivalent condition is

4. Martingales, stopping times, and the strong Markov property

(4.16)

    Ex F (ϕ)ψ = Ex F (ϕ) Ex (ψ),

429

∀ ψ ∈ L1 (P0 , Bt , dWx ),

given F (ϕ)ψ ∈ L1 (P0 , dWx ), and another equivalent condition is     (4.17) Ex F (ϕ)|Bt = Ex F (ϕ) . In turn, this identity holds whenever the left side is constant. From Corollary 4.3 we deduce: Corollary 4.4. For s ≥ 0, ϑs ω(t) = ω(t + s) − ω(s) is independent of Bs . Proof. By (4.13), (4.18)

     Ex F (ω(t + s) − ω(s))Bs = E0 F (ω(t)) ,

which is constant. The Markov property gives rise to martingales. By definition (valid in general for an increasing family Bt of σ-fields), a martingale is a family Ft ∈ L1 (P0 , Bt , dWx,t ) such that (4.19)

   Ex Ft Bs = Fs when s < t.

If Ex (Ft |Bs ) ≥ Fs for s < t, {Ft } is called a submartingale over Bt . The following is a very useful class of martingales. Proposition 4.5. Let h(t, x) be smooth in t ≥ 0, x ∈ Rn , and satisfy |h(t, x)| ≤ 2 Cε eε|x| for all ε > 0, and the backward heat equation ∂h = −Δh. ∂t

(4.20)

Then ht (ω) = h(t, ω(t)) is a martingale over Bt . Proof. The hypothesis on h(t, x) implies that, for t, s > 0,  (4.21)

h(s, x) =

p(t, y)h(t + s, x − y) dy,

where p(t, x) = etΔ δ(x) is given by (1.5). Now (4.22)

      Ex ht+s Bs = Ex h(t + s, ω(t + s))Bs   = Eω(s) h(t + s, ω(t)) ,

for Wx -almost all ω, by (4.5). This is equal to  (4.23) p(t, y − ω(s)) h(t + s, y) dy,

430 11. Brownian Motion and Potential Theory

by the characterization (1.9) of expectation, adjusted as in (1.36), and by (4.21) this is equal to h(s, ω(s)) = hs (ω). Corollary 4.6. For one-dimensional Brownian motion, the following are martingales over Bt : xt (ω) = ω(t),

(4.24)

qt (ω) = ω(t)2 − 2t,

2

zt (ω) = eaω(t)−a t ,

given a > 0. One important property of martingales is the following martingale maximal inequality. Proposition 4.7. If Ft is a martingale over Bt , then, given any countable set {tj } ⊂ R+ , the “maximal function” F ∗ (ω) = sup Ftj (ω)

(4.25)

j

satisfies, for all λ > 0,   1 Wx {ω : F ∗ (ω) > λ} ≤ Ft L1 (P0 ,dWx ) . λ

(4.26)

Of course, the assumption that Ft is a martingale implies that Ft L1 is independent of t. Proof. It suffices to demonstrate this for an arbitrary finite subset {tj } of R+ . Thus we can work with fj (ω) = Ftj (ω), Bj = Btj , 1 ≤ j ≤ N , and take t1 < t2 < · · · < tN , and the martingale hypothesis is that Ex (fk Bj ) = fj when j < k. There is no loss in assuming fN (ω) ≥ 0, so all fj (ω) ≥ 0. Now consider (4.27)

Sλ = {ω : f ∗ (ω) > λ} = {ω : some fj (ω) > λ}.

There is a pairwise-disjoint decomposition (4.28)

Sλ =

N

Sλj ,

Sλj = {ω : fj (ω) > λ but f (ω) ≤ λ for < j}.

j=1

Note that Sλj is Bj -measurable. Consequently, we have

4. Martingales, stopping times, and the strong Markov property

431

 fN (ω) dWx (ω) Sλ

(4.29)

=

N  

fN (ω) dWx (ω) =

j=1S λj



N 

N  

fj (ω) dWx (ω)

j=1S λj

  λ Wx Sλj = λ Wx (Sλ ).

j=1

This yields (4.26), in this special case, and the proposition is hence proved. 2

Applying the martingale maximal inequality to zt (ω) = eaω(t)−a t , we obtain the following. Corollary 4.8. For one-dimensional Brownian motion, given t > 0, (4.30)

  W0 {ω ∈ P0 : sup ω(s) − as > λ} ≤ e−aλ . 0≤s≤t

Proof. The set whose measure is estimated in (4.30) is 2

{ω ∈ P0 : sup eaω(s)−a 0≤s≤t

s

> eaλ }.

Since paths in P0 are continuous, one can take the sup over [0, t] ∩ Q, which is countable, so (4.26) applies. Note that E0 (zt ) = 1. We turn to a discussion of the strong Markov property of Brownian motion. For this, we need the notion of a stopping time. A function τ on P0 with values in [0, +∞] is called a stopping time provided that, for each t ≥ 0, {ω ∈ P0 : τ (ω) < t} belongs to the σ-field Bt . It follows from (3.39) that ϑO is a stopping time. So is ϑO . Given a stopping time τ , define Bτ + to be the σ-algebra of sets S ∈ B∞ such that S ∩ {ω : τ (ω) < t} belongs to Bt for each t ≥ 0. Note that τ is measurable with respect to Bτ + . The hypothesis that τ is a stopping time means precisely that the whole set P0 satisfies the criteria for membership in Bτ + . We note that any t ∈ [0, ∞), regarded  as a constant function on P0 , is a stopping time and that, in this case, Bt+ = s>t Bs . The following analogue of Propositions 4.1 and 4.2 is one statement of the strong Markov property. Proposition 4.9. If τ is a stopping time such that τ (ω) < ∞ for Wx -almost all ω, and if t > 0, then       (4.31) Ex f ω(τ + t) Bτ + = Eω(τ ) f (ω(t)) , for Wx -almost all ω. More generally, with

432 11. Brownian Motion and Potential Theory

(στ ω)(t) = ω(t + τ ), and ϕ bounded and B∞ -measurable, we have    (4.32) Ex ϕ ◦ στ Bτ + = Eω(τ ) (ϕ), for Wx -almost all ω. As in (4.6), the content of (4.31) is that         (4.33) f ω(τ + t) dWx (ω) = f ω # (t) dWω(τ ) (ω # ) dWx (ω), S

S

given S ∈ Bτ + . In other words, given that S ∩ {ω : τ (ω) < t } ∈ Bt , for each t ≥ 0. There is no loss in taking x = 0, and we can rewrite (4.33) as        (4.34) f ω(τ + t) dW (ω) = f ω # (t) + ω(τ ) dW (ω # ) dW (ω). S

S

It is useful to approximate τ by discretization: (4.35)

τν (ω) = 2−ν k, if 2−ν (k − 1) ≤ τ (ω) < 2−ν k.

Thus (4.36)

{ω : τν (ω) < t} = {ω : τ (ω) < 2−ν k} ∈ Bt ,

so each τν is a stopping time. Note that (4.37)

Aνk = {ω : τν (ω) = 2−ν k} = {ω : τ (ω) < 2−ν k} \ {ω : τ (ω) < 2−ν (k − 1)}

belongs to B2−ν k . If τ is replaced by τν , the left side of (4.34) becomes     (4.38) f ω(t + 2−ν k) dW (ω), ν,k S∩A

νk

and the right side of (4.34) becomes      (4.39) f ω # (t) + ω(2−ν k) dW (ω # ) dW (ω). ν,k S∩A

νk

Note that if S ∈ Bτ + , then S ∩ Aνk ∈ B2−ν k . Thus, the fact that each term in the sum (4.38) is equal to the corresponding term in (4.39) follows from (4.6).

4. Martingales, stopping times, and the strong Markov property

433

Consequently, we have 

  f ω(τν + t) dW (ω) =

(4.40) S

 

  f ω # (t) + ω(τν ) dW (ω # ) dW (ω),

S

for all ν, if S ∈ Bτ + . The desired identity (4.34) follows by taking ν → ∞, if f ∈ C(R˙ n ). Passing from this to (4.32) is then done as in the proof of Proposition 4.2. In particular, the extension of (4.31) analogous to (4.8), in the special case F (x1 , x2 ) = f (x2 − x1 ), yields the identity  (4.41)

  f ω(τ + t) − ω(τ ) dW (ω) =

S

 

  f ω # (t) dW (ω # ) dW (ω)

S

  = E f (ω(t)) · W (S),

given S ∈ Bτ + . This, together with the extension to F (x1 , . . . xK ), says that ω(τ + t) − ω(τ ) = β(t) has the probability distribution of a Brownian motion, independent of Bτ + . This is a common form in which the strong Markov property is stated. It is sometimes useful to consider stopping times for which {ω : τ (ω) = ∞} has positive measure. In such a case, the extension of Proposition 4.9 is that (4.32) holds for Wx -almost ω in the set {ω : τ (ω) < ∞}. Thus, for example, (4.33) and (4.34) hold, given S ∈ Bτ + and S ⊂ {ω : τ (ω) < ∞}. We next look at some operator-theoretic properties of (4.42)

Qt :L2 (P0 , dW0 ) → L2 (P0 , dW0 ), 2

2

Θt :L (P0 , dW0 ) → L (P0 , dW0 ),

Qt ϕ = E0 (ϕ|Bt ), Θt ϕ(ω) = ϕ(ϑt ω),

where ϑt is given by (4.11). For each t ≥ 0, Qt is an orthogonal projection, and Qs Qt = Qt Qs = Qs , for s ≤ t. Note that (4.13) implies (4.43)

Qt Θt = Q0 ,

since Q0 is the orthogonal projection of L2 (P0 , dW0 ) onto (4.44)

R(Q0 ) = set of constant functions.

Proposition 4.10. The family Θt , t ∈ [0, ∞), is a strongly continuous semigroup of isometries of L2 (P0 , dW0 ), with (4.45)

R(Θt ) ⊂ Ker(Qt − Q0 ) = {ϕ : E0 (ϕ|Bt ) = const.}.

434 11. Brownian Motion and Potential Theory

Proof. That Θt is an isometry follows from the measure-preserving property (4.14). If we apply Q0 to (4.43), we get Q0 Θt = Q0 ; hence (Qt − Q0 )Θt = 0, which yields (4.45). The semigroup property follows from a straightforward calculation: ϑσ ϑs ω = ϑσ+s ω =⇒ Θs+σ = Θs Θσ .

(4.46) The convergence (4.47)

Θs ϕ → Θt ϕ

in L2 (P0 , dW0 ), as s → t,

is easy to demonstrate for ϕ(ω) of the form (1.8), that is,   ϕ(ω) = f ω(t1 ), . . . , ω(tk ) ,

(4.48)

˙ n × · · · × R˙ n (k factors). In fact, ϕ(ϑs (ω)) = ϕs (ω) → with f continuous on R ϕt (ω) boundedly and pointwise on P0 for such ϕ. Since the set of ϕ of the form (4.48) is dense in L2 (P0 , dW0 ), (4.47) follows. Proposition 4.11. The family of orthogonal projections Qt is strongly continuous in t ∈ [0, ∞). Proof. It is easy to verify that, for any ϕ ∈ L2 (P0 , dW0 ), (4.49)

Qs ϕ → Qt− ϕ = E0 (ϕ|Bt− ),

as s t,

provided t > 0, and (4.50)

Qs ϕ → Qt+ ϕ = E0 (ϕ|Bt+ ),

where (4.51)

Bt− = σ

 st

It is also easy to verify that Bt− = Bt , for t > 0, so Qs ϕ → Qt ϕ as s t. On the other hand, it is not true that Bt+ = Bt , so the continuity of Qt ϕ from above requires more work. Suppose tj ∈ Q+ and (4.52)

0 ≤ t1 < t2 < · · · < t ≤ t < t+1 < · · · < t+k .

  Let fj ∈ C R˙ n . Consider any function on P of the form (4.53)

ϕ(ω) = A (ω)Bk (ω)         = f1 ω(t1 ) · · · f ω(t ) · f+1 ω(t+1 ) · · · f+k ω(t+k ) .

Exercises

435

Denote by Cˆ the linear span of the set of such functions. For ϕ of the form (4.53), we have E0 (ϕ|Bt ) = A (ω)E0 (Bk |Bt ).

(4.54)

If t+ν = t + sν , 1 ≤ ν ≤ k, we have, by (4.8), (4.55)

  E0 (Bk |Bt ) = Eω(t) f+1 (ω(s1 )) · · · f+k (ω(sk )) ,

a.e. on P0 .

Now, if t ≤ t + h < t+1 , we also have E0 (ϕ|Bt+h ) = A (ω)E0 (Bk |Bt+h ) = A (ω)Eω(t+h) (ψ ),

(4.56) where

    ψ (ω) = f+1 ω(s1 − h) · · · f+k ω(sk − h) .

(4.57)

Now, as in (1.9),  Ex (ψ ) = (4.58)

 ···

p(s1 − h, x1 )p(s2 − s1 , x2 − x1 ) · · · p(sk − sk−1 , xk − xk−1 ) · f+1 (x1 + x) · · · f+k (xk + x) dxk · · · dx1 .

The continuity in (x, h) is clear. Since paths in P0 are continuous, we have, by linearity, that (4.59)

ϕ ∈ Cˆ =⇒ E0 (ϕ|Bt ) = lim E0 (ϕ|Bt+h ), h0

W0 -a.e.

Now the Stone-Weierstrass theorem implies that Cˆ is dense in C(P), which is dense in L2 (P, dW0 ) = L2 (P0 , dW0 ). Thus we have (4.60)

E0 (ϕ|Bt+ ) = E0 (ϕ|Bt ),

W0 -a.e.,

for every ϕ ∈ L2 (P0 , dW0 ), and the proposition is proved.

Exercises 1. Show that the martingale maximal inequality applied to xt (ω) = ω(t) yields    1 ≤ . W0 ω ∈ P0 : sup ω(s) > b 4t/π b 0≤s≤t Compare with the precise result in (3.29).

436 11. Brownian Motion and Potential Theory 2. With Bt− characterized by (4.51), show that Bt− = Bt , as stated in the proof of Proposition 4.11. (Hint: In the characterization (4.1) of Bt , one can restrict attention to E open in Rn .) 3. Using (4.60), show that S ∈ B0+ =⇒ W0 (S) = 0 or 1. This is called Blumenthal’s 01 law. If E ∈ Rn is a closed set, show that {ω ∈ P0 : ω(tν ) ∈ E for some tν  0} is a set in B0+ . (Hint: Consider {ω ∈ P0 : dist(ω(t), E) ≥ δ > 0 for t ∈ [2−ν ε, ε] ∩ Q} = S(E, δ, ε, ν).) 4. Let N be the collection of (W0 -outer measurable) subsets of P0 with W0 -measure zero. Form the family of σ-algebras B# t = Bt ∪ N , called the augmentation of Bt . Show that B# t ⊃ Bt+ and, with notation parallel to (4.51), # # B# t− = Bt = Bt+ .

Note: The augmentation of Bt is bigger than the completion of Bt . t be the σ-algebra of subsets of P0 generated by sets of the form (4.1) for s ≥ t, 5. Let F  t . Using Blumenthal’s 01 law and Exercise 2 of §1, show that and set A∞ = t>0 F S ∈ A∞ =⇒ W0 (S) = 0 or 1. If E ⊂ Rn is a closed set, show that {ω ∈ P0 : ω(tν ) ∈ E for some tν ∞} is a set in A∞ .

5. First exit time and the Poisson integral At the end of §3 we produced a formula for PI f , giving the solution u to (5.1)

Δu = 0 in Ω,

u = f on ∂Ω,

at least in case Ω is a bounded domain in Rn with smooth boundary. Here we produce a formula that is somewhat neater than (3.36) and that is also amenable to extension to general bounded, open Ω ⊂ Rn , with no smoothness assumed on ∂Ω. In the smooth case, the formula is   (5.2) PI f (x) = Ex f (ω(ϑΩ )) , x ∈ Ω, where ϑΩ (ω) is the first exit time defined by (3.24). From an intuitive point of view, the formula (5.2) has a very easy and natural justification. To show that the right side of (5.2), which we denote by u(x), is harmonic on Ω, it suffices to verify the mean-value property. Let x ∈ Ω be the

5. First exit time and the Poisson integral

437

center of a closed ball B ⊂ Ω. We claim that u(x) is equal to the mean value of u|∂B . Indeed, a continuous path ω starting from x and reaching ∂Ω must cross ∂B, say at a point y = ω(ϑB ). The future behavior of such paths is independent of their past, so the probability distribution of the first contact point ω(ϑΩ ), when averaged over starting points in ∂B, should certainly coincide with the probability distribution of such a first contact point in ∂Ω, for paths starting at x (the distribution of whose first contact point with ∂B must be constant, by symmetry). The key to converting this into a mathematical argument is to note that the time ϑB (ω) is not constant, so one needs to make use of the strong Markov property as a tool to establish the mean-value property of the function u(x) defined by the right side of (5.2). Let us first make some comments on the right side u(x) of (5.2). By (3.40) we have        f ω(ϑΩ ) dWx (ω) ≤ Cf L∞ (∂Ω) N −1 . (5.3) u(x) − PΩ,N

Let us extend f ∈ C(∂Ω) to an element f ∈ C0 (Rn ), without increasing the sup norm. By (3.41), we have     (5.4) f ω(ϑΩ ) = lim f ω(ϑν,N (ω)) , for ω ∈ PΩ,N , ν→∞

  / Ω} . Thus, if the integral in where ϑν,N (ω) = min N, inf {s ∈ 2−ν Z+ : ω(s) ∈ (5.3) is denoted by uN (x), then  (5.5)

uN (x) = lim uN ν (x) = lim ν→∞

ν→∞

  f ω(ϑν,N (ω)) dWx (ω).

PΩ,N

Here the limit exists pointwise in x ∈ Ω. Now each uN ν is continuous on Ω, indeed on Rn . Consequently, u(x) given by the right side of (5.2) is at least a bounded, measurable function of x. To continue the analysis, given x ∈ Ω, we define a probability measure νx,Ω on ∂Ω by    (5.6) Ex f (ω(ϑΩ )) = f (y) dνx,Ω (y), Ω

for f ∈ C(∂Ω). Lemma 5.1. If x ∈ O ⊂⊂ Ω and O and Ω are open, then  νy,Ω dνx,O (y). (5.7) νx,Ω = ∂O

438 11. Brownian Motion and Potential Theory

Proof. The identity (5.4) is equivalent to the statement that, for f ∈ C(∂Ω),      Ey f (ω(ϑΩ )) dνx,O (y). (5.8) Ex f (ω(ϑΩ )) = ∂O

The right side is equal to   (5.9) Ex g(ω(ϑO )) ,

  g(y) = Ey f (ω(ϑΩ )) .

In other words, g(ω(ϑO )) = Eω(ϑO ) (ϕ),

(5.10)

  ϕ(ω) = f ω(ϑΩ (ω)) .

Now we use the strong Markov property, in the form (4.32), namely,    Eω(τ ) (ϕ) = Ex ϕ ◦ στ Bτ + , for Wx -almost all ω, where (στ ω)(t) = ω(t + τ ) and τ is a stopping time. This implies      (5.11) Eω(τ ) (ϕ) dWx (ω) = Ex ϕ ◦ στ Bτ + dWx (ω) = Ex (ϕ ◦ στ ). P0

P0

Applied to τ = ϑO , this shows that (5.9) is equal to Ex (ϕ ◦ σϑO ). Now, with ω ) = ϑΩ (ω) − ω

(t) = σϑO ω(t) = ω(t + ϑO (ω)), we have, for O ⊂⊂ Ω, ϑΩ ( ϑO (ω), as long as ω is a continuous path starting in O. Hence     (5.12) ϕ( ω) = f ω

(ϑΩ (ω) − ϑO (ω)) = f ω(ϑΩ (ω)) = ϕ(ω). Thus (5.9) is equal to Ex (ϕ), which is the left side of (5.6), and the lemma is proved. Consequently, the right side u(x) of (5.2) is a bounded, measurable function of x satisfying the mean-value property. An integration yields that such u(x) is equal to the mean value of u over any ball D ⊂ Ω, centered at x, from which it follows that u(x) is continuous in Ω. Then the mean-value property guarantees that u is harmonic on Ω. To verify (5.2), it remains to show that u(x) has the correct boundary values. Lemma 5.2. Assume ∂Ω is smooth. Given y ∈ ∂Ω, we have u(y) = f (y), and u is continuous at y ∈ Ω. Proof. That u(y) = f (y) follows from the fact that ϑΩ (ω) = 0 for Wy -almost all ω, according to Proposition 3.7. To show that u(x) → u(y) as x → y from within Ω, we argue as follows. By (3.23), for x ∈ Ω, Ex (ϑΩ ) = −Δ−1 1(x). Hence this quantity approaches 0 as x → y. Thus, given εj > 0, there exists δ > 0 such that

5. First exit time and the Poisson integral

(5.13)

439

  |x − y| ≤ δ =⇒ Wx {ω : ϑΩ (ω) > ε1 } < ε2 .

Meanwhile, in a short time, 0 ≤ s ≤ ε1 , a path ω(s) is not likely to wander far. In fact, by (3.28) plus a scaling argument, (5.14)

Wε1 = {ω ∈ P0 : sup

0≤s≤ε1

1/3

|ω(s) − ω(0)| ≥ ε1 }

=⇒ Wx (Wε1 ) ≤ ψ(ε1 ), where ψ(ε) → 0 as ε → 0. Thus, if |x − y| ≤ δ, with probability > 1 − ε2 − ψ(ε1 ), a path starting at 1/3 x will, within time ε1 , hit ∂Ω, without leaving the ball Bε1/3 (x) of radius ε1 1 centered at x. Now, a given f ∈ C(∂Ω) varies only a little over {z ∈ ∂Ω : 1/3 |z − y| ≤ ε1 + δ} if ε1 and δ are small enough. Therefore, indeed u(x) → u(y), as x → y. We have completed the demonstration of the following. Proposition 5.3. If Ω is a bounded region in Rn with smooth boundary and f ∈ C(∂Ω), then PI f is given by (5.2). Recall from §5 of Chap. 5 the construction of (5.15)

PI : C(∂Ω) −→ L∞ (Ω) ∩ C ∞ (Ω)

when Ω is an arbitrary bounded, open subset of Rn , with perhaps a very nasty boundary. As shown there, we can take (5.16)

Ω1 ⊂⊂ Ω2 ⊂⊂ · · · ⊂⊂ Ωj Ω

such that each boundary ∂Ωj is smooth, and, if f is extended from ∂Ω to an element of Co (Rn ), then (5.17)

x ∈ Ω =⇒ PI f (x) = lim uj (x), j→∞

where uj ∈ C(Ωj ) is the Poisson integral of f |∂Ωj . In (5.17) one has uniform convergence on compact sets K ⊂ Ω, the right side being defined for j ≥ j0 , where K ⊂ Ωj0 . The details were carried out in Chap. 5 for f ∈ C ∞ (Rn ), but approximation by smooth functions plus use of the maximum principle readily extends this to f ∈ Co (Rn ). If we apply Proposition 5.3 to Ωj , we conclude that, for f ∈ Co (Rn ), x ∈ Ω, (5.18)

   PI f (x) = lim Ex f ω(ϑΩj ) . j→∞

On the other hand, it is straightforward from the definitions that

440 11. Brownian Motion and Potential Theory

(5.19)

ϑΩj (ω) ϑΩ (ω), for all ω ∈ P0 .

Therefore, via the dominated convergence theorem, we can pass to the limit in (5.18), proving the following. Proposition 5.4. If Ω is any bounded, open region in Rn and f ∈ C(∂Ω), then (5.20)

   PI f (x) = Ex f ω(ϑΩ ) ,

x ∈ Ω.

We recall from Chap. 5 the notion of a regular boundary point. A point y ∈ ∂Ω is regular provided PI f is continuous at y, for all f ∈ C(∂Ω). We discussed several criteria for a boundary point to be regular, particularly in Propositions 5.11–5.16 of Chap. 5. Here is another criterion. Proposition 5.5. If Ω ⊂ Rn is a bounded open set, y ∈ ∂Ω, then y is a regular boundary point if and only if (5.21)

Ex (ϑΩ ) → 0, as x → y, x ∈ Ω.

Proof. Recall from (3.20) that Ex (ϑΩ ) = −Δ−1 1(x). Thus (5.21) holds if and only if this function is a weak barrier at y ∈ ∂Ω, as defined in Chap. 5, right after (5.26). Therefore, (5.21) here implies y is a regular point. On the other hand,  Δ−1 1(x) can be written as the sum x21 /2+u0 (x), where u0 = −(1/2) PI (x21 ∂Ω ), so if (5.21) fails, y is not a regular point. One might both compare and contrast this proof with that of Lemma 5.2. In that case, where ∂Ω was assumed smooth, the known regularity of each boundary point was exploited to guarantee that Ex (ϑΩ ) → 0 as x → y ∈ ∂Ω, which then was exploited to show that u(x) → u(y) as x → y. In the next section, we will derive another criterion for y to be regular, in terms of “capacity.”

Exercises 1. Explore connections between the formulas for PI f (x), for f ∈ C(∂Ω), when Ω is bounded and ∂Ω smooth, given by (3.36) and by (5.2), respectively.

6. Newtonian capacity The (Newtonian) capacity of a set is a measure of size that is very important in potential theory and closely related to the probability of a Brownian path hitting that set. In our development here, we restrict attention to the case n ≥ 3 and define the capacity of a compact set K ⊂ Rn . We first assume that K is the closure of an open set with smooth boundary.

6. Newtonian capacity

441

Proposition 6.1. Assume n ≥ 3. If K ⊂ Rn is compact with smooth boundary ∂K, then there exists a unique function UK , harmonic on Rn \ K, such that UK (x) → 1 as x → K and UK (x) → 0 as |x| → ∞. Proof. We can assume that the origin 0 ∈ Rn is in the interior of K. Then the inversion ψ(x) = x/|x|2 interchanges 0 and the point at infinity, and the transformation v(x) = |x|−(n−2) w(|x|−2 x)

(6.1)

preserves harmonicity. We let w be the unique harmonic function on the bounded domain ψ(Rn \ K), with boundary value w(x) = |x|−(n−2) on ψ(∂K). Then v, defined by (6.1), is the desired solution. The uniqueness is immediate, via the maximum principle. Note that the construction yields (6.2)

|UK (x)| ≤ C|x|−(n−2) ,

|∂r UK (x)| ≤ C|x|−(n−1) ,

|x| → ∞.

The n = 3 case of this result was done in §1 of Chap. 9. Another approach to the proof of Proposition 6.1 would be to represent UK (x) as a single-layer potential, as in (11.44) of Chap. 7. This was noted in a remark after the proof of Proposition 11.5 in that chapter. Now that we have established the existence of such UK , Exercises 7–9 of §3 apply, to yield (6.3)

t (x) UK (x), as t ∞, UK

where, for x ∈ O = Rn \ K, (6.4)

t UK (x) = 1 − etΔO 1(x)   = Wx {ω : ϑO (ω) ≤ t} .

Here, ΔO is the Laplace operator on O, with Dirichlet boundary condition. The last identity follows from (3.27). We can replace the first exit time ϑO by the first hitting time: (6.5)

hK (ω) = ϑRn \K (ω).

Consequently, (6.6)

  UK (x) = Wx {ω : hK (ω) < ∞} ;

that is, for x ∈ O, UK (x) is the probability that a Brownian path ω, starting at x, eventually hits K.

442 11. Brownian Motion and Potential Theory

We set UK (x) = 1 for x ∈ K. Then (6.6) holds for x ∈ K also. It follows that UK ∈ Co (Rn ), and ΔUK is a distribution supported on ∂K. In fact, Green’s formula yields, for ϕ ∈ C0∞ (Rn ),  (UK , Δϕ) = −

(6.7)

ϕ(y)

∂ UK (y) dS(y), ∂ν

∂K

where ν is the unit normal to ∂K, pointing into K. By Zaremba’s principle, ∂ν UK (y) > 0, for all y ∈ ∂K, so we see that ΔUK = −μK , where μK is a positive measure supported on ∂K. The total mass of μK is called the capacity of K:  (6.8) cap K = dμK (x). K

Since, with Cn = (n − 2) · Area(S n−1 ), (6.9)

UK (x) = −Δ−1 μK = Cn



|x − y|−(n−2) dμK (y),

we have  (6.10)

Cn

dμK (x) dμK (y) = |x − y|n−2

 UK (x) dμK (x) = cap K,

the left side being proportional to the potential energy of a collection of charged particles, with density dμK , interacting by a repulsive force with potential Cn |x− y|−(n−2) . The function UK (x) is called the capacitary potential of K. Note that we can also use Green’s theorem to get (6.11)

∇UK 2L2 (Rn )

 =

UK (x) dμK (x) = cap K. K

Note that if K1 ⊂ K2 have capacitary potentials Uj , ΔUj = −μj , then U2 = 1 on K1 , so  cap K1 = U2 (x) dμ1 (x) = −(U2 , ΔU1 )  (6.12) = U1 (x) dμ2 (x) ≤ cap K2 , since U1 (x) ≤ 1. Thus capacity is a monotone set function. Before establishing more formulas involving capacity, we extend it to general  compact K ⊂ Rn . We can write K = Kj , where K1 ⊃⊃ K2 ⊃⊃ · · · ⊃⊃

6. Newtonian capacity

443

Kj  K, each Kj being compact with smooth boundary. Clearly, Uj = UKj is a decreasing sequence of functions ≤ 1, and by (6.11), ∇Uj is bounded in L2 (Rn ). Furthermore, ΔUj = −μj , where μj is a positive measure supported on ∂Kj , of total mass cap Kj , which is nonincreasing, by (6.12). Consequently, we have a limit: (6.13)

lim Uj = UK ,

j→∞

defined a priori pointwise, but also holding in various topologies, such as the weak∗ topology of L∞ (Rn ). We have UK ∈ L∞ (Rn ), 0 ≤ UK (x) ≤ 1; ∇UK ∈ L2 (Rn ), and ΔUK = −μ, where μ is a positive measure, supported on K. Furthermore, μj → μ in the weak∗ topology, and UK = −Δ−1 μ. Any neighborhood of K contains some Kj . Thus, if K1 ⊃⊃ K2 ⊃⊃ · · · ⊃⊃ Kj  K is another choice, one is seen to obtain the same limit UK , hence the same measure μ, which we denote as μK . We set  (6.14) cap K = dμK (x).  Note that, as  in (6.12), cap K = Uj (x) dμK (x), for each j. Thus, as before, cap K = UK (x) dμK (x), this time by the monotone convergence theorem. Consequently, (6.15)

UK (x) = 1 μK -almost everywhere.

Clearly, cap K ≤ inf cap Kj . In fact, we claim (6.16)

cap K = inf cap Kj .

This is easy to see; μj converges to μK pointwise on Co (Rn ); choose g ∈ Co (Rn ), equal to 1 on K1 ; then (6.17)

cap K = (g, μK ) = lim (g, μj ) = lim capKj ,

proving (6.16). We consequently extend the monotonicity property: Proposition 6.2. For general compact K ⊂ L, we have cap K ≤ cap L. Proof. We can take compact approximants with smooth boundary, Kj  K, Lj  L, such that Kj ⊂ Lj . By (6.12) we have cap Kj ≤ cap Lj , and this persists in the limit by (6.16). We also have UK (x) ≤ UL (x) for all x. Using (6.15), we obtain  (6.18) cap K = UL (x) dμK (x).

444 11. Brownian Motion and Potential Theory

One possibility is that cap K = 0. This happens if and only if μK = 0, thus if and only if UK = 0 almost everywhere. If cap K > 0, we continue to call UK the capacitary potential of K. We record some more ways in which Uj → UK . First, it certainly holds in the weak∗ topology on L∞ (Rn ). Hence ∇Uj → ∇UK in D (Rn ). By (6.11), ∇Uj is bounded in L2 (Rn ); hence ∇Uj → ∇UK weakly in L2 (Rn ). Since also Uj ∈ Co (Rn ), we have

(6.19)

∇UK 2L2 = lim (∇Uj , ∇UK ) = lim −(Uj , ΔUK ) j→∞ j→∞  = lim Uj (x) dμ(x) = cap K, j→∞

the last identity holding as in the derivation of (6.15). Thus (6.11) is extended to general compact K. Furthermore, this implies (6.20)

∇Uj −→ ∇UK

in L2 (Rn )-norm.

Hence (6.21)

μj −→ μK

in H −1 (Rn )-norm.

We now extend the identities (6.3) and (6.6) to general compact K, in reverse order. Proposition 6.3. The identity (6.6) holds for general compact K ⊂ Rn . Proof. Since (6.6) has been established for the compact Kj with smooth boundary, we have (6.22)

1 − Uj (x) = Wx (AKj ),

AKj = {ω ∈ P0 : ω(R+ ) ⊂ Rn \ Kj }.

K , where A

K is a proper Clearly, if Kj  K, AK1 ⊂ AK2 ⊂ · · · ⊂ AKj A + n subset of AK = {ω ∈ P0 : ω(R ) ⊂ R \ K}. However, for n ≥ 3, Brownian motion is nonrecurrent, as was established in Exercise 10 of §3. Thus |ω(t)| → ∞  

K = 0, and hence as t → ∞, for Wx -almost all ω, so in fact Wx AK \ A 1 − UK (x) = Wx (AK ), which is equivalent to (6.6). Proposition 6.4. The identity (6.3) holds for general compact K ⊂ Rn . t (x) to be 1 − etΔO 1(x), as in (6.4); the second identity in Proof. We define UK (6.4) continues to hold, by (3.27). Now, clearly, the family of sets St = {ω ∈ P0 : hK (ω) ≤ t} is increasing as t ∞, with union



St = {ω ∈ P0 : hK (ω) < ∞},

6. Newtonian capacity

445

and this gives (6.3). We next establish the subadditivity of capacity. Proposition 6.5. If K and L are compact, then UK∪L (x) ≤ UK (x) + UL (x)

(6.23) and

    cap(K ∪ L) ≤ cap K + cap L .

(6.24)

Proof. The inequality (6.23) follows directly from (6.6) and the subadditivity of Wiener measure. Now, as in (6.12), we have  UK (x) dμK∪L (x) = −(UK , ΔUK∪L )  (6.25) = UK∪L (x) dμK (x) = cap K, the last identity by (6.18), with L replaced by K ∪ L. Hence  cap K + cap L =

  UK (x) + UL (x) dμK∪L (x),

so the estimate (6.23) implies (6.24). Note that even if K and L are disjoint, typically there is inequality in (6.23), hence in (6.24). In fact, if K and L are disjoint compact sets,

(6.26)

(cap K) + (cap L) = cap(K ∪ L) + R,   R = UK (x) dμK∪L (x) + UL (x) dμK∪L (x), L

K

the quantity R being > 0 unless either cap K = 0 or cap L = 0. Unlike measures, the capacity is not an additive set function on disjoint compact sets. We began this section with the statement that the capacity of K is closely related to the probability of a Brownian path hitting K. We have directly tied UK (x) to this probability, via (6.6). We now provide a two-sided estimate on UK (x) in terms of cap K. Proposition 6.6. Let δ(x) = sup {|x − y| : y ∈ K}, and let d(x) denote the distance of x ∈ Rn from K. Then (6.27)

  Cn  Cn  cap K ≤ UK (x) ≤ cap K . δ(x)n−2 d(x)n−2

446 11. Brownian Motion and Potential Theory

 Proof. The formula UK (x) = Cn |x − y|−(n−2) dμK (y) represents UK (x) as Cn (cap K) times a weighted average of |x − y|−(n−2) over K. Now, for y ∈ K, d(x) ≤ |x − y| ≤ δ(x), so (6.27) follows. We want to compare this with the probability that a Brownian path hits ∂K in the interval [0, t]. It t is large, we know that |ω(t)| is probably large, given that n ≥ 3, and hence ω(s) probably will not hit K for any s > t. Thus we t (x)) to be close to UK (x). We derive expect this probability (which is equal to UK t a quantitative estimate as follows. Since 1 − UK (x) = etΔO 1(x), we have, for s ≥ 0, (6.28)

t+s t s UK (x) − UK (x) = etΔO 1(x) − e(t+s)ΔO 1(x) = etΔO UK (x),

and taking s ∞, we get (6.29)

t UK (x) − UK (x) = etΔO UK (x).

Hence, if we denote the heat kernel on O = Rn \ K by pO (t, x, y), and that on Rn by p(t, x − y), as in (1.5),

(6.30)

t UK (x) − UK (x)   = pO (t, x, y)UK (y) dy ≤ p(t, x − y)UK (y) dy  p(t, x − y) = Cn dy dμK (z) ≤ (cap K)σK (t, x), |y − z|n−2

where (6.31)

 σK (t, x) = Cn sup z∈K

p(t, x − y) dy = sup |y − z|n−2 z∈K





p(s, x − z) ds,

t

the last integral being another way etΔ (−Δ)−1 δ(x − z) when n ≥ 3.  ∞ of writing −n/2 ds, so we have An upper bound on σK (t, x) is t (4πs) (6.32)

t 0 ≤ UK (x) − UK (x) ≤

 2  −n/2 −n/2+1  cap K . 4π t n−2

There is an interesting estimate on the smallest eigenvalue of −Δ on the complement of a compact set K, in terms of cap K, which we now describe. Let Q = {x ∈ Rn : 0 ≤ xj ≤ 1} be the closed unit cube in Rn , and let K ⊂ Q be compact. We consider the boundary condition on functions on Q \ K: (6.33)

u = 0 on ∂K,

∂u = 0 on ∂Q \ ∂K. ∂ν

To define this precisely, let H 1 (Q, K) denote the closure in H 1 (Q) of the set of functions in C ∞ (Q) vanishing on a neighborhood of K. Then the quadratic

6. Newtonian capacity

447

form (du, dv)L2 restricted to H 1 (Q, K) × H 1 (Q, K) defines an unbounded, selfadjoint operator L, which we denote −ΔQ,K , with D(L1/2 ) = H 1 (Q, K) ⊂ H 1 (Q). Hence −ΔQ,K has compact resolvent and thus a discrete spectrum. Let λ0 (K) be its smallest eigenvalue. Proposition 6.7. The smallest eigenvalue λ0 (K) of −Δ on Q\K, with boundary condition (6.33), satisfies the estimate λ0 (K) ≥ γn cap K,

(6.34) for some γn > 0.

Proof. Let pQ,K (t, x, y) denote the heat kernel of ΔQ,K . With O = Rn \ K, let pO (t, x, y) denote the heat kernel of Δ on O, with Dirichlet boundary condition, as in (6.30). We claim that   (6.35) pQ,K (t, x, y) dy ≤ pO (t, x, y) dy, x ∈ Q. Rn

Q

by the method of images, so in each unit cube with integer To see this, define K

= Rn \ K,

vertices we have a reflected image of K, and, with O  (6.36) pQ,K (t, x, y) = pO (t, x, Rj y), x, y ∈ Q, j

where the transformations Rj are appropriate reflections. Then (6.35) follows from the obvious pointwise estimate pO (t, x, y) ≤ pO (t, x, y). Now, if we set  M (t) = sup

(6.37)

x∈Q

pO (t, x, y) dy, Rn

it follows that  (6.38) sup pQ,K (t, x, y) dy ≤ M (t), x

Q

 pQ,K (t, x, y) dx ≤ M (t),

sup y Q

the latter by symmetry. It is well known that the operator norm of etΔQ,K is bounded by the quantities (6.38). (See Proposition 5.1 in Appendix A.) Thus (6.39)

etΔQ,K  ≤ M (t).

To relate this to capacity, note that (6.40)

  t (x) . M (t) = sup 1 − UK x∈Q

448 11. Brownian Motion and Potential Theory

Now, applying the first estimate of (6.27), in concert with the estimate (6.32), we have   (6.41) M (t) ≤ 1 − Cn n−n/2+1 cap K +

  2 (4π)−n/2 t−n/2+1 cap K . n−2

In particular, there exists a finite T = Tn and κ > 0 such that (6.42)

M (T ) ≤ 1 − κ(cap K) ≤ e−κ cap K .

Since this is an upper bound on eT ΔQ,K , we have λ0 (K) ≥ (κ/T ) cap K, proving (6.34). As an application of this, we establish the following result of Molchanov on a class of Dirichlet problems with compact resolvent. Proposition 6.8. Let Ω be an unbounded, open subset of Rn , with complement S. Suppose that there exists ψ(a) ∞ as a  0, such that, for each a ∈ (0, 1], if Rn is tiled by cubes Qaj of edge a, we have (6.43)

cap(Qaj ∩ S) ≥ ψ(a)a2(n−2) ,

for all but finitely many j. Then the Laplace operator Δ on Ω, with Dirichlet boundary condition, has compact resolvent. Proof. By scaling Qaj to a unit cube, we see that if (6.43) holds, then −Δ on Qaj \ S, with Dirichlet boundary condition on ∂S, Neumann on ∂Qaj \ S, has smallest eigenvalue ≥ γn (cap Qaj ∩ S)a−2(n−2) , which, by hypothesis (6.43) is ≥ γn ψ(a) for all but finitely many j. The variational characterization of the spectrum implies that the spectral subspace of L2 (Ω) on which −Δ has spectrum in [0, γn ψ(a)] is finite-dimensional, for each a > 0, and this implies that Δ has compact resolvent. In our continued study of which boundary points of a region Ω are regular, it will be useful to have the following variant of Proposition 6.6. Here, Br is the ball of radius r centered at the origin in Rn ; see Fig. 6.1. Proposition 6.9. Let K be a compact subset of the ball B1 . Let VK (x) denote the probability that a Brownian path, starting at x ∈ Rn , hits K before hitting the

n > 0 such that shell ∂B4 = {x : |x| = 4}. Then there is a constant γ (6.44)

 

n cap K . x ∈ B1 =⇒ VK (x) ≥ γ

Proof. Note that, by (5.20), VK is also defined by (6.45)

ΔVK = 0 on B4 \ K,

VK = 1 on K,

VK = 0 on ∂B4 .

6. Newtonian capacity

449

F IGURE 6.1 The Set K

We will compare VK (x) with UK (x). By (6.27), we have (6.46)

  x ∈ B1 =⇒ UK (x) ≥ 2−(n−2) Cn cap K

and (6.47)

  x ∈ ∂B4 =⇒ UK (x) ≤ 3−(n−2) Cn cap K .

By (6.47) and the maximum principle, we have, for x ∈ B4 \ K, (6.48)

VK (x) ≥

UK (x) − q(K) , 1 − q(K)

  q(K) = 3−(n−2) Cn cap K .

Now Cn (cap K) ≤ Cn (cap B1 ) = 1 (compare with Exercise 1 at the end of this section), so using (6.46) we readily obtain (6.44), with (6.49)

−1    2−(n−2) − 3−(n−2) Cn . γ

n = 1 − 3−(n−2)

In particular, γ

3 = C3 /4 = π. Of course, since VK (x) ≤ UK (x), we also have (6.50)

  x ∈ B4 , dist(x, K) ≥ ρ =⇒ VK (x) ≤ Cn ρ−(n−2) cap K .

This upper bound is valid for K ⊂ B4 ; we don’t need K ⊂ B1 . Now suppose y ∈ K is the center of concentric balls Bj , of radius 2−j r, where r > 0 is fixed, 0 ≤ j ≤ ν. See Fig. 6.2. Pick x ∈ Bν . We want to estimate the probability that a Brownian path starting at x will exit B0 before hitting K. Let’s call the probability pmiss (x, K). Using Proposition 6.9 and scaling, we see that, given x ∈ Bj , the probability that it hits ∂Bj−2 before hitting K ∩ Bj is

450 11. Brownian Motion and Potential Theory

F IGURE 6.2 Setup for the Wiener Test −(n−2)

≤ 1−γ

n rj · cap(K ∩ Bj ), where rj = 2−j r. Using the independence of this event and of the event that, given x ∈ ∂Bj−2 , the path will hit ∂Bj−4 before hitting K ∩ Bj−2 , which follows from the strong Markov property, we have an upper bound    1−γ

n r−(n−2) 2(n−2)j · cap K ∩ Bj , (6.51) pmiss (x, K) ≤ j∈Sν

where Sν = {j : 0 ≤ j ≤ ν, j = ν mod 2}. A similar argument dominates pmiss (x, K) by a product over {1, . . . ν} \ Sν , so (6.52)

pmiss (x, K)2 ≤

ν  

  1−γ

n r−(n−2) 2(n−2)j · cap K ∩ Bj .

j=0

Note that, as ν → ∞, the right side of (6.52) tends to zero, precisely when the sum (6.53)

∞ 

  2(n−2)j · cap K ∩ Bj

j=0

is infinite. We are now ready to state the Wiener criterion for regular points. Proposition 6.10. Let Ω be a bounded, open set in Rn , and let y ∈ ∂Ω. If Ω is

set K = B

\ Ω. Then y is a regular point for Ω if and only if the inside a ball B, infinite series (6.53) is divergent, where Bj = {x ∈ Rn : |x − y| ≤ 2−j }. Proof. First suppose (6.53) is divergent. Fix f ∈ C(∂Ω), and look at (6.54)

   u(x) = PI f (x) = Ex f ω(hK ) .

6. Newtonian capacity

451

Given ε > 0, fix r > 0 so that f varies by less than ε on {z ∈ ∂Ω : |z − y| ≤ r}. By (6.52), if δ > 0 is small enough and |x − y| ≤ δ, then the probability that a Brownian path ω(t), starting at x, crosses ∂B0 = {z : |z − y| = r} before hitting K is < ε. Consequently, (6.55)

       |x − y| ≤ δ =⇒ Ex f ω(hK ) − f (y) ≤ ε + ε · sup |f |.

This shows that P I f (x) → f (y) as x → y, for any f ∈ C(∂Ω), so y is regular. For the converse, if (6.53) converges, we claim there is a J < ∞ such that there exist points in Ω ∩ BJ , arbitrarily close to y, which are starting points of Brownian paths whose probability of hitting K before exiting BJ is ≤ 1/2. Consider the shells Aj = {x : 2−j−1 ≤ |x − y| ≤ 2−j }; Bj = ≥j A . We will estimate the probability that a point picked at random in A is the starting point of a Brownian path that hits K before exiting BJ , where is chosen > J. Since we are assuming n ≥ 3, by the analysis behind nonrecurrence in Exercises 7–10 of §3, the probability that a path starting in A ever hits B+3 is ≤ 1/4. Thus if we alter K to K = K \ B+3 , the probability that a Brownian path starting in A hits K before ∂BJ is not decreased by more than 1/4. We aim to show that this new probability is ≤ 1/4 if J is chosen large enough. Now there is no further decrease in probability that the path hits K before ∂BJ if we instead have it start at a random point in B+5 , since almost all such paths will pass into A , in a uniformly distributed fashion through its inner boundary. So we deal with the modified problem of estimating the probability p that a Brownian path, starting at a random point in B+5 , hits K = K \ B+3 before exiting BJ . We partition the set {j : J ≤ j ≤ + 3} into two sets, where j is even or odd; call these subsets J0 and J1 , respectively. Then form (6.56)

A0 =



Aj ,

A1 =

j∈J0



Aj .

j∈J1

We estimate the probability pμ that a path starting in B+5 hits K ∩ Aμ before hitting ∂BJ . We have (6.57)

pμ (x) ≤



pμj ,

j∈Jμ

where pμj is the probability that, given |x − y| = (3/4) · 2−j−1 (i.e., x is on a shell Sj+1 halfway between the two boundary components of Aj+1 ), then a path starting at x hits K ∩ Aj before hitting Sj−1 . By (6.50) and a dilation argument, we have an estimate of the form (6.58)

pμj ≤ γn 2(n−2)j cap(K ∩ Aj ).

Thus the probability p that we want to estimate satisfies

452 11. Brownian Motion and Potential Theory

(6.59)

p ≤ γn

+3 

2(n−2)j cap(K ∩ Aj ).

j=J

Of course, cap(K ∩ Aj ) ≤ cap(K ∩ Bj ), so if (6.53) is assumed to converge, we can pick J sufficiently large that the right side of (6.59) is guaranteed to be ≤ 1/4. From here it is easy to pick f ∈ C(∂Ω) such that f (y) = 1 but (6.54) does not converge to 1 as x → y. This completes the proof of Proposition 6.10 and also shows that the hypothesis of convergence or divergence of (6.53) can be replaced by such a hypothesis on (6.60)

∞ 

  2(n−2)j · cap K ∩ Aj .

j=0

We can extend capacity to arbitrary sets S ⊂ Rn . The inner capacity cap− (S) is defined by (6.61)

cap− (S) = sup {cap K : K compact, K ⊂ S}.

Clearly, cap− (K) = cap K for compact K. If U ⊂ Rn is open, we also set cap U = cap− (U ). Now the outer capacity cap+ (S) is defined by (6.62)

cap+ (S) = inf {cap U : U open, S ⊂ U }.

It is easy to see that cap+ (S) ≥ cap− (S) for all S. If cap+ (S) = cap− (S), then S is said to be capacitable, and the common quantity is denoted cap S. The analysis leading to (6.16) shows that every compact set is capacitable; also, by definition, every open set is capacitable. G. Choquet proved that every Borel set is capacitable; in fact, his capacitability theorem extends to a more general class of sets, known as Souslin sets. We refer to [Mey] for a detailed presentation of this result. The outer capacity can be shown to satisfy the property that, for any increasing sequence of sets Sj ⊂ Rn , Sj S =⇒ cap+ (Sj ) cap+ (S). We establish a useful special case of this. Proposition 6.11. If Uj and U are open and Uj U , then cap Uj cap U. Proof. Given ε > 0, pick a compact K ⊂ U such that cap K ≥ cap U − ε. Then K ⊂ Uj for large j, so cap Uj ≥ cap U − ε for large j.

6. Newtonian capacity

453

We next present a result, due to M. Brelot, to the effect that the set of irregular boundary points of a given bounded, open set is rather small. Proposition 6.12. If Ω ⊂ Rn is open and bounded, the set I of irregular boundary points in ∂Ω has inner capacity zero. Proof. The claim is that if K ⊂ I is compact, then cap K = 0. By subadditivity, it suffices to show the following: Given y ∈ ∂Ω, there is a neighborhood B of y in Rn such that any compact K ⊂ I ∩ B has capacity zero. We prove the result in the case that Ω is connected. Let L = B\Ω, and consider the capacitary potential UL (x). In this case, Rn \ L is connected. The function 1−UL (x) is a weak barrier at any z ∈ L∩∂Ω with the property that UL (x) → 1 as x → z, x ∈ Rn \L. Thus it suffices to show that the set J = {z ∈ L : UL (z) < 1} has inner capacity zero. Let K ⊂ J be compact. We know that UK (x) ≤ UL (x) for all x ∈ Rn . Thus UK (x) < 1 on K. Now, by (6.15), UK (x) = 1 for μK -almost all x, so we conclude that μK = 0, hence cap K = 0. This completes the proof when Ω is connected. The general case can be done as follows. If Ω is not connected, it has at most countably many connected components. One can connect the various components via little tubes whose total (inner) capacity can be arranged, via Proposition 6.11, to be arbitrarily small, say < ε. Then the set of irregular points is decreased by a set of inner capacity < ε. The reader is invited to supply the details. As noted in Proposition 5.5, the set of irregular points of ∂Ω can be characterized as the set of points of discontinuity of a function E, defined on Ω to be −Δ−1 1(x) for x ∈ Ω and to be 0 on ∂Ω. Such a set of points of discontinuity is a Borel subset of Ω, in fact an Fσδ -set. Thus the capacitability theorem applies: If Ω ⊂ Rn is a bounded open set, the set of irregular points of ∂Ω has capacity zero. This sharpening of Proposition 6.12 was first established by H. Cartan. As we stated at the beginning of this section, we have been working under the assumption that n ≥ 3. Two phenomena that we have exploited fail when n = 2. One is that Δ has a fundamental solution ≤ 0 on all of Rn . The other is that Brownian motion is nonrecurrent. (Of course, these two phenomena are related.) There is a theory of logarithmic capacity of planar sets. One way to approach things is to consider capacities only of subsets of some fixed disk, of large radius R, and use the Laplace operator on this disk, with the Dirichlet boundary condition. Then one looks at Brownian paths only up to the first exit time from this disk. The results of this section extend. In particular, the Wiener criterion for n = 2 is the convergence or divergence of (6.63)

∞  j=1

  j · cap K ∩ Aj .

454 11. Brownian Motion and Potential Theory

Exercises 1. If K ⊂ Rn is compact, show that lim

|x|→∞

|x|n−2 UK (x) = Cn cap K.

If K = Ba is a ball of radius a, show that cap Ba = an−2 /Cn . Show generally that if a > 0 and Ka = {ax : x ∈ K}, then cap Ka = an−2 cap K. 2. Show that cap K = cap ∂K. Show that the identity cap ∂Ba = an−2 /Cn follows from (6.27), with x the center of Ba . 3. Let Car be the union of two balls of radius a, with centers separated by a distance r. Show that cap Car 2 cap Ba , as r → ∞. Estimate the rate of convergence. 4. The task here is to estimate the capacity of a cylinder in Rn , of height b and radius a. Suppose C(a, b) = {x ∈ Rn : 0 ≤ xn ≤ b, x21 + · · · + x2n−1 ≤ a2 }. Show that there are positive constants αn and βn such that cap C(a, 1) ∼ αn an−3 , cap C(a, 1) ∼ βn a

n−2

,

a → 0, n ≥ 4, a → ∞, n ≥ 3.

Derive an appropriate result for n = 3, a → 0. 5. Let ν be a positive measure supported on a compact set K ⊂ Rn , such that dν(x) −1 Uν (x) = −Δ ν(x) = Cn ≤ 1. |x − y|n−2 Show that Uν (x) ≤ UK (x) for all x ∈ Rn . Taking the limit as |x| → ∞,  deduce from the asymptotic behavior of Uν (x) and UK (x) (as in Exercise 1) that dν(x) ≤ cap K. 6. Show that, for compact K ⊂ Rn ,

(6.64) cap K = inf |∇f (x)|2 dx : f ∈ C0∞ (Rn ), f = 1 on nbd of K . (Hint: Show that a minimizing sequence fj approaches UK .) Show that the condition f = 1 on a neighborhood of K can be replaced by f ≥ 1 on K. Show that if f ∈ C01 (Rn ), λ > 0,   (6.65) cap {x ∈ Rn : |f (x)| ≥ λ} ≤ λ−2 ∇f 2L2 . 7. Show that, for compact K ⊂ Rn ,

dλ(x) dλ(y) 1 + (6.66) , : λ ∈ PK = inf Cn n−2 cap K |x − y| + where PK denotes the space of probability measures supported on K. (Hint: Consider the sesquilinear form γ(μ, λ) = Cn |x − y|−n+2 dμ(x) dλ(y) = −(Δ−1 μ, λ) −1 as a (positive-definite) inner product on the Hilbert space HK (Rn ) = {u ∈ −1 n H (R ) : supp u ⊂ K}. Thus

7. Stochastic integrals

455

|γ(μ, λ)| ≤ γ(μ, μ)1/2 γ(λ, λ)1/2 . + Take μ = (cap K)−1 μK ∈ PK , where μK is the measure in (6.8)–(6.10). Show that (at least, when ∂K is smooth), 1 1 + −1 ∩ HK (Rn ) =⇒ γ(μ, λ) = UK (y) dλ(y) = , λ ∈ PK cap K cap K

and conclude that γ(λ, λ) ≥ 1/(cap K). Then use some limiting arguments.) 8. If K ⊂ R3 is compact, relate cap K to the zero frequency limit of the scattering amplitude, defined in Chap. 9, §1. 9. Try to establish directly the equivalence between the regularity criteria given by Propositions 5.5 and 6.10. 10. In Chap. 5, §5, a compact set K ⊂ Rn was called “negligible” provided there is no nonzero u ∈ H −1 (Rn ) supported on K. Show that if K is negligible, then cap K = 0. Try to prove the converse. 11. Sharpen the subadditivity result (6.24) to cap(K ∪ L) + cap(K ∩ L) ≤ (cap K) + (cap L), for compact sets K and L. This property is called “strong subadditivity.” (Hint: By (6.6), UK (x) = Wx (SK ), where SK = {ω : hK (ω) < ∞}. Show that SK∪L = SK ∪ SL and SK∩L = SK ∩ SL , and deduce that UK∪L (x) + UK∩L (x) ≤ UK (x) + UL (x). Extending the reasoning used in the proof of Proposition 6.5, deduce that   UK (x) + UL (x) dμK∪L (x) cap K + cap L =   ≥ UK∪L (x) + UK∩L (x) dμK∪L (x) = cap(K ∪ L) + cap(K ∩ L).)

7. Stochastic integrals We will motivate the introduction of the stochastic integral by modifying the Feynman–Kac formula, to produce a formula for the solution operator et(Δ+X) to (7.1)

∂u = Δu + Xu, ∂t

u(0) = f ;

Xu =



Xj (x)

As in (2.2), we use the Trotter product formula to write (7.2)

 k et(Δ+X) f = lim e(t/k)X e(t/k)Δ f. k→∞

∂u . ∂xj

456 11. Brownian Motion and Potential Theory

If we assume that each coefficient Xj of the vector field X is bounded and uniformly Lipschitz, then Proposition A.2 applies to (7.2), given f ∈ Lp (Rn ), 1 ≤ p < ∞, or f ∈ Co (Rn ), in view of Proposition 9.13 in Appendix A. Now, for k  any k, e(t/k)X e(t/k)Δ f can be expressed as a k-fold integral:  (7.3)

k e(t/k)X e(t/k)Δ f (x)      t t = · · · f xk p , xk − xk−1 − ξk−1 k k t t  t t  · · · p , x2 − x1 − ξ1 p , x1 − x − ξ0 dx1 · · · dxk , k k k k

where (with x0 = x) ξj = X(xj ) + rj ,

(7.4)

rj = O(k −1 ).

Now we can write (7.5) p

t t  2 t  , xj+1 − xj − ξj = p , xj+1 − xj eξj ·(xj+1 −xj )/2−(t/k)|ξj | /4 . k k k

Consequently, parallel to (2.4),  (7.6)

e(t/k)X e(t/k)Δ

k f (x) = Ex (ϕk ),

where   ϕk (ω) = f ω(t) eAk (ω)−Bk (ω) ,

(7.7) with Ak (ω) = (7.8)

k−1   j + 1   j  1    j  X ω t + rj · ω t −ω t 2 j=0 k k k

Bk (ω) =

k−1 2 1 t    j  X ω t + rj . 4 k j=0 k

Thus we expect to establish a formula of the form (7.9) where

    et(Δ+X) f (x) = Ex f ω(t) eA(t,ω)−B(t,ω) ,

7. Stochastic integrals

B(t, ω) =

(7.10)



1 4

t

457

 2 X ω(s) ds,

0

and (7.11)

A(t, ω) =

k−1    j    j + 1   j  1 lim t −ω t . X ω t · ω 2 k→∞ j=0 k k k

 Xj (ω)2 . If the coefficients Xj are real-valued, this In (7.10), X(ω)2 denotes 2 is equal to |X(ω)| . Certainly Bk (ω) → B(t, ω) nicely for all ω ∈ P0 . The limit we now need to investigate is (7.11), which we would like to write as 1 A(t, ω) = 2

(7.12)



t

0

  X ω(s) · dω(s).

However, ω(s) has unbounded variation for Wx -almost all ω, so there remains some analysis to be done on this object, which is a prime example of a stochastic integral. We aim to make sense out of stochastic integrals of the form  (7.13) 0

t

  g s, ω(s) · dω(s),

beginning with  (7.14) 0

t

g(s) · dω(s) = lim

k→∞

k−1  j=0

g

 j   j   j + 1  t · ω t −ω t . k k k

2 This is readily seen to  be well defined  in L (P0 , dWx ), in view of the fact that the terms θj (ω) = ω (j + 1)t/k − ω jt/k satisfy

(7.15)

t θj 2L2 (P0 ,dWx ) = 2 , k

(θj , θ )L2 (P0 ,dWx ) = 0, for j = ,

the first by (1.38). Thus (7.16)

k−1 k−1   t   j 2  j  2   j   j + 1  g t  . t −ω t  2 g t ω =2  k k k k k L (P0 ,dWx ) j=0 j=0

For continuous g, this is a Riemann sum approximating Thus we obtain the following:

t 0

|g(s)|2 ds, as k → ∞.

458 11. Brownian Motion and Potential Theory

  Proposition 7.1. Given g ∈ C [0, t] , the right side of (7.14) converges in L2 (P0 , dW x ). The resulting correspondence  g → √

extends uniquely to

t

g(s) dω(s) 0

  2 times an isometry of L2 [0, t], dt into L2 (P0 , dWx ).

We next consider (7.17)

Sk (ω) =

k−1 

     k−1 g tj , ω(tj ) · ω(tj+1 ) − ω(tj ) = gj (ω) · θj (ω),

j=0

j=0

where θj (ω) = ω(tj+1 ) − ω(tj ), tj = (j/k)t. Following [Si], Chap. 5, we compute (7.18)



Sk 2L2 (P0 ,dWx ) =

  Ex gj (ω)θj (ω)g (ω)θ (ω) .

j,

If > j, θ (ω) = ω(t+1 ) − ω(t ) is independent of the other factors in parentheses on the right side of (7.18), so the expectation of the product is equal to Ex (gj θj g )Ex (θ ) = 0 since Ex (θ ) = 0. Similarly the terms in the sum in (7.18) vanish when < j, so Sk 2L2 (P0 ,dWx ) =

 j

(7.19) =2

    Ex |gj (ω)|2 Ex |θj |2



  Ex |g(tj , ω(tj ))|2 (tj+1 − tj ).

j

If g and ω are continuous, this is a Riemann sum approximating the integral  t 2 2 0 Ex |g(s, ω(s))| ds, and we readily obtain the following result.   Proposition 7.2. Given g ∈ BC [0, t] × Rn , the expression (7.17) converges as k → ∞, in L2 (P0 , dWx ), to a limit we denote by (7.13). Furthermore, the map  g → is

t

0

  g s, ω(s) · dω(s)

√ 2 times an isometry into L2 (P0 , dWx ), when g has the square norm 

(7.20)

Qx (g) =

0

t

  2  Ex g s, ω(s)  ds.

7. Stochastic integrals

459

t Note that Qx (g) = 0 Rn |g(s, y)|2 p(s, x − y) dy ds. In case g = g(ω(s)), we have Qx (g) given as the square of a weighted L2 -norm:  (7.21)

Qx (g) =

|g(y)|2 rt (x − y) dy = Rt (D)|g|2 (x),

Rn

where (7.22)

Rt (D) = Δ−1 (etΔ − I),

rt (x) = Rt (D)δ(x).

We see that Rt (D) ∈ OP S −2 (Rn ). The convolution kernel rt (x) is smooth on Rn \ 0 and rapidly decreasing as |x| → ∞. More precisely, one easily verifies that (7.23)

rt (x) ≤ C(n, t)|x|−2 e−|x|

2

/4t

, for |x| ≥

1 , 2

and (7.24)

rt (x) ≤ C(n, t)|x|2−n , for |x| ≤

1 , 2

n ≥ 3,

with |x|2−n replaced by log 1/|x| for n = 2 and by 1 for n = 1. Of course, rt (x) > 0 for all t > 0, x ∈ Rn \ 0. In particular, the integral in (7.21) is absolutely convergent and Qx (g) is a continuous function of x provided   2 (7.25) g ∈ Lploc (Rn ), for some p > n, and g ∈ L2 Rn , x−2 e−|x| /4t dx . Proposition 7.2 is adequate to treat the case where the coefficients Xj are in BC(Rn ) and purely imaginary. Since Ak (ω) → A(t, ω) in L2 (P0 , dWx ), (7.26)

eAk (ω) −→ eA(t,ω) in measure,

and boundedly, since the terms in (7.26) all have absolute value 1. Then convergence of (7.6) follows from the dominated convergence theorem. In such a case, X(ω)2 in (7.10) is equal to −|X(ω)|2 . We have the following. Proposition 7.3. If X = iY is a vector field on Rn with coefficients that are bounded, continuous, and purely imaginary, then (7.27)

   t t  2 et(Δ+iY ) f (x) = Ex f ω(t) e(i/2) 0 Y (ω(s))·dω(s)+(1/4) 0 |Y (ω(s))| ds .

One final ingredient is required to prove Proposition 7.3, since in this case etX is not a semigroup of bounded operators, so we cannot apply Proposition A.2.

460 11. Brownian Motion and Potential Theory

However, we can apply Proposition A.3, with  S(t)f (x) =

  f (y)p t, y − x − tX(x) dy.

If X = iY is purely imaginary, then, parallel to (7.5), we have   2 p t, y − x − itY (x) = p(t, y − x)eiY (x)·(y−x)/2+t|Y (x)| /4 . If V is bounded and continuous, a simple modification of the analysis above, combining techniques of §2, yields (7.28)

   t  et(Δ+X−V ) f (x) = Ex f ω(t) eA(t,ω)/2−B(t,ω)/4− 0 V (ω(s)) ds

when X is purely imaginary. For another interpretation of this, consider H= (7.29)

 j

=−

−i

2 ∂ − Aj (x) + V ∂xj

  ∂ 2 ∂ ∂Aj 2 − 2iA − i − A j j + V. ∂x2j ∂xj ∂xj j

Assume each Aj is real-valued, and Aj , ∂Aj /∂xj ∈ BC(Rn ). Then

(7.30)

    e−tH f (x) = Ex f ω(t) eS(t,ω) ,  t   S(t, ω) = i A ω(s) · dω(s) 0  t  t      div A ω(s) ds − V ω(s) ds. −i 0

0

Compare with the derivation in [Si], Chap. 5. If the coefficients of X are not assumed to be purely imaginary, we need some more estimates. More generally, we will derive further estimates on the approxit mants Sk (ω) to 0 g(s, ω(s)) · dω(s), defined by (7.17). Lemma 7.4. If g is bounded and continuous, then (7.31)

  2 Ex eSk ≤ etγ ,

γ = gL∞ ,

and (7.32)

  2 2 Ex eλ|Sk | ≤ 2etλ γ .

Proof. The left side of (7.31) is

7. Stochastic integrals

461

∞      1 ν ν Ex eg0 θ0 · · · egk−2 θk−2 gk−1 θk−1 Ex eg0 θ0 · · · egk−1 θk−1 = ν! ν=0

(7.33)



∞  γν ν=0

ν!

   ν  Ex eg0 θ0 · · · egk−2 θk−2 Ex θk−1

    = Ex eg0 θ0 · · · egk−2 θk−2 Ex eγθk−1 , by independence arguments such as used in the analysis of (7.18). Note that the sums over ν above have terms that vanish for odd ν. Now Ex (eγθj ) = 2 e(tj+1 −tj )γ . An inductive argument leads to (7.31), and (7.32) follows from this |u| plus e ≤ eu + e−u . We next estimate the L2 (P0 , dWx )-norm of S2k − Sk . Another calculation, parallel to (7.18)–(7.19), yields

(7.34)

S2k − Sk 2L2 (P0 ,dWx )      2  = Ex g tj+1/2 , ω(tj+1/2 ) − g tj , ω(tj )  (tj+1 − tj+1/2 ), j

where tj = jt/k as in (7.17), and tj+1/2 = (j+1/2)t/k. If we assume a Lipschitz condition on g, we obtain the following estimate. Lemma 7.5. Assume that (7.35)

|g(t, x) − g(s, y)|2 ≤ C0 |t − s|2 + C1 |x − y|2 .

Then t2 t + 2C1 . k2 k   Proof. This follows from (7.34) plus Ex |ω(t) − ω(s)|2 = 2|t − s|.

(7.36)

S2k − Sk 2L2 (P0 ,dWx ) ≤ C0

We can now make an estimate directly relevant to the limiting behavior of (7.7). Lemma 7.6. Given the bound gL∞ ≤ γ, we have (7.37)

eS2k − eSk L1 (P0 ,dWx ) ≤



2

2S2k − Sk L2 (P0 ,dWx ) e32tγ .

Proof. Using eu − ev = (u − v)Φ(u, v), with |Φ(u, v)| ≤ e2|u|+2|v| , we have (7.38)

1/2

eS2k − eSk L1 (P0 ) ≤ S2k − Sk L2 (P0 ) · e4|S2k |+4|Sk | L1 (P0 ) ;

and the estimate (7.32), plus 2eu+v ≤ e2u + e2v , then yields (7.37).

462 11. Brownian Motion and Potential Theory

With these estimates, we can pass to the limit in (7.6)–(7.7), obtaining the following result. Proposition 7.7. If X is a real vector field on Rn whose coefficients are bounded and uniformly Lipschitz, and if f ∈ C0∞ (Rn ), then (7.39)    t t  2 et(Δ+X) f (x) = Ex f ω(t) e(1/2) 0 X(ω(s))·dω(s)−(1/4) 0 |X(ω(s))| ds . Now that the identity (7.39) is established for X and f such as described above, one can use limiting arguments to extend the identity to more general cases. Such extensions are left to the reader. t We now evaluate the stochastic integral 0 ω(s) dω(s) in the case of onedimensional Brownian motion. One might anticipate that it should be ω(t)2 /2 − ω(0)2 /2. However, guesses based on what should happen if ω had bounded variation can be misleading, and the truth is a little stranger. Let us begin with ω(t)2 − ω(0)2 =

k−1 

  ω(tj+1 )2 − ω(tj )2

j=0

(7.40) =



   ω(tj+1 ) + ω(tj ) · ω(tj+1 ) − ω(tj ) ,

j

where tj = (j/k)t, as in (7.17). We also use θj (ω) = ω(tj+1 ) − ω(tj ) below. t  Recalling that 0 ω(s) dω(s) is the limit of ω(tj )[ω(tj+1 ) − ω(tj )], we write (7.40) as (7.41)

ω(t)2 − ω(0)2 = 2

k−1 

ω(tj )θj (ω) +

j=0

k−1 

θj (ω)2 .

j=0

The next result is the key to the computation. Lemma 7.8. Given t > 0, (7.42)

Θk (ω) =

k−1  j=0

ω

j + 1   j 2 t −ω t −→ 2t in L2 (P0 , dWx ), k k

as k → ∞. Proof. We have

(7.43)

   t 2   Ex |Θk − 2t|2 = Ex  θj (ω)2 − 2  k j   t 2  = Ex θj (ω)2 − 2 , k j

7. Stochastic integrals

463

the last identity by independence of the different θj . Now we know that Ex (θj2 ) =   2t/k; furthermore, generally Ex [F − Ex (F )]2 ≤ Ex (F 2 ), so it follows that    t2 Ex |Θk − t|2 ≤ Ex (θj4 ) = 12 . k j

(7.44)

This proves the lemma. t 0

Thus, as k → ∞, the right side of (7.41) converges in L2 (P0 , dWx ) to ω(s) dω(s) + t. This gives the identity 

 1 ω(t)2 − ω(0)2 − 2t , 2

t

ω(s) dω(s) =

(7.45) 0

for Wx -almost all ω. More generally, for sufficiently smooth f , we can write (7.46)

     k−1    f ω(t) − f ω(0) = f ω(tj+1 ) − f ω(tj ) 

j=0

and use the expansion

(7.47)

    f ω(tj+1 ) − f ω(tj )     1   = θj (ω)f ω(tj ) + θj (ω)2 f

ω(tj ) + O |θj (ω)|3 2

to generalize (7.45) to Ito’s fundamental identity: (7.48)

    f ω(t) − f ω(0) =



t

  f ω(s) dω(s) +

0



t

  f

ω(s) ds,

0

for one-dimensional Brownian motion. For n-dimensional Brownian motion and functions of the form f = f (t, x), this generalizes to

(7.49)

    f t, ω(t) − f 0, ω(0)  t    ∇x f s, ω(s) · dω(s) = 0  t  t      Δf s, ω(s) ds + ft s, ω(s) ds. + 0

0

Another way of writing this is (7.50)

  df t, ω(t) = (∇x f ) · dω + (Δf ) dt + ft dt.

464 11. Brownian Motion and Potential Theory

We remind the reader that our choice of etΔ rather than etΔ/2 to define the transition probabilities for Brownian paths leads to formulas that sometimes look different from those arising from the latter convention, which for example would replace (Δf ) dt by (1/2)(Δf ) dt in (7.50). Note in particular that  2  2 d eλω(t)−λ t = λ eλω(t)−λ t dω(t); in other words, we have a solution to the “stochastic differential equation”: (7.51)

dX = λX dω(t),

2

X(t) = eλω(t)−λ t ,

for W0 -almost all ω. Recall from (4.16) that this is the martingale zt (ω). We now discuss a dynamical theory of Brownian motion due to Langevin, whose purpose was to elucidate Einstein’s work on the motion of a Brownian particle. Langevin produced the following equation for the velocity of a small particle suspended in a liquid, undergoing the sort of random motion investigated by R. Brown: (7.52)

dv = −βv + ω (t), dt

v(0) = v0 .

Here, the term −βv represents the frictional force, tending to slow down the particle as it moves through the fluid. The term ω (t), which contributes to the force, is due to “white noise,” a random force whose statistical properties identify it with the time derivative of ω, which is defined, not classically, but through Propositions 7.1 and 7.2. Thus we rewrite (7.52) as the stochastic differential equation (7.53)

dv = −βv dt + dω,

v(0) = v0 .

As in the case of ODE, we have d(eβt v) = eβt (dv + βv dt), so (7.50) yields d(eβt v) = eβt dω, which integrates to −βt

t

e−β(t−s) dω(s)  t e−β(t−s) ω(s) ds. = v0 e−βt + ω(t) − β

v(t) = v0 e (7.54)



+

0

0

The actual path of such a particle is given by  (7.55)

x(t) = x0 +

In the case x0 = 0, v0 = 0, we have

t

v(s) ds. 0

7. Stochastic integrals

 t

s

x(t) = 0

(7.56) =

1 β

465

e−β(s−r) dω(r) ds

0 t



 1 − e−β(t−s) dω(s).

0

Via the identity in (7.54), we have  x(t) =

(7.57)

t

e−β(t−s) ω(s) ds.

0

Of course, the path x(t) taken by such a particle is not the same as the “Brownian path” ω(t) we have been studying, but it is approximated by ω(t) in the following sense. It is observed experimentally that the frictional force component in (7.52) acts to slow down a particle in a very short time (∼ 10−8 sec.). In other words, the dimensional quantity β in (7.52) is, in terms of units humans use to measure standard macroscopic quantities, “large.” Now (7.57) implies lim βxβ (t) = ω(t),

(7.58)

β→∞

where xβ (t) denotes the path (7.57). There has been further work on the dynamics of Brownian motion, particularly by L. Ornstein and G. Uhlenbeck [UO]. See [Nel3] for more on this, and references to other work.

Exercises   1. If g ∈ C 1 [0, t] , show that the integral of Proposition 7.1 is given by 0

t

g(s) dω(s) = g(t)ω(t) − g(0)ω(0) −

0

t

g  (s)ω(s) ds.

Show that this yields the second identity in (7.54) and the implication (7.56) ⇒ (7.57). 2. With θj as in (7.15), show that Ex

k−1 

 |θj (ω)|3 → 0, as k → ∞.

j=0

(Hint: Use 2|θj |3 ≤ ε|θj |2 + ε−1 |θj |4 and (7.44).) 3. Making use of Exercise 2, give a detailed proof of Ito’s formula (7.48). Assume f ∈ C 2 (R) and 2 |Dα f (x)| ≤ Cε eε|x| , ∀ ε > 0, |α| ≤ 2. More generally, establish (7.49). Warning: The estimate of the remainder term in (7.47) is valid only when |ω(tj+1 − ω(tj )| is bounded (say ≤ K). But the probability that |ω(tj+1 ) − ω(tj )| is ≥ K is very small.

466 11. Brownian Motion and Potential Theory 4. Show that (7.42) implies that Wx -almost all paths ω have locally unbounded variation, on any interval[s, t] ⊂ [0, ∞).  t  5. If ψ(t, ω) = 0 g s, ω(s) · dω(s) is a stochastic integral given by Proposition 7.2, show that   Ex ψ(t, ·) = 0.   Show that ψ(t, ·) is a martingale, that is, Ex ψ(t, ·)|Bs = ψ(s, ·), for s ≤ t. Compare Exercise 2 of §8.

8. Stochastic integrals, II In §7 we considered stochastic integrals of the form  t   g s, ω(s) · dω(s), (8.1) h(t, ω) = 0

where g is defined on [0, ∞) × Rn . This is a special case of integrals of the form  t (8.2) ψ(t, ω) = ϕ(s, ω) · dω(s), 0

where ϕ is defined on [0, ∞)×P  0 . There  are important examples of such ϕ which are not of the form ϕ(s, ω) = g s, ω(s) , such as the function h in (8.1), typically. It is important to be able to handle more general integrals of the form (8.2), for a certain class of functions ϕ on [0, ∞) × P0 called “adapted,” which will be defined below. To define (8.2), we extend the analysis in (7.17)–(7.19). Thus we consider (8.3)

Sk (t, ω) =

k−1 

   k−1 ϕ(tj , ω) · ω(tj+1 ) − ω(tj ) = ϕj (ω) · θj (ω),

j=0

j=0

where, as before, θj (ω) = ω(tj+1 ) − ω(tj ), tj = (j/k)t. As in (7.18), we want to compute  (8.4) Sk (t, ·)2L2 (P0 ,dWx ) = Ex (ϕj θj ϕ θ ). j,

Following the analysis of (7.18), we want θ to be independent of the other factors in the parentheses on the right side of (8.4) when > j. Thus we demand of ϕ that (8.5)

ϕ(s, ·) is independent of ω(t + h) − ω(t),

∀ t ≥ s, h > 0.

Granted this, we see that the terms in the sum in (8.4) vanish when j = , and

8. Stochastic integrals, II

Sk (t, ·)2L2 (P0 ,dWx ) =



    Ex |ϕj |2 Ex |θj |2

j

(8.6) =2

467



  Ex |ϕ(tj , ·)|2 (tj+1 − tj ).

j

  If ϕ ∈ C R+ , L2 (P0 , dWx ) , this is a Riemann sum approximating 

t

2 0

  Ex |ϕ(s, ·)|2 ds = 2ϕ2L2 ([0,t]×P0 ) .

We use the following spaces: (8.7)

     C I, R(Q) = ϕ ∈ C I, L2 (P0 , dWx ) : ϕ(t) = Qt ϕ(t), ∀t ∈ I ,      L2 I, R(Q) = ϕ ∈ L2 I, L2 (P0 , dWx ) : ϕ(t) = Qt ϕ(t), ∀t ∈ I ,

where I = [0, T ], and, as in §4, Qt ϕ = Ex (ϕ|Bt ). Elements of these spaces satisfy (8.5), by Corollary 4.4.   Proposition 8.1. Given ϕ ∈ C I, R(Q) , the expression (8.3) converges as k =   2ν → ∞, in the space C I, R(Q) , to a limit we denote (8.2). Furthermore, ψ = I(ϕ) extends uniquely to a linear map     I : L2 I, R(Q) → C I, R(Q) ,

(8.8) satisfying (8.9)

I(ϕ)(t, ·)L2 (P0 ,dWx ) =



2 ϕL2 ([0,t)×P0 ,dt dWx ) .

Regarding continuity, note that (8.10) I(ϕ)(t+h, ·)−I(ϕ)(t, ·)L2 (P0 ,dWx ) =



2 ϕL2 ([t,t+h]×P0 ,dt dWx ) .

We need to verify that I(ϕ)(t, ·) ∈ R(Qt ). But clearly, each term ϕ(tj , ω) · [ω(tj+1 ) − ω(tj )] in (8.3) belongs to R(Qt ) in this case, so we have the desired result. We mention an approach to (8.8) just slightly different from that described above. Define a simple function to be a function ϕ(t, ω) that is constant in t for t in intervals of the form [ 2−ν , ( + 1)2−ν ), with values in R(Qs ), s = 2−ν , for some ν ∈ Z+ . For a simple function ϕ, the stochastic integral has a form similar to (8.3), namely,  (8.11)

0

t

ϕ(s, ω) · dω(s) =

−1 

  ϕ(tj , ω) · ω(tj+1 ) − ω(tj )

j=0

  + ϕ(t , ω) · ω(t) − ω(t ) ,

468 11. Brownian Motion and Potential Theory

  where tj = j2−ν and t ∈ 2−ν , ( +1)2−ν . An identity similar to (8.6), together with the denseness of the set of simple functions in L2 (I, R(Q)), yields (8.8). There is the following generalization of Ito’s formula (7.49)–(7.50). Suppose 



t

X(t) = X0 +

(8.12)

t

u(s, ω) ds + t0

v(s, ω) dω(s), t0

    where u, v ∈ L2 I, R(Q) . Then X ∈ C I, R(Q) . We write dX = u dt + v dω.

(8.13)

We might assume X, u, and ω take values in Rn and v is n × n matrix-valued. n m n m More generally, let ω take  values in R , X and u in R , and v in Hom(R , R ). If Y(t) = g t, X(t) , with g(t, x) real-valued and smooth in its arguments, then   dY(t) = (∇x g) t, X(t) · dX(t)      (8.14) + (D2 g) t, X(t) dX(t), dX(t) + gt t, X(t) dt, where (D2 g)(dX, dX) = the rules (8.15)

 2 (∂ g/∂xj ∂xk ) dXj · dXk is computed, via (8.13), by

dt · dt = dt · dωj = dωj · dt = 0,

dωj · dωk = δjk dt.

  There is also an integral formula for g t, X(t) − g(t0 , X0 ), parallel to (7.49): 

 t ∂2g  vj vk ds t0 ∂xj ∂xk  t  t    ∂g  + uj ds + vj dω . gt s, X(s) ds + t0 t0 ∂xj

 g t, X(t) = g(t0 , X0 ) + (8.16)

Here, we sum over repeated indices. The formulas (7.49) and (7.50) cover the special case u = 0, v = I. The proof of (8.16) is parallel to that of (7.49). If we apply (8.14) to g(x) = eλx , m = 1, we obtain for  t   Y(t) = exp λX(t) − λ2 |v(s, ω)|2 ds , 

(8.17)

t0 t

X(t) =

v(s, ω) · dω(s), t0

the stochastic differential equation (8.18)

dY = λY v · dω,

8. Stochastic integrals, II

469

generalizing the identity (7.51). There is another important property that Y(t), defined by (8.17), has in com2 mon with zt (ω) = eλω(t)−λ t .   Proposition 8.2. Given v ∈ L2 I, R(Q) , with values in Rn , the function Y(t) defined by (8.17) is a supermartingale; that is, for s ≤ t, (8.19)

   Ex Y(t)Bs ≤ Y(s),

Wx -a.e. on P0 .

Proof. We treat the case t0 = 0. First suppose vν is a simple function, constant as a function of t on intervals of the form [ 2−ν , ( + 1)2−ν ), with values in R(Q2−ν ), and Yν is given by (8.17), with v = vν . We claim that Yν is a martingale, that is, (8.20)

   Ex Yν (t)Bs = Yν (s), for s ≤ t.

Suppose, for example, that 0 ≤ t < 2−ν , so vν (s) = vν (0), for s ≤ t. Now vν (0) is independent of ω(t) − ω(s), so in this case      2 2 Ex Yν (t)Bs = Ex eλvν (0)[ω(t)−x]−λ t|vν (0)| Bs   2 2 2 2 = eλvν (0)[ω(s)−x]−λ s|vν (0)| · Ex eλvν (0)[ω(t)−ω(s)]−λ (t−s)|vν (0)| Bs , and the last conditional expectation is 1. A similar argument in the case 2−ν ≤ s ≤ t ≤ ( + 1)2−ν , using (8.11), gives      2 2 Ex Yν (t)Bs = Yν (tν )Ex eλvν [ω(t)−ω(tν )]−λ (t−tν )|vν | Bs = Yν (s), where tν = 2−ν , vν = vν (tν ). The identity (8.20), for general s ≤ t, follows easily from this.   vν converging to v in For general v ∈ L2 I, R(Q) , we can take simple  the norm of this space, and then Xν → X in C I, R(Q) , where Xν (t) = t v (s, ω) · dω(s). Passing to a subsequence, we can assume (for fixed s, t) 0 ν that Xν (s) → X(s) and Xν (t) → X(t), Wx -a.e.; hence Yν (s) → Y(s) and Yν (t) → Y(t), Wx -a.e. Then (8.19) follows, by Fatou’s lemma. The case of general t0 ≥ 0 is easily obtained from this; one can extend v(s, ω) to be 0 for 0 ≤ s < t0 . Note in particular that s = 0 in (8.19) implies (8.21)

   λX(t)−λ2 tt |v(s,·)|2 ds 0 ≤ 1. Ex e

Using Cauchy’s inequality, we deduce that

470 11. Brownian Motion and Potential Theory

(8.22)

   2t 1/2 λ |v(s,·)|2 ds Ex eλX(t)/2 ≤ Ex e t0 .

We get a similar estimate upon replacing v(s, ω) by −v(s, ω), which converts X(t) to −X(t). Since e|x| ≤ ex + e−x , we have (replacing λ by 2λ) (8.23)

   2t 1/2 4λ t |v(s,·)|2 ds 0 Ex eλ|X(t)| ≤ 2Ex e .

Compare with Lemma 7.4. Note that the convexity of the exponential function implies (8.24)

 −1  t  1 t   Ex et 0 F (s,·) ds ≤ Ex eF (s,·) ds. t 0

Therefore, (8.23) implies   Ex eλ|X(t)| ≤ 2 (8.25)



1 t − t0





t

4λ2 t|v(s·)|2



Ex e

1/2

ds

t0

 2  2 1/2 ≤ 2 max Ex e4λ t|v(s,·)| . t0 ≤s≤t

λX (t)−λ2

t

|v (s,·)|2 ds

ν t0 If we expand Yν (t) = e ν in powers of λ, the coefficient j of each λ is a martingale. The coefficient of λ4 , for example, is

 1 1 |Xν (t)|4 − Xν (t)2 (8.26) 24 2



t

t0

1 |vν (s, ω)| ds + 2 2





t

|vν (s, ω)|2 ds

2

.

t0

This has expectation zero; hence

(8.27)

 t    1  1 4 2 Ex |Xν (t)| ≤ Ex Xν (t) |vν (s, ·)|2 ds 24 2 t0  t 2    1 , Ex |Xν (t)|4 + 48Ex |vν (s, ·)|2 ds ≤ 48 t0

so    Ex |Xν (t)|4 ≤ 482 Ex



t

|vν (s, ·)|2 ds

t0

(8.28)



2 ≤ 48|t − t0 | 

1 t − t0



t

t0

2 

  Ex |vν (s, ·)|4 ds

  2 ≤ 48|t − t0 | max Ex |vν (s, ·)|4 , t0 ≤s≤t

8. Stochastic integrals, II

471

where the second inequality here uses convexity, as in (8.24). Again a use of Fatou’s lemma yields for  t (8.29) X(t) = v(s, ω) · dω(s) t0

the estimate (8.30)

 1/2 max v(s, ·)L4 (P0 ) . X(t)L4 (P0 ) ≤ 48|t − t0 | t0 ≤s≤t

Similarly we obtain, for t1 < t2 , (8.31)

X(t1 ) − X(t2 )L4 (P0 ) ≤ C1 |t1 − t2 |1/2

max v(s, ·)L4 (P0 ) ,

t1 ≤s≤t2

√ with C1 = 48, when X(t) is given by (8.29). If X(t) is given more generally by (8.12), we have X(t1 ) − X(t2 )L4 (P0 ) ≤ C0 |t1 − t2 | max u(s, ·)L4 (P0 ) t1 ≤s≤t2

(8.32)

+ C1 |t1 − t2 |1/2

max v(s, ·)L4 (P0 ) .

t1 ≤s≤t2

The martingale maximal inequality of Proposition 4.7 extends to submartingales, but it is not obvious that it applies to the supermartingale Y(t). However, it does apply to Yν (t), so, for each ν ∈ Z+ , we have (8.33)  t ! " |vν (s, ω)|2 ds > β Wx ω ∈ P0 : sup Xν (t) − Xν (t0 ) − λ t∈I(t0 ,t1 )

t0 −λβ

≤e

,

where I(t0 , t1 ) = [t0 , t1 ] ∩ Q. It follows that (8.34)

! Wx ω ∈ P0 :

 sup t∈I(t0 ,t1 )

|Xν (t) − Xν (t0 )| > λ

t1

|vν (s, ω)|2 ds + β

t0 −λβ

≤ 2e

.

Thus, if we have 

t1

(8.35) t0

then

|vν (s, ω)|2 ds




≤ ε + e−1/δ .

Since Xν (t) converges to X(t) in measure, locally uniformly in t, we have (8.40)

Wx

 ω ∈ P0 :

sup t∈I(t0 ,t1 )

|X(t) − X(t0 )| > 2δ



≤ ε + e−1/δ

whenever 

t

(8.41) t0

v(s, ·)2L2 (P0 ) ds < δ 3 ε.

The estimate (8.40) enables us to establish the following important result.   t 2 Proposition 8.3. Let  I = [0, T ]. Given v ∈ L I, R(Q) , so 0 v(s, ω)·dω(s) = X(t) belongs to C I, R(Q) , you can define X(t, ω) so that t → X(t, ω) is continuous in t, for Wx -a.e. ω. Proof. Start with any measurable function on I × P0 representing X(t); call it Xb (t, ω), so for each t ∈ I, Xb (t, ·) = X(t), Wx -a.e. on P0 . Set X(t, ω) = Xb (t, ω), for t ∈ I ∩ Q. From (8.40)–(8.41) it follows that there is a set N ⊂ P0 such that Wx (N ) = 0 and σω (t) = X(t, ω) is uniformly continuous in t ∈ I ∩ Q for each ω ∈ P0 \ N . Then, for ω ∈ P0 , t ∈ I \ Q, define X(t, ω) by continuity: (8.42)

X(t, ω) =

lim

I∩Qtν →t

Xb (tν , ω),

ω ∈ P0 \ N.

If ω ∈ N , define X(t, ω) arbitrarily. To show that this works, it remains to check that, for each t ∈ I,

9. Stochastic differential equations

X(t, ·) = X(t),

(8.43)

473

Wx -a.e. on P0 .

Indeed, since Xb (tν , ·) → X(t) in L2 -norm, passing to a subsequence we have Xb (tνj , ·) → X(t) Wx -a.e. Comparing with (8.42), we have (8.43).

Exercises 1. Generalize (8.30) to show that X(t) =

t t0

v(s, ω) · dω(s) satisfies

k−1 X(t)2k L2k (P0 ) ≤ Ck |t − t0 |

t t0

v(s, ·)2k L2k (P0 ) ds,

for k ∈ Z+ .   2. Given ϕ ∈ L2 [0, ∞), R(Q) , show that, for t ≥ s, 

t

Ex

  ϕ(τ, ω) · dω(τ )Bs = 0.

s

t Deduce that the stochastic integral ψ(t, ω) = 0 ϕ(s, ω) · dω(s) is a martingale, so that, for t ≥ s,    Ex ψ(t, ·)Bs = ψ(s, ·). 3. Show that if v(s, ω) satisfies the hypotheses of Proposition 8.2, then the supermartingale Y(t) in (8.17) is a martingale if and only if Ex Y(t) = 1,

∀ t ≥ 0.

9. Stochastic differential equations In this section we treat stochastic differential equations of the form (9.1)

dX = b(t, X) dt + σ(t, X) dω,

X(t0 ) = X0 .

The function X is an unknown function on I × P0 , where I = [t0 , T ]. We assume t0 ≥ 0. As in the case of ordinary differential equations, we will use the Picard iteration method, to obtain the solution X as the limit of a sequence of approximate solutions to (8.1), which we write as a stochastic integral equation:  (9.2)

t

X(t) = X0 + t0

  b s, X(s) ds +



t

  σ s, X(s) dω(s) = ΦX(t).

t0

The last identity defines the transformation Φ, and we look for a fixed point of Φ. As usual, X(t) is shorthand for X(t, ω). If ω is a Brownian path in Rn , we can let X and b(t, x) take values in Rm and let σ(t, x) be an m × n matrix-valued function.

474 11. Brownian Motion and Potential Theory

Let us assume that σ(t, x) and b(t, x) are continuous in their arguments and satisfy |b(t, x)| ≤ K0 (1 + |x|),

(9.3)

|b(t, x) − b(t, y)| ≤ L0 |x − y|,

2 1/2

|σ(t, x)| ≤ K1 (1 + |x| )

|σ(t, x) − σ(t, y)| ≤ L1 |x − y|.

,

We will use results of §8 to show that     Φ : L2 I, R(Q) −→ C I, R(Q) ,

(9.4) where, as in (8.7),

     C I, R(Q) = ϕ ∈ C I, L2 (P0 , dW0 ) : ϕ(t) ∈ R(Qt ), ∀t ∈ I ,   and L2 I, R(Q) is similarly defined. Note that X(s) belongs to R(Qs ) if and only if X(s) is (equal W0 -a.e. to) a Bs -measurable   function on P0 , so if X(s) ∈ R(Qs ), then also σ s, X(s) and b s, X(s) belong to R(Qs ). Thus Proposition 8.1 applies to the second integral in (9.2), and if X0 ∈ R(Qt0 ), we have (9.4). Applying (8.9) to estimate the second integral in (9.2), we have ΦX(t) − X0 2L2 (P0 ) ≤ 2K02



t

t0

(9.5) + 4K12



 1 + X(s)L2 (P0 ) ds

t

t0

2

 1 + X(s)2L2 (P0 ) ds.

Also (8.9) applies to an estimate of the second integral in  ΦX(t) − ΦY(t) = (9.6)

t

    b s, X(s) − b s, Y(s) ds

t0



+

t

    σ s, X(s) − σ s, Y(s) ds.

t0

We get ΦX(t) − ΦY(t)2L2 (P0 ) ≤ 2L20



(9.7) + 4L21

2

t

t0



X(s) − Y(s)L2 (P0 ) ds t

t0

X(s) − Y(s)2L2 (P0 ) ds.

To solve (9.2), we take X0 (t, ω) = X0 (ω), the given initial value, and inductively define Xj+1 = ΦXj . Note that

9. Stochastic differential equations

 (9.8)

t

X1 (t, ω) = X0 (ω) +

  b s, X0 (ω) ds +



t0

t

475

  σ s, X0 (ω) dω(s)

t0

contains a stochastic integral of the form (7.14), provided X0 (ω) is constant. On the other hand, the stochastic integral yielding X2 (t, ω) is usually not even of the form (7.13), but rather of the more general form (8.2). The following estimate will readily yield convergence of the sequence Xj . Lemma 9.1. For some M = M (T ) < ∞, we have (9.9)

Xj+1 (t) − Xj (t)2L2 (P0 ) ≤

(M |t − t0 |)j+1 , (j + 1)!

t0 ≤ t ≤ T.

Proof. We establish this estimate inductively. For j = 0, we can use (9.5), with X = X1 , and the j = 0 case of (9.9) follows. Assume that (9.9) holds for j = 0, . . . , k−1; we need to get it for j = k. To do this, apply (9.7) with X = Xk , Y = Xk−1 , to get Xk+1 (t) − Xk (t)2L2 (P0 ) ≤ (9.10)

2L20 M k k!



4L21 M k + k!

2

t

|s − t0 |k/2 ds t0



t

|s − t0 |k ds. t0

 k+1 This is ≤ M |t − t0 | /(k + 1)! as long as M is sufficiently large for (9.9) to hold for j = 0 and also M ≥ 2L20 max(1, T ) + 4L21 . These estimates immediately yield an existence theorem: Theorem 9.2. Given 0 ≤ t0 < T < ∞, I = [t0 , T ], if b and σ are continuous on I × Rn and satisfy the estimates  (9.3), and  if X0 ∈ R(Qt0 ), then the equation (9.2) has a unique solution X ∈ C I, R(Q) . Only the uniqueness remains to be demonstrated. But if X and Y are two such solutions, we have ΦX = X and ΦY = Y, so (9.7) implies X(t) − Y(t)2L2 (P0 ) ≤ right side of (9.7), and a Gronwall argument implies X(t) − Y(t)L2 = 0, for all t ∈ I. Of course, the hypothesis that b and σ are continuous in t can be weakened in ways that are obvious from an examination of (9.4)–(9.7). Allowing b and σ to be piecewise continuous in t, still satisfying (9.3), we can reduce (9.1) to the case t0 = 0, by setting b(t, x) = 0 and σ(t, x) = 0 for 0 ≤ t < t0 . If X0 has higher integrability, so does the solution X(t). To see this, in case X0 ∈ L4 (P0 ), we can exploit (8.26)–(8.30) to produce the following estimate, parallel to (9.7):

476 11. Brownian Motion and Potential Theory

(9.11)

ΦX(t) − ΦY(t)4L4 (P0 ) ≤  t 4 8L40 X(s) − Y(s)L4 (P0 ) ds t0

2

+ 8(48

)L41 |t



t

− t0 | t0

X(s) − Y(s)4L4 (P0 ) ds.

Using this, assuming X0 ∈ L4 (P0 , dW0 ), we can obtain the following analogue of (9.9): 

(9.12)

Xj+1 (t) −

Xj (t)4L4 (P0 )

j+1 M |t − t0 |2 , ≤ (j + 1)!

for M = M (T ), on any interval t ∈ [t0 , T ]. We have the following: Proposition 9.3. Under the hypotheses of Theorem 9.2, if also X0 ∈ L4 (P0 ,   dW0 ), then X ∈ C I, L4 (P0 , dW0 ) .   More generally, one can establish that X ∈ C I, L2k (P0 ) , provided X0 ∈ L2k (P0 ), k ≥ 1. The case 2k = 4 enables us to prove part of the following important result. Proposition 9.4. The solution X(t) to (9.2) given by Theorem 9.2 can be represented as X(t, ω) such that, for W0 -a.e. ω ∈ P0 , the map t → X(t, ω) is continuous in t. Proof. First we assume X0 ∈ L4 (P0 , dW0 ) and give a demonstration that is somewhat parallel to that of Theorem 1.1. Given ε > 0, δ > 0, and s, t ∈ R+ such that |t − s| < δ, we estimate the probability that |X(t) − X(s)| > ε. We use the estimate X(t) − X(s)4L4 (P0 ) ≤ C|t − s|2 ,

(9.13)

C = C(T ), for s, t ∈ [0, T ], which follows (when t > s) from X(t) − X(s)4L4 (P0 ) ≤ C



(9.14)

t

b(τ, X(τ ))L4 dτ

4

s



+C

t

σ(τ, X(τ ))4L4 dτ,

s

together with the estimate X(s)L4 ≤ C(τ ). Consequently, given s, t ∈ R+ , (9.15)

W0



 C ω ∈ P0 : |X(t, ω) − X(s, ω)| > ε ≤ 4 |t − s|2 . ε

Now an argument parallel to that of Lemma 1.2 gives

9. Stochastic differential equations

W0



ω ∈ P0 : |X(t1 , ω) − X(tj , ω)| > ε, for some j = 2, . . . , ν

(9.16) ≤ Cr

477



ε  ,δ , 2

when {t1 , . . . , tν } is any finite set of numbers in Q+ such that 0 ≤ t1 < · · · < tν and tν − t1 ≤ δ, where (9.17)

  r(ε, δ) = min 1, Cδ 2 ε−4 .

The function r(ε, δ) takes the place of ρ(ε, δ) in (1.23); as in (1.21), we have (9.18)

r(ε, δ) → 0, as δ → 0, δ

for each ε > 0. From here, one shows just as in the proof of Theorem 1.1 that, for some Z ⊂ P0 such that W0 (Z) = 0, the map t → X(t, ω) is uniformly continuous on t ∈ Q+ , for each ω ∈ P0 \Z. the rest of the proof of Proposition 9.4 can be carried out just like the proof of Proposition 8.3. We now give another demonstration of Proposition 9.4, not requiring X0 to be in L4 (P0 ), but only in L2 (P0 ). In such acase, under the hypothe  ses, and conclusions, of Theorem 9.2, we have σ t, X(t) ∈ C I, R(Q) . Hence Proposition 8.3 applies to the second integral in (9.2), so A(t, ω) =  t  σ s, X(s) dω(s) can be represented as a continuous function of t, for W0 -a.e. t0       ω ∈ P0 . Furthermore, we have b t, X(t) ∈ C I, L2 (P0 ) ⊂ C I, L1 (P0 ) . Thus, by Fubini’s theorem, the first integral in (9.2) is absolutely integrable, hence continuous in t, for W0 -a.e. ω. This establishes the desired property for the left side of (9.2). We next investigate the dependence of the solution to (9.2) on the initial data X0 , in a fashion roughly parallel to the method used in §6 of Chap. 1. Thus, let Y solve  t  t     b s, Y(s) ds + σ s, Y(s) dω(s). (9.19) Y(t) = Y0 + t0

t0

Proposition 9.5. Assume that b(t, x) and σ(t, x) satisfy the hypotheses of Theorem 9.2 and are also C 1 in x. If X(t) and Y(t) solve (9.2) and (9.19), respectively, then (9.20)

X(t) − Y(t)L2 (P0 ) ≤ C(t, L0 , L1 )X0 − Y0 L2 (P0 ) .

Proof. Consider Z(t) = X(t) − Y(t), which satisfies the identity

478 11. Brownian Motion and Potential Theory



t

Z(t) = Z0 + t0 t



(9.21) +

  b s, X(s), Y(s) Z(s) ds   σ s, X, Y(s) Z(s) dω(s),

t0

with Z0 = X0 − Y0 . Here 



1

b (s, x, y) =

(9.22)

0

  Dx b s, ux + (1 − u)y du,

so b (s, x, y)(x − y) = b(s, x) − b(s, y), and similarly



σ (s, x, y) =

(9.23)

0

1

  Dx σ s, ux + (1 − u)y du.

We estimate the right side of (9.21) in L2 (P0 ). By (9.3), |b (s, x, y)| ≤ L0 , so (9.24)

  t    

 b s, X(s), Y(s) Z(s) ds  



L2

t0

t

≤ L0

Z(s)L2 ds. t0

Since |σ (s, x, y)| ≤ L1 and σ (s, X(s), Y(s))Z(s) ∈ R(Qs ), we have (9.25)

 t 2  t    

  ≤ L21 s, X(s), Y(s) Z(s) dω(s) σ Z(s)2L2 ds.   L2

t0

t0

Thus the identity (9.21) implies   (9.26) Z(t)2L2 ≤ 3X0 − Y0 2L2 + 3 L20 (t − t0 )2 + L21



t

Z(s)2L2 ds.

t0

Now Gronwall’s inequality applied to this estimate yields (9.20). Note that (9.21) is a linear stochastic equation for Z(t), of a form a little different from (9.2), if X(s) and Y(s) are regarded as given. On the other hand, we can regard X, Y, and Z as solving together a system of stochastic equations, of the same form as (9.2). An important special case of (9.2) is the case X0 = x, a given point of Rm , so let us look at Xx,s (t), defined for t ≥ s as the solution to  (9.27)

X

x,s

(t) = x + s

t



 b r, X(r) dr +



t

  σ r, X(r) dω(r).

s

In this case we have the following useful property, which is basically the Markov property. Let Bts denote the σ-algebra of subsets of P0 generated by all sets of the form

9. Stochastic differential equations

(9.28)

{ω ∈ P0 : ω(t1 ) − ω(s1 ) ∈ A},

479

s ≤ s1 ≤ t1 ≤ t, A ⊂ Rm Borel,

plus all sets of W0 -measure zero. Proposition 9.6. For any fixed t ≥ s, the solution Xx,s (t) to (9.27) is Bts measurable. Proof. By the proof of Theorem 9.2, we have Xx,s (t) = limk→∞ Xk (t), where X0 (t) = x and, for k ≥ 0, 

t

Xk+1 (t) = x +

  b r, Xk (r) dr +



s

t

  σ r, Xk (r) dω(r).

s

It follows inductively that each Xk (t) is Bts -measurable, so the limit also has this property. The behavior of Xx,s (t) will be important for the next section. We derive another useful property here. Proposition 9.7. For s ≤ τ ≤ t, we have Xx,s (t, ω) = Xq,τ (t, ω),

(9.29)

q = Xx,s (τ, ω),

for W0 -a.e. ω ∈ P0 . Proof. Let Y(t) denote the right side of (9.29). Thus Y(τ ) = Xx,s (τ ). The stochastic equation satisfied by Xx,s (t) then implies  Y(t) = Xx,s (τ ) +

t

  b r, Y(r) dr +



τ

t

  σ r, Y(r) dω(r).

τ

Now (9.27) implies that Xx,s (t) satisfies this same stochastic equation, for t ≥ τ . The identity Y(t) = Xx,s (t) a.e. on P0 follows from the uniqueness part of Theorem 9.2.

Exercises 1. Show that the solution to dX = a(t)X(t) dt + b(t)X(t) dω(t), in case m = n = 1, is given by

t [a(s) − b(s)2 ] ds + (9.30) X(t) = X(0) exp 0

t 0

b(s) dω(s) = X(0)eZ(t) .

In this problem and the following one, X(t) depends on ω, but a(t) and b(t) do not depend on ω, nor do f (t) and g(t) below.

480 11. Brownian Motion and Potential Theory 2. Show that the solution to     dX(t) = f (t) + a(t)X(t) dt + g(t) + b(t)X(t) dω(t), in case m = n = 1, is given by X(t) = eZ(t) Y(t), where eZ(t) is as in (9.30) and t t  e−Z(s) f (s) − g(s)b(s) ds + g(s)e−Z(s) dω(s). Y(t) = X(0) + 0

0

3. Consider the system (9.31)

  dX(t) = A(t)X(t) + f (t) dt + g(t) dω(t),

where A(t) ∈ End(Rm ), f (t) ∈ Rm , and g(t) ∈ Hom(Rn , Rm ). Suppose S(t, s) is the solution operator to the linear m × m system of differential equations dy = A(t)y, dt

S(t, t) = I,

as considered in Chap. 1, §5. Show that the solution to (9.31) is t t S(t, s)f (s) ds + S(t, s)g(s) dω(s). X(t) = S(t, 0)X(0) + 0

0

4. The following Langevin equation is more general than (7.52):   (9.32) x (t) = −∇V x(t) − βx (t) + ω  (t). Rewrite this as a first-order system of the form (9.1). Using Exercise 3, solve this equation when V (x) is the harmonic oscillator potential, V (x) = ax2 .

10. Application to equations of diffusion Let Xx,s (t) solve the stochastic equation  (10.1)

Xx,s (t) = x +

t

  b Xx,s (r) dr +

s



t

  σ Xx,s (r) dω.

s

As in (9.2), x and b can take values in Rm and σ values in Hom(Rn , Rm ). We want to study the transformations on functions on Rm defined by (10.2)

  Φts f (x) = E0 f Xx,s (t) ,

Clearly, Xx,s (s) = x, so (10.3)

Φtt f (x) = f (x).

0 ≤ s ≤ t.

10. Application to equations of diffusion

481

We assume b(x) and σ(x) are bounded and satisfy the Lipschitz conditions of (9.3). For simplicity we have taken b and σ to be independent of t in (10.1). We claim this implies the following: (10.4)

Φt0 f (x) = Φt+s s f (x),

for s, t ≥ 0. In fact, it is clear that (10.5)

Xx,s (t + s, ω) = Xx,0 (t, ϑs ω),

where ϑs ω(τ ) = ω(τ + s) − ω(s), as in (4.11). The measure-preserving property of the map ϑs : P0 → P0 then implies     E0 f Xx,0 (t, ϑs ω) = E0 f Xx,0 (t) = Φt0 f (x), so we have established (10.4). Let us set (10.6)

  P t f = Φt0 f = E0 f Xx (t) ,

where for notational convenience we have set Xx (t) = Xx,0 (t). We will study the action of P t on the Banach space Co (Rm ) of continuous functions on Rm that vanish at infinity. Proposition 10.1. For each t ≥ 0, (10.7)

P t : Co (Rm ) −→ Co (Rm ),

and P t forms a strongly continuous semigroup of operators on Co (Rm ). Proof. If f ∈ Co (Rm ), then f is uniformly continuous, that is, it has a modulus of continuity: (10.8)

  |f (x) − f (y)| ≤ ωf |x − y| ,

where ωf (δ) is a bounded, continuous function of δ such that ωf (δ) → 0 as δ → 0. Then     t    P f (x) − P t f (y) ≤ E0 f Xx (t) − f Xy (t)    (10.9) ≤ E0 ωf Xx (t) − Xy (t) . Now if x is fixed and y = xν → x, then, for each t ≥ 0, Xx (t) − Xxν (t) → 0 in L2 (P0 ), by Proposition 9.5. Hence Xx (t) − Xxν (t) → 0 in measure on P0 , so the Lebesgue dominated convergence theorem implies that (10.9) tends to 0 as y → x. This shows that P t f ∈ C(Rm ) if f ∈ Co (Rm ).

482 11. Brownian Motion and Potential Theory

To show that P t f (x) vanishes at infinity, for each t ≥ 0, we note that, for x most ω ∈ P0 (in a sense that will be quantified below),  x |X (t)  − x| ≤ Ct if m C is large, so if f ∈ Co (R ) and |x| is large, then f X (t, ω) is small for most ω ∈ P0 . In fact, subtracting x from both sides of (10.1) and estimating L2 -norms, we have (10.10)

Xx (t) − x2L2 (P0 ) ≤ 2B 2 t2 + 2S 2 t,

B = sup |b|, S = sup |σ|.

Hence (10.11)

W0



ω ∈ P0 : |Xx (t, ω) − x| > λ





2B 2 t2 + 2S 2 t . λ2

The mapping property (10.7) follows. We next examine continuity in t. In fact, parallel to (10.9), we have (10.12)

   t  P f (x) − P s f (x) ≤ E0 ωf Xx (t) − Xx (s) .

  We know from §9 that Xx (t) ∈ C R+ , L2 (P0 ) , and estimates from there readily yield that the modulus of continuity can be taken to be independent of x. Then the vanishing of (10.12), uniformly in x, as s → t, follows as in the analysis of (10.9). There remains the semigroup property, P s P t−s = P t , for 0 ≤ s ≤ t. By (10.4), this is equivalent to Φs0 Φts = Φt0 . To establish this, we will use the identity (10.13)

      E0 f Xx,s (t) Bs = E0 f Xx,s (t) = Φts f (x),

which is an immediate consequence of Proposition 9.6. If we replace s by τ in (10.13), and then replace x by Xx,s (τ ), with s ≤ τ ≤ t, and use the identity (10.14)

Xq,τ (t) = Xx,s (t),

q = Xx,s (τ ),

established in Proposition 9.7, we obtain (10.15)

      E0 f Xx,s (t) Bτ = Φtτ f Xx,s (τ ) .

We thus have, for s ≤ τ ≤ t,

(10.16)

    Φτs Φtτ f (x) = E0 Φtτ f Xx,s (τ ) Bs       = E0 E0 f Xq,τ (t) Bτ Bs     = E0 f Xq,τ (t) Bs ,

10. Application to equations of diffusion

483

and again using (10.14) we see that this is equal to the left side of (10.13), hence to Φts f (x), as desired. This completes the proof of Proposition 10.1. We want to identify the infinitesimal generator of P t . Assume now that Dα f , for |α| ≤ 2, are bounded and continuous on Rm . Then Ito’s formula implies  t ∂2f  σj σk dr ∂xj ∂xk 0  t  ∂f  bj dr + σj dω , + 0 ∂xj

  f Xx (t) = f (x) + (10.17)

using the summation convention. Let us apply E0 to both sides. Now  E0

(10.18)

t

0

 ∂f σj dω = 0, ∂xj

so we have 

 ∂2f  Ajk dr ∂xj ∂xk 0  t  ∂f  + E0 bj dr, ∂xj 0

  E0 f (Xx (t)) = f (x) + (10.19)

t

E0

where Ajk in the first integral is given by Ajk (y) =

(10.20)



σj (y)σk (y),

y = Xx (r).



In matrix notation, A = σσ t .

(10.21)

We can take the t-derivative of the right side of (10.16), obtaining ∂ t P f (x) = ∂t          E0 Ajk Xx (t) ∂j ∂k f Xx (t) + bj Xx (t) ∂j f Xx (t) .

(10.22)

In particular,

(10.23)

   ∂ t P f (x)t=0 = Ajk (x) ∂j ∂k f (x) + bj (x) ∂j f (x) = Lf (x), ∂t j j,k

484 11. Brownian Motion and Potential Theory

where the last identity defines the second-order differential operator L, acting on functions of x. This is known as Kolmogorov’s diffusion equation. We have shown that the infinitesimal generator of the semigroup P t , acting on Co (Rm ), is a closed extension of the operator   bj (x) ∂j , (10.24) L= Ajk (x) ∂j ∂k + defined initially, let us say, on C02 (Rm ). It is clear from (10.6) that P t f L∞ ≤ f L∞ for each f ∈ Co (Rm ), so P t is a contraction semigroup on Co (Rm ). It is also clear that (10.25)

f ≥ 0 =⇒ P t f ≥ 0 on Rm ,

that is, P t is “positivity preserving.” For given x ∈ Rn , t ≥ 0, f → P t f (x) is a positive linear functional on Co (Rm ). Hence there is a uniquely defined positive Borel measure μx,t on Rm , of mass ≤ 1, such that  (10.26)

t

P f (x) =

f (y) dμx,t (y).

In fact, by the construction (10.6), μx,t = F(x,t)∗ W0 ,

(10.27)

 −1  (U ) for a where F(x,t) (ω) = Xx (t, ω), and (10.27) means μx,t (U ) = W0 F(x,t) Borel set U ⊂ Rm . This implies that, for each x, t, μx,t is a probability measure on Rm , since |Xx (t)| is finite for W0 -a.e. ω ∈ P0 . We will use the notation (10.28)

P (s, x, t, U ) = μx,t−s (U ),

0 ≤ s ≤ t, U ⊂ Rm , Borel.

We can identify P (s, x, t, U ) with the probability that Xx,s (t) is in U . We can rewrite (10.26) as  t (10.29) P f (x) = f (y) P (0, x, t, dy) or  (10.30)

Φts f (x)

=

f (y) P (s, x, t, dy).

The semigroup property on P t implies  (10.31)

P (s, x, t, U ) =

P (s, x, τ, dy) P (τ, y, t, U ),

0 ≤ s ≤ τ ≤ t,

10. Application to equations of diffusion

485

which is known as the Chapman–Kolmogorov equation. Let us denote by L the extension of (10.24) that is the infinitesimal generator of P t . If V is a bounded, continuous function on Rm , then L − V generates a semigroup on Co (Rm ), and an application of the Trotter product formula similar to that done in §2 yields (10.32)

    t x et(L−V ) f (x) = E0 f Xx (t) e− 0 V (X (s)) ds .

This furnishes an existence result for weak solutions to the initial-value problem

(10.33)

 ∂u  = Ajk (x) ∂j ∂k u + bj (x) ∂j u − V u, ∂t u(0) = f ∈ Co (Rm ),

under the hypotheses that V is bounded and continuous, the coefficients bj are bounded and uniformly Lipschitz, and Ajk has the form (10.20), with σj bounded and uniformly Lipschitz. As for the last property, we record the following fact: Proposition 10.2. If A(x) is a C 2 positive-semidefinite, matrix-valued function on Rm with Dα A(x) bounded on Rm for |α| ≤ 2, then there exists a bounded, uniformly Lipschitz, matrix-valued function σ(x) on Rm such that A(x) = σ(x)σ(x)t . This result is quite easy to prove in the elliptic case, that is, when for certain λj ∈ (0, ∞), λ0 |ξ|2 ≤

(10.34)



Ajk (x)ξj ξk ≤ λ1 |ξ|2 ,

but a careful argument is required if A(x) is allowed to degenerate. See the exercises for more on this. If Ajk (x) has bounded, continuous derivatives of order ≤ 2, we can form the formal adjoint of (10.24): (10.35)

Lt f =



    

− V f, ∂j ∂k Ajk (x)f − ∂j bj (x)f = Lf

has the same second-order derivatives as L, though perhaps a different where L  

has an extenfirst-order part, and V (x) = − ∂j ∂k Ajk (x)+ ∂k bj (x). Thus L

generating a contraction semigroup on Co (Rm ), with sion, which we denote as L, the positivity-preserving property. Furthermore, L − V generates a semigroup on  Co (Rm ), and there is a formula for et(L−V ) f parallel to (10.32). Thus we obtain a weak solution to the initial-value problem (10.36)

     ∂u  = ∂j ∂k Ajk (x)u − ∂j bj (x)u , ∂t

u(0) = f ∈ Co (Rm ),

486 11. Brownian Motion and Potential Theory

provided that Ajk (x) satisfies the conditions of Proposition 10.2, and that each bj is bounded, with bounded, continuous first derivatives. Equation (10.36) is called the Fokker-Planck equation. To continue, we shall make a further simplifying hypothesis, namely that the ellipticity condition (10.34) hold. We will also assume Ajk (x) and bj (x) are C ∞ , and that Dα Ajk (x) and Dα bj (x) are bounded for all α. In such a case, (gjk ) = (Ajk )−1 defines a Riemannian metric on Rm , and if Δg denotes its Laplace operator, we have (10.37)

Lf = Δg f + Xf,

 for some smooth vector field X = ξj (x) ∂j , such that Dα ξj (x) is bounded for |α| ≤ 1. Note that if we use the inner product  (10.38)

(f, g) =

f (x)g(x) dV (x),

where dV is the Riemannian volume element determined by the Riemannian metric gjk , then this puts the same topology on L2 (Rm ) as the inner product  f (x)g(x) dx. We prefer the inner product (10.38), since Δg is then self-adjoint. Now consider the closed operator L2 on L2 (Rm ) defined by (10.39)

L2 f = Lf on D(L2 ) = H 2 (Rm ).

It follows from results on Chap. 6, §2, that L2 generates a strongly continuous semigroup etL2 on L2 (Rn ). To relate this semigroup to the semigroup P t = etL on Co (Rm ) described above, we claim that (10.40)

etL2 f = etL f, for f ∈ C0∞ (Rm ).

To see this, let u0 (t, x) and u1 (t, x) denote the left and right sides, respectively. These are both weak solutions to ∂t uj = Luj , for which one has regularity results. Also, estimates discussed in §2 of Chap. 6 imply that u0 (t, x) vanishes as |x| → ∞, locally uniformly in t ∈ [0, ∞). Thus the maximum principle applies to u0 (t, x) − u1 (t, x), and we have (10.40). From here a simple limiting argument yields (10.41)

etL2 f = etL f, for f ∈ Co (Rm ) ∩ L2 (Rm ).

Now the dual semigroup (etL2 )∗ is a strongly continuous semigroup on L (Rn ), with infinitesimal generator Lt2 defined by 2

(10.42)

Lt2 f = Lt f on D(Lt2 ) = H 2 (Rm ),

10. Application to equations of diffusion

487

where Lt is given by (10.35). An argument parallel to that used to establish (10.41) shows that (10.43)

 tL2 ∗ t  f = etL2 f = et(L−V ) f, for f ∈ Co (Rm ) ∩ L2 (Rm ). e

On the other hand, (P t )∗ = (etL )∗ is a weak∗ -continuous semigroup of operators on M(Rm ), the space of finite Borel measures on Rm ; it is not strongly continuous. Using (10.43), we see that (10.44)



    f, etL2 g = et(L−V ) f, g , for f, g ∈ C0∞ (Rm ),

and bringing in (10.40) we have 

(10.45)

∗  etL f = et(L−V ) f,

for f∈ C0∞ (Rm ), hence for f ∈ C0 (Rm ) ∩ L1 (Rm ). From here one can deduce ∗ that etL preserves L1 (Rm ) and acts as a strongly continuous semigroup on this space. Let us return to the family of measures P (s, x, t, ·). Under our current hypotheses, regularity results for parabolic PDE imply that, for s < t, there is a smooth function p(s, x, t, y) such that  (10.46)

p(s, x, t, y) dy.

P (s, x, t, U ) = U

We have  (10.47)

Φts f (x)

=

f (y) p(s, x, t, y) dy,

s < t,

and (10.48)

 t ∗ Φs f (y) =

 f (x) p(s, x, t, y) dx,

s < t.

Furthermore, we have for p(s, x, t, y) the “backward” Kolmogorov equation (10.49)

  ∂2p ∂p ∂p =− Ajk (x) − bj (x) ∂s ∂xj ∂xk ∂xj j j,k

and the Fokker-Planck equation (10.50)

  ∂   ∂p  ∂ 2  = Ajk (y)p − bj (y)p . ∂t ∂yj ∂yk ∂yj j j,k

488 11. Brownian Motion and Potential Theory

While we have restricted attention to the smooth elliptic case for the last set of results, it is also interesting to relax the regularity required on the coefficients as much as possible, and to let the coefficients depend on t, and also to allow degeneracy. See [Fdln] and [StV] for more on this. Exercise 5 below illustrates the natural occurence of degenerate L. We mention that, working with (10.32), we can obtain the solution to

(10.51)

∂u = Lu, for t ≥ 0, x ∈ Ω, ∂t u(t, x) = 0, for x ∈ ∂Ω, u(0, x) = f (x),

by considering a sequence Vν → ∞ on Rm \ Ω, as in the analysis in §3, when Ω is an open domain in Rm , with smooth boundary, or at least with the regularity property used in Proposition 3.3. In analogy with (3.8), we get (10.52)

    u(t) = E0 f Xx (t) ψΩ (Xx , t) ,

where (10.53)

ψΩ (Xx , t) = 1 0

  if Xx [0, t] ⊂ Ω, otherwise.

The proof can be carried out along the same lines as in the proof of Proposition 3.3, provided L2 (defined in (10.39)) is self-adjoint. Otherwise a different approach is required. Also, when L2 is self-adjoint, the analysis leading to Proposition 3.5 extends to (10.51), for any open Ω ⊂ Rm , with no boundary regularity required. For other approaches to these matters, and also to the Dirichlet problem for Lu = f on Ω, in both the elliptic and degenerate cases, see [Fdln] and [Fr]. We end this section with a look at a special case of (10.1), namely when σ = I, so we solve  t   b Xx (r) dr. (10.54) Xx (t) = x + ω(t) + 0

Assume as before that b is bounded and uniformly Lipschitz. Then the analysis of (10.6) done above implies (10.55)

  et(Δ+X) f (x) = E0 f Xx (t) ,

X=



On the other hand, in §7 we derived the formula (10.56)

    et(Δ+X) f (x) = Ex f ω(t) eZ(t) ,

bj (x) ∂j .

10. Application to equations of diffusion

where Z(t) =

1 2

 0

t

  1 b ω(s) · dω(s) − 4



489

t

  b ω(s) 2 ds.

0

We conclude that the right-hand sides of (10.55) and (10.56) coincide. We can restate this identity as follows. Given x ∈ Rn , we have a map Ξx : P0 → P0 ,

(10.57)

Ξx (ω)(t) = Xx (t).

Then Wiener measure W0 on P0 gives rise to a measure Ξx∗ W0 on P0 , by   Ξx∗ W0 (S) = W0 (Ξx )−1 (S) .

(10.58)

For example, if 0 ≤ t1 < · · · < tk ,  (10.59)

    F ω(t1 ), . . . , ω(tk ) dΞx∗ W0 = E0 F Xx (t1 ), . . . , Xx (tk ) .

P0

Thus the identity of (10.55) and (10.56) can be written as 

  f ω(t) dΞx∗ W0 =

(10.60) P0



  f ω(t) eZ(t) dWx .

P0

This is a special case of the following result of Cameron-Martin and Girsanov:  Proposition 10.3. Given t ∈ (0, ∞), Ξx∗ W0 B is absolutely continuous with t  respect to Wx B , with Radon-Nikodym derivative t

(10.61)

dΞx∗ W0 = eZ(t) . dWx

  Note that by taking fν 1 in (10.56), we have Ex eZ(t) = 1, so the supermartingale eZ(t) is actually a martingale in this case. To prove the proposition, it suffices to show that, for 0 ≤ t1 < · · · < tk ≤ t, and a sufficiently large class of continuous functions fj ,

(10.62)

     E0 f1 Xx (t1 ) · · · fk Xx (tk )       = Ex f1 ω(t1 ) · · · fk ω(tk ) eZ(t) .

We will get this by extending (10.55) and (10.56) to formulas for the solution operators to time-dependent equations of the form (10.63)

∂u = (Δ + X)u − V (t, x)u, ∂t

u(0) = f.

490 11. Brownian Motion and Potential Theory

Only the coefficient V (t, x) depends on t; X does not. Parallel to (2.16), we can extend (10.55) to     t x u(t) = E0 f Xx (t) e− 0 V (s,X (s)) ds ,

(10.64)

and we can extend (10.56) to (10.65)

    t u(t) = Ex eZ(t) f ω(t) e− 0 V (s,ω(s)) ds .

Now we can pick V (s, x) to be highly peaked, as a function of s, near s = t1 , . . . , tk , in such a way as to get (10.66)

e−

t 0

V (s,ω(s)) ds

≈ e−V1



ω(t1 )



· · · e−Vk



ω(tk )

 .

Thus having the identity of (10.64) and (10.65) for a sufficiently large class of functions V (s, x) can be seen to yield (10.62). We leave the final details to the reader. For further material on the Cameron-Martin-Girsanov formula (10.61), see [Fr], [Kal], [McK], and [Øk].

Exercises 1. As an alternative derivation of (10.13), namely,     E0 f Xx,s (t) Bs = P t−s f (x), via the Markov in light of the identity (10.5), it follows by applying  property, show that   (4.12) to E0 f Xx (t − s, ϑs ω) Bs . 2. Under the hypotheses of Proposition 10.1, show that, for λ > 0,   x 2 2 E0 eλ|X (t)−x| ≤ 2e2λ S t+λBt . (Hint:  If Z(t)  denotes the last integral in (10.1), use (8.23) to estimate the quantity E0 eλ|Z(t)| .) Using this estimate in place of (10.10), get as strong a bound as you can on the behavior of P t f (x), for fixed t ∈ R+ , as |x| → ∞, given f ∈ C0 (Rn ), that is, f continuous with compact support.  ∗  3. Granted the hypotheses under which the identity etL = et(L−V ) on the space  ∗ Co (Rm ) ∩ L1 (Rm ) was established in (10.45), show that if P(t) denotes etL restricted to L1 (Rm ), then P(t) = P(t)∗ : L∞ (Rm ) → L∞ (Rm ) is given by the same formula as (10.6):   P(t)f (x) = E0 f Xx (t) , f ∈ L∞ (Rm ). Show that P (s, x, t, U ) = P(t − s)χU (x).

11. Diffusion on Riemannian manifolds

491

4. Assume A(x) is real-valued, A ∈ C 2 (Rm ), and A(x) ≥ 0 for all x. Show that

2A(x) . |∇A(x)|2 ≤ 4A(x) sup |D2 A(y)| : |x − y| < |∇A(x)|  Use this to show that A(x) is uniformly Lipschitz on Rm , establishing the scalar case of Proposition 10.2. (Hint: Reduce to the case m = 1; show that if A (c) > 0, then A must change by at least A (c)/2 on an interval of length ≤ 2A(c)/A (c), to prevent A from changing sign. Use the mean-value theorem to deduce |A (ζ)| ≥ |A (c)|2 /4A(c) for some ζ in this interval.) For the general case of Proposition 10.2, see [Fdln], p. 189. 5. Suppose (10.1) is the system arising in Exercise 4 of §9, for X = (x, v). Show that the generator L for P t is given by L=

(10.67)

  ∂ ∂ ∂2 − βv + V  (x) +v . ∂v 2 ∂v ∂x

6. Using methods produced in Chap. 8, §6, to derive Mehler’s formula, compute the integral kernel for etL when L is given by (10.67), with V (x) = ax2 . Remark: This integral kernel is smooth for t > 0, reflecting the hypoellipticity of ∂t −L. This is a special case of a general phenomenon analyzed in [Ho]. A discussion of this work can also be found in Chap. 15 of [T3].

11. Diffusion on Riemannian manifolds We extend the construction of Wiener measure done in §1 from the setting of path space on Rn to that of path space on a family of complete Riemannian manifolds. Thus, let M be a complete Riemannian manifold, with Laplace Beltrami operator Δ. We begin with some preliminaries on the heat semigroup. As shown in §2 of Chapter 8, Δ is essentially self adjoint on C0∞ (M ). Let Δ also denote the self adjoint extension. Then the heat semigroup {etΔ : t ≥ 0} is a strongly continuous semigroup on L2 (M ). As seen in Chapter 6, §2, this heat semigroup has the positivity preserving property f ≥ 0 =⇒ etΔ f ≥ 0.

(11.1)

It also follows from the construction there that, for g ∈ C0∞ (M ), 0 ≤ g ≤ 1 =⇒ etΔ g ≤ 1.

(11.2)

Taking such g, we have, for f ∈ C0∞ (M ), 

 f ≥ 0 =⇒

g(x)etΔ f (x) dV (x) =

(etΔ g(x))f (x) dV (x) 

(11.3) ≤

f (x) dV (x),

492 11. Brownian Motion and Potential Theory

and a limiting argument gives 

 etΔ f (x) dV (x) ≤

f ≥ 0 =⇒

(11.4)

M

f (x) dV (x). M

We also have  (11.5)



e

f (x) =

h(t, x, y)f (y) dV (y), M

for t > 0, with h(t, x, y) smooth, satisfying estimates such as (2.30) of Chapter 6, i.e., (11.6)

2  2 0 ≤ h(t, x, y) ≤ Cκ(x, δ)κ(y, δ) 1 + t−k t−1 ρ2 k e−ρ /4t ,

with dist(x, y) = ρ + 2δ, k > n/4. Furthermore, as shown in Proposition 2.4 of Chapter 6, we have, for f ∈ C0∞ (M ), 

 (11.7)

f ≥ 0 =⇒



e

f (x) dV (x) =

M

f (x) dV (x), M

improving (11.4), if M is complete and, for some p ∈ M, β < ∞, (11.8)

2

Vol Bs (p) ≤ Cp eβs ,

∀ s < ∞.

Note that (11.4) implies etΔ extends uniquely to (11.9)

etΔ : L1 (M ) −→ L1 (M ),

etΔ f L1 ≤ f L1 .

Hence, by duality and interpolation, (11.10)

etΔ : Lp (M ) −→ Lp (M ),

1 ≤ p ≤ ∞,

etΔ f Lp ≤ f Lp .

Furthermore, as long as (11.7) holds, (11.11)

etΔ 1(x) ≡ 1,

∀ t > 0.

The following is an important complement to (11.10). Proposition 11.1. Let M be a complete Riemannian manifold. Then, for p ∈ [1, ∞), {etΔ : t ≥ 0} is a strongly continuous semigroup on Lp (M ). Proof. To start, we take f, g ∈ C0∞ (M ) and write

11. Diffusion on Riemannian manifolds

 (11.12)

(etΔ f − f, g) =

493

t

(esΔ Δf, g) ds. 0

In this setting, (esΔ Δf, g) is continuous in s ≥ 0, and we have the pointwise bound (11.13)

|(esΔ Δf, g)| ≤ Δf Lp gLp ,

for 1 ≤ p ≤ ∞.

Hence (11.14)

|(etΔ f − f, g)| ≤ tΔf Lp gLp ,

1 ≤ p ≤ ∞,

so (11.15)

etΔ f − f Lp ≤ tΔf Lp ,

1 ≤ p ≤ ∞,

for all f ∈ C0∞ (M ). If 1 ≤ p < ∞, C0∞ (M ) is dense in Lp (M ), and the uniform operator bound (11.10) yields the asserted strong continuity on Lp (M ). We also get a result valid for p = ∞. To state it, set C∗ (M ) = C(M ) if M is compact, and (11.16)

C∗ (M ) = {f ∈ C(M˙ ) : f (∞) = 0}

if M is not compact, where M˙ is the one point compactification of M . (Set M˙ = M if M is compact.) Proposition 11.2. In the setting of Proposition 11.1, we have for u(t) = etΔ f that (11.17)

f ∈ C∗ (M ) =⇒ u ∈ C([0, ∞), L∞ (M )).

Proof. The estimate (11.15) implies that if f ∈ C0∞ (M ), then u is continuous at t = 0, with values in L∞ (M ). Since C0∞ (M ) is dense in C∗ (M ) in L∞ -norm and etΔ has operator norm ≤ 1, we get (11.18)

f ∈ C∗ (M ) =⇒ etΔ f − f L∞ → 0 as t  0.

To proceed, we take t > 0. We have, for h > 0, f ∈ C∗ (M ), (11.19)

e(t+h)Δ f − etΔ f L∞ = etΔ (ehΔ f − f )L∞

as h  0. If also h < t,

≤ ehΔ f − f L∞ → 0

494 11. Brownian Motion and Potential Theory

(11.20)

etΔ f − e(t−h)Δ f L∞ = e(t−h)Δ (ehΔ f − f )L∞ ≤ ehΔ f − f L∞ → 0.

This proves (11.17). R EMARK . If M is compact, it is elementary that (11.21)

f ∈ C(M ) =⇒ u ∈ C([0, ∞), C(M )).

When M is complete but not compact, desired improvements on Proposition 11.2 take more work. We prove one such result here. For simplicity, we assume M has C ∞ bounded geometry. That is to say, there exists r0 > 0 such that for each p ∈ M the exponential map Expp : Tp M → M gives a diffeomorphism of Br0 (0) ⊂ Tp M onto its image, and the pull back of the metric tensor on M to Br0 (0) forms a C ∞ bounded family of metric tensors, as p varies over M . In such a case, we can take δ = r0 /2 and κ(x, δ) independent of x in (11.6), giving (11.22)

 2 2 0 ≤ h(t, x, y) ≤ C 1 + t−k t−1 ρ2 k e−ρ /4t .

Also we have, not only (11.8), but actually an exponential volume bound, (11.23)

Vol Bs (p) ≤ CeKs ,

∀ s < ∞,

with C and K independent of p ∈ M . Proposition 11.3. Assume M has C ∞ bounded geometry. Then (11.24)

etΔ : C∗ (M ) −→ C∗ (M ), strongly continuous semigroup.

Proof. To get etΔ : C∗ (M ) → C∗ (M ), it suffices to show that (11.25)

etΔ : C0∞ (M ) −→ C∗ (M ).

This is a straightforward consequence of (11.22). The strong continuity in (11.24) then follows from (11.17). Corollary 11.4. In the setting of Proposition 11.3, (11.26)

etΔ : C(M˙ ) −→ C(M˙ ), strongly continuous semigroup.

Proof. Given (11.23), we have etΔ 1 ≡ 1, by (11.11), and (11.26) follows from this plus (11.25). Wiener measure. Parallel to §1, we pick x0 ∈ M and construct Wiener measure

11. Diffusion on Riemannian manifolds

495

Wx0 as a probability measure on the “path space” P=

(11.27)



M˙ ,

t∈Q+

where (as stated below (11.16)) M˙ is the one point compactification of M (M˙ = M is M is compact). Then P is a compact metrizable space. In order to construct Wx0 , we construct a positive linear functional Ex0 : C(P) → R on the space C(P) of real valued continuous functions on P, satisfying Ex0 (1) = 1. One motivation is that, given 0 < t1 < · · · < tk (rational) and given Borel sets Ej ⊂ M , we want to arrange that the probability that a path, starting at x0 at time t = 0, lies in Ej at time tj , for each j ∈ {1, . . . , k} is 

 ···

(11.28) E1

h(tk − tk−1 , xk , xk−1 ) · · · h(t1 , x1 , x0 ) dV (xk ) · · · dV (x1 ).

Ek

To implement this, we define Ex0 on the space C # consisting of continuous functions on P of the form (11.29)

ϕ(ω) = F (ω(t1 ), . . . ω(tk )),

t 1 < · · · < tk ,

k where F is continuous on 1 M˙ and tj ∈ Q+ , to be (11.30)   Ex0 (ϕ) = · · · h(t1 , x1 , x0 )h(t2 − t1 , x2 , x1 ) · · · h(tk − tk−1 , xk , xk−1 ) F (x1 , . . . , xk ) dV (xk ) · · · dV (x1 ). If ϕ(ω) in (11.29) actually depends on ω(tν ) for some proper subset {tν } of {t1 , . . . , tk }, there arises a formula for Ex0 (ϕ) with a different appearance from (11.30). The fact that these two expressions are equal follows from the identity  (11.31)

h(t, x, y)h(s, y, z) dV (y) = h(t + s, x, z).

From this it follows that Ex0 : C # → R is well defined. It is also a positive linear functional, satisfying Ex0 (1) = 1, this last identity thanks to (11.11). Now, by the Stone-Weierstrass theorem, C # is dense in C(P). It follows that, for each x0 ∈ M , Ex0 has a unique continuous linear extension to C(P), preserving these properties. We therefore have: Theorem 11.5. Assume M is a complete Riemannian manifold, with heat semigroup etΔ , and that (11.7) holds (yielding (11.11)). Then, given x0 ∈ M , there is a unique probability measure Wx0 on P such that (11.30) is given by

496 11. Brownian Motion and Potential Theory

 Ex0 (ϕ) =

(11.32)

ϕ(ω) dWx0 (ω), P

k for each ϕ of the form (11.29) with F continuous on 1 M˙ . In such a case, (11.30) and (11.32) then hold for ϕ as in (11.29), for bounded Borel function each k F , and also for each positive Borel function F on 1 M˙ . Stochastic continuity. Let us set (11.33)

Xt : P −→ M˙ ,

Xt (ω) = ω(t),

for t ∈ Q+ .

We want to investigate when Xt and Xs are “close,” for |s − t| small. To do this, we define Mx0 (P, M˙ ) to be the set of equivalence classes of Borel measurable maps from P to M˙ , with (11.34)

ϕ ∼ ψ provided ϕ(ω) = ψ(ω) for Wx0 -a.e. ω.

This is a metric space, with distance function  (11.35)

˜ d(ϕ(ω), ψ(ω)) dWx0 (ω),

Dx0 (ϕ, ψ) = P

where we set ˜ y) = ρ(d(x, y)), d(x,

(11.36)

ρ(τ ) = 1 − e−τ .

If M is not compact, d˜ is discontinuous at (∞, ∞), but this is a Borel function on M˙ × M˙ . Now, if s, t ∈ Q+ and 0 < s < t, then (11.37)  ˜ Dx0 (Xt , Xs ) = d(ω(t), ω(s)) dWx0 (ω) P



=  =

˜ 1 , x2 ) dV (x1 ) dV (x2 ) h(s, x1 , x0 )h(t − s, x2 , x1 )d(x h(s, x1 , x0 ) e(t−s)Δ d˜x1 (x1 ) dV (x1 ),

M

where we set (11.38) Hence

˜ y). d˜x (y) = d(x,

11. Diffusion on Riemannian manifolds

(11.39)

497

Dx0 (Xt , Xs ) ≤ sup e(t−s)Δ d˜x1 (x1 ). x1

This yields the following result. Proposition 11.6. Take M as in Theorem 11.5, and assume s, t ∈ Q+ . Then (11.40)

Dx0 (Xt , Xs ) ≤ ϑ(|s − t|),

where, for t > 0, ϑ(t) = sup etΔ d˜x (x).

(11.41)

x∈M

We combine this with the following elementary result. Lemma 11.7. The space Mx0 (P, M˙ ) with distance function (11.35) is a complete metric space. Proof. If (ϕj ) is Cauchy in Mx0 (P, M˙ ), it has a subsequence (ψj ) such that Dx0 (ψj , ψj+k ) ≤ 4−j .  ˜ j (ω), ψj+1 (ω)) < ∞ for Wx -a.e. ω. Hence (ψj ) converges Then j≥0 d(ψ 0 pointwise a.e., say ψj → ψ. The dominated convergence theorem implies Dx0 (ψj , ψ) → 0. Hence Dx0 (ϕj , ψ) → 0. With this in hand, we have the following. Proposition 11.8. Take M as in Theorem 11.5. Assume in addition that ϑ(t) −→ 0 as t  0,

(11.42)

for ϑ(t) defined as in (11.41). Then the map t → Xt defined above for t ∈ Q+ has a unique continuous extension to (11.43)

R+ −→ Mx0 (P, M˙ ),

t → Xt .

One has (11.40) for all s, t ∈ R+ . Note that if M = Rn , then (11.41) yields (11.44)

ϑ(t) ≤ (4πt)−n/2



Rn

Here is a natural extension.

e−|x|

2

/4t

|x| dx = Ct1/2 .

498 11. Brownian Motion and Potential Theory

Proposition 11.9. Assume the Riemannian manifold M has C ∞ bounded geometry. Then ϑ(t), defined by (11.41), satisfies ϑ(t) ≤ C(M )t1/2 .

(11.45)

Recall that the notion of M having C ∞ bounded geometry was defined right below (11.21). Note that (11.44) is much stronger than what is needed to verify (11.45) in the case M = Rn . More generally we have, for p ∈ [1, ∞), 

(4πt)−n/2

(11.46)

e−|x|

2

/4t

|x|p dx = Ctp/2 .

Rn

This motivates the following. For p ∈ [1, ∞), set (11.47)

Dxp0 (ϕ, ψ)

 =

1/p d(ϕ(ω), ψ(ω))p dWx0 (ω) .

P

˜ y). In place of (11.37) we have, for Note that here we use d(x, y) rather than d(x, s, t ∈ Q+ , s < t,  p p Dx0 (Xt , Xs ) = d(ω(t), ω(s))p dWx0 (ω) P



(11.48) =

h(s, x1 , x0 )e(t−s)Δ dpx1 (x1 ) dV (x1 ),

M

hence  1/p Dxp0 (Xt , Xs ) ≤ sup e(t−s)Δ dpx (x) ,

(11.49)

x

where  (11.50)

dpx (y)

p

= d(x, y) ,

etΔ dpx (x)

h(t, x, y)d(x, y)p dV (y).

= M

This establishes the following counterpart of Proposition 11.6. Proposition 11.10. In the setting of Proposition 11.6, for s, t ∈ Q+ , 1 ≤ p < ∞, (11.51) where, for t > 0,

Dxp0 (Xt , Xs ) ≤ ϑp (|s − t|),

11. Diffusion on Riemannian manifolds

499

 1/p ϑp (t) = sup etΔ dpx (x) .

(11.52)

x∈M

To proceed, we investigate the t-dependence, in various function spaces, of Ft (ω) = f (Xt (ω)),

(11.53)

given f : M˙ → R. As a first estimate of this sort, we assume f is Lipschitz on M and use (11.54)

|Ft (ω) − Fs (ω)| ≤ Lip(f ) d(Xt (ω), Xs (ω)).

Taking the pth power and integrating, we have (11.55)   |Ft (ω) − Fs (ω)|p dWx0 (ω) ≤ Lip(f )p d(Xt (ω), Xs (ω))p dWx0 (ω), P

P

hence (11.56)

Ft − Fs Lp (P,Wx0 ) ≤ Lip(f ) Dxp0 (Xt , Xs ) ≤ Lip(f ) ϑp (|t − s|),

for p ∈ [1, ∞). With this, we can establish the following. Proposition 11.11. Assume M has the property that ϑp (t) → 0 as t  0. Take f ∈ C(M˙ ), and define Ft by (11.53). Then Ft is a continuous function of t ∈ [0, ∞) with values in Lp (P, Wx0 ). Proof. Given ε > 0, write f = f0 + f1 , with f0 ∈ Lip(M ), sup |f1 | < ε. Then (11.57)

Ft − Fs Lp (P,Wx0 ) ≤ Lip(f0 )ϑp (|s − t|) + 2 sup |f1 |.

Hence (11.58)

lim sup Ft − Fs Lp (P,Wx0 ) ≤ 2ε, t→s

∀ ε > 0,

yielding the result. Feynman-Kac formula. We extend some of the results of §2 to the manifold setting. Thus, consider (11.59) with

∂u = Δu − V u, ∂t

u(0) = f,

500 11. Brownian Motion and Potential Theory

f, V ∈ C(M˙ ).

(11.60)

We assume M satisfies the hypotheses of Theorem 11.5 and Proposition 11.10, and that, for all p ∈ [1, ∞), ϑp (t) → 0 as t  0, so Proposition 11.11 holds. As in §2, we use the Trotter product formula to write et(Δ−V ) f = lim

(11.61)

k→∞



e(t/k)Δ e−(t/k)V

k f,

with 

(t/k)Δ −(t/k)V

e

(11.62)



k

e

f (x) =

ϕk (ω) dWx (ω) = Ex (ϕk ), P

where ϕk (ω) = F (Xt (ω))e−Sk (ω) ,

(11.63) and

Sk =

(11.64)

k t F(j/k)t , k j=1

Fτ = V (Xτ ).

Proposition 11.11 applies, to give Fτ continuous in τ , with values in Lp (P, Wx ).

(11.65) It follows that

 Sk −→

(11.66)

t

0

V (Xτ ) dτ in Lp (P, Wx ) norm, ∀ p < ∞.

We also have pointwise bounds: sup |Sk | ≤ tv0 = t sup |V |.

(11.67)

P

M

We hence have (11.68)

ϕk −→ f (Xt )e−

t 0

V (Xτ ) dτ

in Lq (P, Wx ) norm, ∀ q < ∞.

This gives the following conclusion. Proposition 11.12. Assume M satisfies the hypotheses of Theorem 11.5 and Proposition 11.10 and that ϑp (t) → 0 as t  0, for all p ∈ [1, ∞). Assume f and V satisfy (11.60). Then, for each x ∈ M, t > 0,

11. Diffusion on Riemannian manifolds



(11.69)

et(Δ−V ) f (x) = Ex f (Xt )e−

t 0

V (Xτ ) dτ

501

 .

Diffusion with drift. We turn to semigroups arising as solution operators to ∂u = Δu + Xu, ∂t

(11.70)

where Δ is the Laplace Beltrami operator on a complete Riemannian manifold M and X is a real vector field on M . We start with the assumption that X is bounded and continuous, say X ≤ C. In such a case, if g ∈ C0∞ (M ), (11.71)

Xg2L2 ≤ C 2 ∇g2L2 = C 2 (−Δg, g) ≤ C 2 ΔgL2 gL2 .

Since completeness implies C0∞ (M ) is dense in D(Δ), we have (11.71) for all g ∈ D(Δ), hence, for f ∈ L2 (M ), t > 0, (11.72)

XetΔ f 2L2 ≤ C 2 ΔetΔ f L2 etΔ f L2 ≤ C 2 t−1 f 2L2 .

By the result of Exercise 10 in §9 of Appendix A (Outline of Functional Analysis), we have: Proposition 11.13. If M is complete and X is a real vector field that is continuous and bounded, then D(X) ⊂ D(Δ) and Δ + X, with domain D(Δ), generates a strongly continuous semigroup on L2 (M ). To proceed, we augment our hypotheses on X: (11.73)

X is C 1 , with a bound ∇X ≤ L.

Hence X generates a global flow F t by C 1 diffeomorphisms on M , giving rise to a group of operators, etX f (x) = f (F t x).

(11.74) Note that, for g ∈ C0∞ (M ), (11.75) hence

vt (x) = etX g(x) =⇒

d vt (x) = X(x) · ∇vt (x), dt

502 11. Brownian Motion and Potential Theory

d dt





X(x) · ∇vt (x) dV (x)

t

g(F x) dV (x) = M

(11.76)



=−

div X(x) g(F t x) dV (x), M

with div X defined as in Chapter 2, §2. In addition, for a ∈ R, (11.77)

d at e dt



 (a − div X)g(F t x) dV (x),

g(F t x) dV (x) = eat M

M

hence, for g ∈ C0∞ (M ), g ≥ 0, t > 0, 

 (11.78)

g(F x) dV ≤

a − div X ≤ 0 =⇒ e

at

t

M

g(x) dV. M

A limiting argument gives this for all non-negative g ∈ L1 (M ). Taking f ∈ Lp (M ), g = |f |p , we have: Proposition 11.14. Assume X is a bounded vector field on M satisfying (11.73). Pick a ∈ R such that a − div X ≤ 0.

(11.79) Then, for p ∈ [1, ∞),

eat etX f pLp ≤ f pLp ,

(11.80)

∀ t ≥ 0,

and B = X + (a/p)I generates a strongly continuous contraction semigroup on Lp (M ). We can now apply the Trotter product formula, as presented in Appendix A to this chapter, to establish the following. Proposition 11.15. Under the hypotheses of Propositions 11.13–11.14, we have that Δ+X +(a/2)I generates a contraction semigroup on L2 (M ) and, for t ≥ 0, (11.81)

 k et(Δ+X) f = lim e(t/k)Δ e(t/k)X f, in L2 -norm, k→∞

for all f ∈ L2 (M ). Note that, for f ∈ L2 (M ), t ≥ 0,

11. Diffusion on Riemannian manifolds

(11.82)

503

0 ≤ f ≤ 1 ⇒ 0 ≤ e(t/k)Δ f ≤ 1 and 0 ≤ e(t/k)X f ≤ 1, so 0 ≤ et(Δ+X) f ≤ 1.

Let us denote the operator on the right side of (11.81) by Pk (t). We have, for p ∈ [1, ∞), f ∈ L1 (M ) ∩ L∞ (M ) (11.83)

⇒ etΔ f Lp ≤ f Lp , etX f Lp ≤ e−(a/p)t f Lp ⇒ Pk (t)f Lp ≤ e−(a/p)t f Lp ,

hence et(Δ+X) f Lp ≤ e−(a/p)t f Lp ,

(11.84)

for f ∈ L1 (M ) ∩ L∞ (M ). Since L1 (M ) ∩ L∞ (M ) is dense in Lp (M ), (11.84) extends to all f ∈ Lp (M ), for each p ∈ [1, ∞). This leads to the following result, extending Proposition 11.1. Proposition 11.16. Let M be a complete Riemannian manifold and X a bounded vector field satisfying (11.73). Then {et(Δ+X) : t ≥ 0} is a strongly continuous semigroup on Lp (M ), for each p ∈ [1, ∞). Proof. To start, we take f, g ∈ C0∞ (M ) and write  (11.85)

t(Δ+X)

(e

f − f, g) =

t

(es(Δ+X) (Δ + X)f, g) ds. 0

The integrand is continuous in s, and we have the pointwise bound (11.86) |(es(Δ+X) f, g)| ≤ e−(a/p)s (Δ + X)f Lp gLp ,

for 1 ≤ p ≤ ∞.

Hence, for t ∈ [0, 1], (11.87)

|(et(Δ+X) f − f, g)| ≤ Ct(Δ + X)f Lp gLp ,

1 ≤ p ≤ ∞,

so (11.88)

et(Δ+X) f − f Lp ≤ Ct(Δ + X)f Lp ,

1 ≤ p ≤ ∞,

for all f ∈ C0∞ (M ). If 1 ≤ p < ∞, C0∞ (M ) is dense in Lp (M ), and the uniform operator bound (11.84) yields the desired strong continuity on Lp (M ). It is also useful to consider the adjoint semigroup: (11.89)

P (t) = et(Δ+X) =⇒ P (t)∗ = et(Δ−X−V ) ,

V = div X.

504 11. Brownian Motion and Potential Theory

Compare (2.22) of Chapter 2. It follows from Proposition 11.16 and material of §9 in the Outline of Functional Analysis appendix (cf. Exercise 9) that P (t)∗ is a strongly continuous semigroup on Lp (M ) for p ∈ (1, ∞). We complement this: Proposition 11.17. The family {P (t)∗ : t ≥ 0} maps L1 (M ) ∩ L2 (M ) to itself and extends uniquely to a strongly continuous contraction semigroup on L1 (M ). Proof. We have from (11.80) that 

et(−X−V ) f Lp ≤ e−(a/p )t f Lp ,

(11.90)

for p ∈ (1, ∞). In addition,  (11.91)

et(−X−V ) f (x) dV (x) =

M

 f (x) dV (x), M

by an argument parallel to (11.76), which also yields (11.90) for p = 1. Taking p = 2, we have as in Proposition 11.15 that (11.92)

 k et(Δ−X−V ) f = lim e(t/k)Δ e(t/k)(−X−V ) f, k→∞

in L2 -norm, for all f ∈ L2 (M ). Denote the operator on the right side of (11.92) by Pk# (t). We have, for p ∈ [1, ∞), f ∈ L1 (M ) ∩ L∞ (M ) (11.93)



⇒ etΔ f Lp ≤ f Lp , et(−X−V ) f Lp ≤ e−(a/p )t f Lp 

⇒ Pk# (t)f Lp ≤ e−(a/p )t f Lp . Hence (11.92) gives (11.94)



P (t)∗ f Lp ≤ e−(a/p )t f Lp ,

for f ∈ L1 (M ) ∩ L∞ (M ), p ∈ [1, ∞). Since L1 (M ) ∩ L∞ (M ) is dense in Lp (M ), (11.94) extends to all f ∈ Lp (M ), for each p ∈ [1, ∞), and in particular for p = 1. Hence P (t)∗ extends to a contraction semigroup on L1 (M ). Strong continuity follows as in Proposition 11.16. By duality, we have a natural extension of P (t) to (11.95)

et(Δ+X) : L∞ (M ) −→ L∞ (M ),

consistent with (11.82). Arguing as in Proposition 11.2 also leads to the following.

11. Diffusion on Riemannian manifolds

505

Proposition 11.18. In the setting of Proposition 11.16, we have for u(t) = et(Δ+X) f that (11.96)

f ∈ C∗ (M ) =⇒ u ∈ C([0, ∞), L∞ (M )).

We next establish the following extension of (11.7) and (11.11). Proposition 11.19. Assume the complete Riemannian manifold M has the property that (11.7) holds for f ∈ C0∞ (M ) (hence (11.11) holds). Take X as in Proposition 11.16. Then, for t > 0, (11.97)



1

f ∈ L (M ) =⇒





P (t) f dV = M

f dV, M

and et(Δ+X) 1 ≡ 1.

(11.98)

Proof. If f ∈ L1 (M ) ∩ L2 (M ), then, under our hypotheses,  (11.99) M

Pk# (t)f dV =

 f dV, M

for each k. Having Proposition 11.17, we also have (11.100)

Pk# (t)f −→ et(Δ−X−V ) f in L1 -norm,

for all f ∈ L1 (M ). This gives (11.97), and (11.98) follows by duality. Heat kernel estimates on et(Δ+Y ) . Here we establish Gaussian upper bounds on the integral kernel of etL , with (11.101)

L = Δ + Y,

where Y is a first order differential operator, of the form Y f = Xf +V f , X a real vector field, V a real valued function on M (or, as in (11.89), Y = −X − V ). As usual, M is a complete Riemannian manifold. In addition, we make the hypothesis that (11.102)

M has C ∞ bounded geometry,

i.e., (for some δ > 0), in exponential coordinates on Bδ (0) ⊂ Tp M , we have C ∞ bounds on the metric tensor and its inverse, and we also assume

506 11. Brownian Motion and Potential Theory

(11.103)

the coefficients of Y are C ∞ bounded on Bδ (0),

independently of p ∈ M . As for the estimate (11.6), we use wave equation techniques, such as done in [CGT], but in the present setting we do not have self adjointness of L, so further work is required. Regarding the spectral behavior of L, the analysis done above implies L generates a strongly continuous semigroup etL , with D(L) = H 2 (M ). Adding a constant to L, we can arrange that 0 is in the resolvent set of L, so ≈

L−1 : L2 (M ) −→ H 2 (M ).

(11.104)

The solution operator to the wave equation we work with is W (t), defined initially on C0∞ (M ) ⊕ C0∞ (M ) by     f u(t) W (t) = , g v(t)

(11.105) solving (11.106)

    ∂ u v = , ∂t v Lu

u(0) = f, v(0) = g.

Note that (11.106) implies (11.107)

∂t2 u = Lu,

u(0) = f, ∂t u(0) = g,

∂t2 v

v(0) = g, ∂t v(0) = Lf.

= Lv,

The fact that u(t), v(t) ∈ C0∞ (M ) for all t follows by finite propagation speed, plus the hypothesis that M is complete. Under the hypotheses (11.102)–(11.103), energy estimates of the sort used in Chapter 6 imply that W (t) extends to a strongly continuous group of operators (11.108)

W (t) : H 1 (M ) ⊕ L2 (M ) −→ H 1 (M ) ⊕ L2 (M ).

We have an operator norm bound (11.109)

W (t)L(H 1 ⊕L2 ) ≤ CeM |t| .

Now W (t) is a 2 × 2 matrix of operators,  (11.110)

W (t) =

 C1 (t) S01 (t) , S10 (t) C0 (t)

with Cj (t) even in t and Sjk (t) odd in t, so

11. Diffusion on Riemannian manifolds

(11.111)

 1 W (t) + W (−t) = 2



507



C1 (t) C0 (t)

,

with (11.112)

C1 (t) : H 1 (M ) −→ H 1 (M ),

C0 (t) : L2 (M ) −→ L2 (M ).

In both cases Cj (t)f = u(t) satisfies (11.113)

∂t2 u = Lu,

u(0) = f, ∂t u(0) = 0.

We have that C0 (t) is the unique continuous linear extension of C1 (t) from H 1 (M ) to L2 (M ), and drop the notational distinction, obtaining (11.114)

C(t) : H 1 (M ) −→ H 1 (M ),

L2 (M ) −→ L2 (M ),

satisfying (11.115)

C(t)L(H 1 ) , C(t)L(L2 ) ≤ CeM |t| .

Note that (11.116)

√ L = Δ =⇒ C(t) = cos t −Δ.

Note that the identity W (t)W (s) = W (t + s) implies (11.117)

C(t + s) = C(t)C(s) + S01 (t)S10 (s).

Similarly (11.118)

C(t − s) = C(t)C(s) − S01 (t)S10 (s),

so (11.119)

C(t + s) + C(t − s) = 2C(t)C(s).

This identity will be useful below. We now define a functional calculus. Assume (11.120)

ϕ(λ) = ϕ(−λ), and |ϕ(t)| ˆ ≤ Ce−K|t| , K > M.

Then we set (11.121)

√ 1 ϕ( −L) = √ 2π





ϕ(t)C(t) ˆ dt, −∞

508 11. Brownian Motion and Potential Theory

a strongly convergent integral in view of the operator estimates (11.115). Here are some basic properties of such a functional calculus. First, if ϕ satisfies (11.120),

(11.122)

ϕs (λ) = (cos sλ)ϕ(λ)  1 ˆ + s) + ϕ(t ˆ − s) ⇒ ϕˆs (t) = ϕ(t 2  ∞ √   1 1 C(t − s) + C(t + s) dt ⇒ ϕs ( −L) = √ 2 2π −∞  ∞ 1 =√ ϕ(t)C(s)C(t) ˆ dt 2π −∞ √ = C(s)ϕ( −L),

the second-to-last identity by (11.119). If ψ(λ) is even and satisfies a similar estimate, we have  1 ˆ cos λt dt ϕ(λ) (ψϕ)(λ) = ψ(λ)ϕ(λ) = √ ψ(t) 2π  (11.123) 1 ˆ ψ(t)ϕ =√ t (λ) dt, 2π and hence

(11.124)

 ∞ √ √ 1 ˆ √ (ψϕ)( −L) = ψ(t)ϕ t ( −L) dt 2π −∞  ∞ √ 1 ˆ √ = ψ(t)C(t) dt ϕ( −L) 2π −∞ √ √ = ψ( −L)ϕ( −L).

For a related identity, suppose ϕ(λ) satisfies (11.120) and also (11.125)

ˆ ≤ Ce−K|t| . |∂t2 ϕ(t)|

Then (11.126)

ˆ = −∂ 2 ϕ(t) ψ(λ) = λ2 ϕ(λ) ⇒ ψ(t) t ˆ √ √ ⇒ ψ( −L) = −Lϕ( −L),

since C(t)f = u(t) solves (11.113). We prepare to tackle the heat semigroup, by considering, for s > 0, (11.127)

2

Es (λ) = e−sλ ,

11. Diffusion on Riemannian manifolds

509

which satisfies #s (t) = √1 e−t2 /4s , E 2s

(11.128)

and gives rise to the family of operators √ 1 Es ( −L) = √ 4πs

(11.129)





2

e−t

/4s

C(t) dt.

−∞

Noting that ∂s Es (λ) = −λ2 Es (λ),

(11.130) we have from (11.126) that (11.131)

√ √ ∂ Es ( −L) = LEs ( −L). ∂s

Meanwhile, (11.129) directly gives (11.132)

√ Es ( −L)f −→ f in L2 -norm, as s  0,

√ for each f ∈ L2 (M ). Hence Es ( −L) = esL . We have the identity (11.133)

1 esL f = √ 4πs





2

e−t

/4s

C(t)f dt,

−∞

for s > 0, f ∈ L2 (M ). We are prepared for integral kernel estimates. To start, suppose ϕ(λ) satisfies (11.120) and let U1 and U2 be disjoint open sets, satisfying (11.134)

xj ∈ Uj =⇒ d(x1 , x2 ) ≥ ρ.

Consequently, thanks first to finite propagation speed and then to (11.109), √  1 f ∈ L2 (U1 ) ⇒ ϕ( −L)f U = √ 2 2π (11.135)



  dt ϕ(t)C(t)f ˆ U 2

|t|≥ρ

√ ⇒ ϕ( −L)f L2 (U2 ) ≤ C



M |t| |ϕ(t)|e ˆ dt · f L2 .

|t|≥ρ

Here the hypothesis f ∈ L2 (U1 ) means f ∈ L2 (M ) and supp f ⊂ U1 . If also λ2m ϕ(λ) satisfies (11.120), then

510 11. Brownian Motion and Potential Theory

(11.136)



√ f ∈ L2 (U1 ) ⇒ Lm ϕ( −L)f L2 (U2 ) ≤ C

|ϕˆ(2m) (t)|eM |t| dt · f L2 .

|t|≥ρ

The integrals that arise for heat kernel estimates are 



(11.137) ρ

C #s (t)eM t dt = √ E s





2

e−t

/4s M t

e

dt,

ρ

and, more generally, 



(11.138)

#s (t)|eM t dt = Im (M, s, ρ), |∂s2m E

ρ

so, when (11.134) holds, f ∈ L2 (U1 ) ⇒ Lm esL f L2 (U2 ) ≤ CIm (M, s, ρ)f L2 .

(11.139)

To proceed, take x, y ∈ M and say d(x, y) = ρ + 2δ,

(11.140)

with δ as in (11.103). Take U1 = Bδ (y), U2 = Bδ (x). With n = dim M , pick k > n/2, and write δy = Lk fy + gy ,

(11.141)

fy , gy ∈ L2 (Bδ (y)),

with L2 -norms ≤ Cδ . Thus, esL δy = Lk esL fy + esL gy .

(11.142)

By elliptic regularity and Sobolev embedding we have (with the same Cδ )   |esL δy (x)| ≤ Cδ Lk esL δy L2 (Bδ (x)) + esL δy L2 (Bδ (x)) ,

(11.143) followed by (11.144)

Lk esL δy L2 (Bδ (x)) ≤ L2k esL fy L2 (Bδ (x)) + Lk esL gy L2 (Bδ (x)) ≤ Cδ I2k (M, s, ρ) + Cδ Ik (M, s, ρ),

and an analogous estimate on esL δy L2 (Bδ (x)) , yielding (11.145)

|esL δy (x)| ≤ Cδ2

 j≤2k

Ij (M, s, ρ).

11. Diffusion on Riemannian manifolds

511

Having this heat kernel estimate, we can now complement Propositions 11.18– 11.19 with the following extension of Proposition 11.3 and Corollary 11.4. Proposition 11.20. Assume M has C ∞ bounded geometry and that (11.103) holds, with Y = X. Then et(Δ+X) : C∗ (M ) −→ C∗ (M )

(11.146) and

et(Δ+X) : C(M˙ ) −→ C(M˙ )

(11.147)

are strongly continuous semigroups. Proof. Given that (11.95) is a contraction and given (11.96) and (11.98), it suffices to show that et(Δ+X) : C0∞ (M ) −→ C∗ (M ).

(11.148)

This follows directly from (11.145), in the same fashion that (11.25) follows from (11.22). Wiener measure for diffusion with drift. Having Propositions 11.18, 11.19, and 11.20, we are in a position to extend the construction of Wiener measure on P =  tΔ ˙ + t∈Q M done in (11.27)–(11.31) in the setting of the diffusion semigroup e . t(Δ+X) , we use the same formulas (11.28)–(11.30), In the expanded setting of e this time with h(t, x, y) given by  (11.149)

t(Δ+X)

e

f (x) =

h(t, x, y)f (y) dV (y). M

Again (11.30) produces a positive linear functional Ex0 : C # → R, satisfying Ex0 (1) = 1, and we have the following variant of Theorem 11.5. Proposition 11.21. Assume M is a complete Riemannian manifold for which (11.7) holds and that X satisfies the conditions of Proposition 11.19. Then the conclusion of Theorem 11.5 holds, with h(t, x, y) as in (11.149). Having the heat kernel estimate (11.145), we can also extend the results on stochastic continuity given in Propositions 11.6–11.11. Relativistic diffusion. Results we have derived on diffusion so far in this chapter have been decidedly non-relativistic. A Brownian path x(t) as described in §1 is nowhere Lipschitz, so does not even have finite velocity, much less below light

512 11. Brownian Motion and Potential Theory

speed. However, there is a relativistic theory of diffusion. We present a simple case here. To start, we recall from §7 that the Langevin theory of Brownian motion actually has the velocity v(t) = x (t) satisfy a stochastic equation, (11.150)

dv = dω − βv dt,

where β is a friction. In case β = 0, we have v(t) = ω(t). In this model, the velocity is at least finite, but it has no upper bound. A further wrinkle is needed to model relativistic diffusion. To see how it works, we recall from §18 of Chapter 1 how a path x in Minkowski space is parametrized by proper time τ , yielding the 4-velocity (11.151)

u=

dx = γ(1, v), dτ

satisfying u, u = −1,

(11.152) i.e., (11.153)

γ=

1 dt = . dτ 1 − |v|2

Then u is a path in the one-sheeted hyperboloid (11.154)

H = {(t, x) ∈ R4 : −t2 + |x|2 = −1, t > 0}.

Given the metric tensor induced from the Lorentz metric on R4 , H is 3D hyperbolic space, of constant sectional curvature −1. In case of zero friction, we therefore take u(τ ) to be given by the hyperbolic space analogue of the Wiener process, so (11.155)

u(τ ) = Xτ ∈ Lp (P, Wu0 ),

depending continuously on τ , where (11.156)

P=



˙ H,

τ ∈Q+

and Wu0 is the Wiener measure on P constructed in Theorem 11.5, with eτ Δ the heat semigroup generated by the Laplace-Beltrami operator Δ on H, and x0 replaced by u0 . More generally, we can scale the rate of diffusion and replace eτ Δ by eατ Δ for some constant α > 0. As seen in Chapter 8, §5, on 3D hyperbolic

11. Diffusion on Riemannian manifolds

513

space the heat kernel is given by (11.157)

2 r e−r /4τ , sinh r

h(τ, x, y) = (4πτ )−3/2 e−τ

r = d(x, y).

With the random 4-velocity so specified, the actual random path x(τ ) in Minkowski space is given by  (11.158)

x(τ ) = x(0) +

τ

u(σ) dσ, 0

where x(0) is the initial position. We now consider relativistic diffusion with friction. A straightforward analogue of the stochastic differential equation (11.150) will not work, since the 4-velocity u is not running over a linear space. For inspiration, we look to §10, Exercise 5, which shows that the equation (11.150) is associated to a diffusion equation ∂f /∂t = Lf , where, for n = 1, (11.159)

L=

∂ ∂2 − βv . ∂v 2 ∂v

For Brownian motion on Rn , we have (11.160)

L = Δ − βv · ∇,

 where Δ = (∂/∂vj )2 is the Laplace operator on Rn and −βv · ∇ = X is a vector field on Rn generating a radial flow in toward the origin (given β > 0). For relativistic diffusion, we bring in a diffusion process with drift on H, replacing the heat semigroup eτ Δ by a semigroup of the form (11.161)

eτ L ,

L = Δ + X,

where Δ is the Laplace Beltrami operator on H and X is a vector field on X. In analogy with (11.160), we pick a distinguished point u1 ∈ H (perhaps u1 = (1, 0, 0, 0), for which the classical component of the velocity vanishes) and take X to generate a flow toward u1 . A natural analogue of the vector field in (11.160) is (11.162)

X = −β grad V,

V (u) = v(d(u1 , u)),

for example with (11.163)

r2 v(r) = √ . 1 + r2

514 11. Brownian Motion and Potential Theory

In such a case, {eτ L : τ ≥ 0} is a strongly continuous, positivity preserving ˙ and satisfying eτ L 1 ≡ semigroup on Lp (H), for p ∈ [1, ∞), also acting on C(H), 1. More closely parallel to (11.160) would be v(r) = r2 , which goes beyond the results developed here on diffusion with drift. For a discussion of that case, as it relates to relativistic diffusion, see [Hab]. Relativistic diffusion has also been studied on non-flat spacetimes. See in particular [FJ] for a treatment on Schwarzschild spacetimes.

A. The Trotter product formula It is often of use to analyze the solution operator to an evolution equation of the form ∂u = Au + Bu ∂t in terms of the solution operators etA and etB , which individually might have fairly simple behavior. The case where A is the Laplace operator and B is multiplication by a function is used in §2 to establish the Feynman–Kac formula, as a consequence of Proposition A.4 below. The following result, known as the Trotter product formula, was established in [Tro]. Theorem A.1. Let A and B generate contraction semigroups etA and etB , on a Banach space X. If A + B is the generator of a contraction semigroup R(t), then (A.1)

 n R(t)f = lim e(t/n)A e(t/n)B f, n→∞

for all f ∈ X. Here, A + B denotes the closure of A+B. A simplified proof in the case where A + B itself is the generator of R(t) is given in an appendix to [Nel2]. We will give that proof. Proposition A.2. Assume that A, B, and A+B generate contraction semigroups P (t), Q(t), and R(t) on X, respectively, where D(A+B) = D(A)∩D(B). Then (A.1) holds for all f ∈ X. Proof. It suffices to prove (A.1) for f ∈ D = D(A + B). In such a case, we have (A.2)

P (h)Q(h)f − f = h(A + B)f + o(h),

since P (h)Q(h)f − f = (P (h)f − f ) + P (h)(Q(h)f − f ). Also, R(h)f − f = h(A + B) + o(h), so

A. The Trotter product formula

515

P (h)Q(h)f − R(h)f = o(h) in X, for f ∈ D. Since A + B is a closed operator, D is a Banach space in the norm f D = (A + B)f  + f . For each f ∈ D, h−1 P (h)Q(h) − R(h) f is a bounded set in X. By the uniform boundedness principle, there is a constant C such that  1 P (h)Q(h)f − R(h)f  ≤ Cf D , h    for all h > 0 and f ∈ D. In other words, h−1 P (h)Q(h) − R(h) : h > 0 is bounded in L(D, X), and the family tends strongly to 0 as h → 0. Consequently,  1 P (h)Q(h)f − R(h)f  −→ 0 h uniformly for f is a compact subset of D. Now, with t ≥ 0 fixed, for any f ∈ D, {R(s)f : 0 ≤ s ≤ t} is a compact subset of D, so (A.3)

    P (h)Q(h) − R(h) R(s)f  = o(h),

 n uniformly for 0 ≤ s ≤ t. Set h = t/n. We need to show that P (h)Q(h) f − R(hn)f → 0, as n → ∞. Indeed, adding and subtracting terms of the form (P (h)Q(h))j R(hn − hj), and using P (h)Q(h) ≤ 1, we have

(A.4)

    P (h)Q(h) n f − R(hn)f     ≤  P (h)Q(h) − R(h) R(h(n − 1))f     +  P (h)Q(h) − R(h) R(h(n − 2))f     + · · · +  P (h)Q(h) − R(h) f .

This is a sum of n terms that are uniformly o(t/n), by (A.3), so the proof is done. Note that the proof of Proposition A.2 used the contractivity of P (t) and of Q(t), but not that of R(t). On the other hand, the contractivity of R(t) follows from (A.1). Furthermore, the hypothesis that P (t) and Q(t) are contraction semigroups can be generalized to P (t) ≤ eat , Q(t) ≤ ebt . If C = A + B generates a semigroup R(t), we conclude that R(t) ≤ e(a+b)t . We also note that only certain properties of S(h) = P (h)Q(h) play a role in the proof of Proposition A.2. We use (A.5)

S(h)f − f = hCf + o(h),

f ∈ D = D(C),

where C is the generator of the semigroup R(h), to get (A.6)

S(h)f − R(h)f = o(h),

f ∈ D.

516 11. Brownian Motion and Potential Theory

As above, we have h−1 S(h)f − R(h)f  ≤ Cf D in this case, and consequently h−1 S(h)f − R(h)f  → 0 uniformly for f in a compact subset of D, such as {R(s)f : 0 ≤ s ≤ t}. Thus we have analogues of (A.3) and (A.4), with P (h)Q(h) everywhere replaced by S(h), proving the following. Proposition A.3. Let S(t) be a strongly continuous, operator-valued function of t ∈ [0, ∞), such that the strong derivative S (0)f = Cf exists, for f ∈ D = D(C), where C generates a semigroup on a Banach space X. Assume S(t) ≤ 1 or, more generally, S(t) ≤ ect . Then, for all f ∈ X, (A.7)

etC f = lim S(n−1 t)n f. n→∞

This result was established in [Chf], in the more general case where S (0) has closure C, generating a semigroup. Proposition A.2 applies to the following important family of examples. Let X = Lp (Rn ), 1 ≤ p < ∞, or let X = Co (Rn ), the space of continuous functions vanishing at infinity. Let A = Δ, the Laplace operator, and B = −MV , that is, Bf (x) = −V (x)f (x). If V is bounded and continuous on Rn , then B is bounded on X, so Δ − V , with domain D(Δ), generates a semigroup, as shown in Proposition 9.12 of Appendix A. Thus Proposition A.2 applies, and we have the following: Proposition A.4. If X = Lp (Rn ), 1 ≤ p < ∞, or X = Co (Rn ), and if V is bounded and continuous on Rn , then, for all f ∈ X, (A.8)

 n et(Δ−V ) f = lim e(t/n)Δ e−(t/n)V f. n→∞

This is the result used in §2. If X = Lp (Rn ), p < ∞, we can in fact take V ∈ L∞ (Rn ). See the exercises for other extensions of this proposition. It will be useful to extend Proposition A.2 to solution operators for timedependent evolution equations: (A.9)

∂u = Au + B(t)u, ∂t

u(0) = f.

We will restrict attention to the special case that A generates a contraction semigroup and B(t) is a continuous family of bounded operators on a Banach space X. The solution operator S(t, s) to (A.9), satisfying S(t, s)u(s) = u(t), can be constructed via the integral equation  (A.10)

tA

u(t) = e f +

t

e(t−s)A B(s)u(s) ds,

0

parallel to the proof of Proposition 9.12 in Appendix A on functional analysis. We have the following result.

A. The Trotter product formula

517

Proposition A.5. If A generates a contraction semigroup and B(t) is a continuous family of bounded operators on X, then the solution operator to (A.9) satisfies  (A.11) S(t, 0)f = lim

t→∞

   e(t/n)A e(t/n)B((n−1)t/n) · · · e(t/n)A e(t/n)B(0) f,

for each f ∈ X. There are n factors in parentheses on the right side of (A.11), the jth from the right being e(t/n)A e(t/n)B((j−1)t/n) . The proof has two parts. First, in close parallel to the derivation of (A.4), we have, for any f ∈ D(A), that the difference between the right side of (A.11) and (A.12)

e(t/n)(A+B((n−1)t/n)) · · · e(t/n)(A+B(0)) f

has norm ≤ n · o(1/n), tending to zero as n → ∞, for t in any bounded interval [0, T ]. Second, we must compare (A.12) with S(t, 0)f . Now, for any fixed t > 0, define v(s) on 0 ≤ s ≤ t by (A.13)

j − 1  ∂v = Av + B t v, ∂s n

j−1 j t ≤ s < t; n n

v(0) = f.

Thus (A.12) is equal to v(t). Now we can write (A.14)

∂v = Av + B(s)v + R(s)v, ∂s

v(0) = f,

where, for n large enough, R(s) ≤ ε, for 0 ≤ s ≤ t. Thus  (A.15)

v(t) = S(t, 0)f +

t

S(t, s)R(s)v(s) ds, 0

and the last term in (A.15) is small. This establishes (A.11). Thus we have the following extension of Proposition A.4. Denote by BC(Rn ) the space of bounded, continuous functions on Rn , with the sup norm. n n Proposition A.6. If X = Lp (R  ), 1 ≤ p < ∞, or X = Co (R ), and if V (t) belongs to C [0, ∞), BC(Rn ) , then the solution operator S(t, 0) to

∂u = Δu − V (t)u ∂t satisfies (A.16)     S(t, 0)f = lim e(t/n)Δ e−(t/n)V ((n−1)t/n) · · · e(t/n)Δ e−(t/n)V (0) f, n→∞

518 11. Brownian Motion and Potential Theory

for all f ∈ X. To end this appendix, we give an alternative proof of the Trotter product formula when Au = Δu and Bu(x) = V (x)u(x), which, while valid for a more restricted class of functions V (x) than the proof of Proposition A.4, has some k  desirable features. Here, we define vk = e(1/n)Δ e−(1/n)V f and set v(t) = esΔ e−sV vk , for t =

(A.17)

1 k + s, 0 ≤ s ≤ . n n

We use Duhamel’s principle to compare v(t) with u(t) = et(Δ−V ) f . Note that v(t) → vk+1 as t (k + 1)/n, and for k/n < t < (k + 1)/n, ∂v = Δv − esΔ V e−sV vk ∂t = (Δ − V )v + [V, esΔ ]e−sV vk .

(A.18)

Thus, by Duhamel’s principle, t(Δ−V )

v(t) = e

(A.19)

 f+

t

e(t−s)(Δ−V ) R(s) ds,

0

where R(s) = [V, eσΔ ]e−σV vk , for s =

(A.20)

1 k + σ, 0 ≤ σ < . n n

We can write [V, eσΔ ] = [V, eσΔ − 1], and hence (A.21)

R(s) = V (eσΔ − 1)e−σV vk − (eσΔ − 1)V e−σV vk .

Now, as long as (A.22)

D(Δ − V ) = D(Δ) = H 2 (Rn ),

we have, for 0 ≤ γ ≤ 1, (A.23)

 t(Δ−V )   t(Δ−V )  e   2 2γ ≤ C(T )t−γ , e = −2γ 2 L(H ,L ) L(L ,H )

for 0 < t ≤ T . Thus, if we take γ ∈ (0, 1) and t ∈ (0, T ], we have for  (A.24)

F (t) = 0

the estimate

t

e(t−s)(Δ−V ) R(s) ds,

Exercises

 F (t)L2 ≤ C

(A.25)

0

t

519

(t − s)−γ R(s)H −2γ ds.

We can estimate R(s)H −2γ using (A.21), together with the estimate   σΔ e − 1 2 −2γ ≤ C σ γ , L(L ,H )

(A.26)

0 ≤ γ ≤ 1.

Since σ ∈ [0, 1/n] in (A.21), we have R(s)H −2γ ≤ Cn−γ ϕ(V )f L2 ,   ϕ(V ) = V L(H 2γ ) + V L∞ esV L∞ .

(A.27)

Thus, estimating v(t) = u(t) at t = 1, we have      (1/n)Δ −(1/n)V n e f − e(Δ−V ) f   e

(A.28)

L2

≤ Cγ ϕ(V )f L2 · n−γ ,

for 0 < γ < 1, provided multiplication by V is a bounded operator on H 2γ (Rn ). Note that this holds if Dα V ∈ L∞ (Rn ) for |α| ≤ 2, and V L(H 2γ ) ≤ C sup Dα V L∞ .

(A.29)

|α|≤2

One can similarly establish the estimate (A.30)

    (t/n)Δ −(t/n)V n  e f − et(Δ−V ) f   e

L2

≤ C(t)ϕ(V )f L2 · n−γ .

Exercises 1. Looking at Exercises 2–4 of §2, Chap. 8, extend Proposition A.4 to any V , continuous on Rn , such that Re V (x) is bounded from below and |Im V (x)| is bounded. (Hint: First apply those exercises directly to the case where V is smooth, real-valued, and bounded from below.) 2. Let H = L2 (R), Af = df /dx, Bf = ixf (x), so etA f (x) = f (x + t), etB f (x) = eitx f (x). Show that Theorem A.1 applies to this case, but not Proposition A.2. Compute both sides of  n epA+qB f = lim e(p/n)A e(q/n)B f, n→∞

and verify this identity directly. Compare with the discussion of the Heisenberg group, in §14 of Chap. 7. 3. Suppose A and B are bounded operators. Show that  t(A+B)  (t/n)A (t/n)B n   ≤ Ct e − e e n and that

520 11. Brownian Motion and Potential Theory  t(A+B)  (t/2n)A (t/n)B (t/2n)A n  e  ≤ ct . − e e e n2 (t/n)A , and so forth.) (Hint: Use the power series expansions for e

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