Mathematical Surveys and Monographs Volume 158 Mathematical Scattering Theory Analytic Theory D. R. Yafaev American Mathematical Society surv-158-yafaev-cov.indd 1 2/4/10 3:47 PM http://dx.doi.org/10.1090/surv/158 Mathematical Scattering Theory Analytic Theory Mathematical Surveys and Monographs Volume 158 Mathematical Scattering Theory Analytic Theory D. R. Yafaev American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen, Chair J. T. Stafford Benjamin Sudakov 2000 Mathematics Subject Classification. Primary 34L25, 35-02, 35P10, 35P25, 47A40, 81U05. For additional information and updates on this book, visit www.ams.org/bookpages/surv-158 Library of Congress Cataloging-in-Publication Data ´IA`faev,D.R.(Dmitri˘ıRauelevich), 1948– Mathematical scattering theory : analytic theory / D.R. Yafaev. p. cm. – (Mathematical surveys and monographs ; v. 158) Includes bibliographical references and index. ISBN 978-0-8218-0331-8 (alk. paper) 1. Scattering (Mathematics) I. Title. QA329.I24 2009 515.724–dc22 2009027382 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 To the memory of my parents Contents Preface xi Basic Notation 1 Introduction 5 Chapter 0. Basic Concepts 17 1. Classification of the spectrum 17 2. Classes of compact operators 20 3. The resolvent equation. Conditions for self-adjointness 23 4. Wave operators (WO) 26 5. The smooth method 29 6. The stationary scheme 33 7. The scattering operator and the scattering matrix (SM) 38 8. The trace class method 42 9. The spectral shift function (SSF) and the perturbation determinant (PD) 45 10. Differential operators 52 11. Function spaces and embedding theorems 56 12. Pseudodifferential operators 58 13. Miscellaneous analytic facts 67 Chapter 1. Smooth Theory. The Schr¨odinger Operator 71 1. Trace theorems 71 2. The free Hamiltonian 75 3. The Schr¨odinger operator 79 4. Existence of wave operators 82 5. Wave operators for long-range potentials 86 6. Completeness of wave operators 93 7. The limiting absorption principle (LAP) 95 8. The scattering matrix 96 9. Absence of the singular continuous spectrum 98 10. General differential operators of second order 101 11. The perturbed polyharmonic operator 103 12. The Pauli and Dirac operators 104 Chapter 2. Smooth Theory. General Differential Operators 109 1. Spectral analysis of differential operators with constant coefficients 109 2. Scalar differential operators 116 3. Nonelliptic differential operators 118 4. Matrix differential operators 122 vii viii CONTENTS 5. Scattering problems for perturbations of a medium 124 6. Strongly propagative systems. Maxwell’s equations 128 Chapter 3. Scattering for Perturbations of Trace Class Type 133 1. Conditions on an integral operator to be trace class 133 2. Perturbations of differential operators with constant coefficients 136 3. The Schr¨odinger operator 139 4. The perturbed polyharmonic operator 145 5. General differential operators of second order 147 6. Scattering problems for perturbations of a medium 154 7. Wave equation 157 8. The scattering matrix and the spectral shift function 159 Chapter 4. Scattering on the Half-line 161 1. Jost solutions. Volterra equations 161 2. Generalized Fourier transform and WO 170 3. Low-energy asymptotics 178 4. High-energy asymptotics 188 5. The SSF for the radial Schr¨odinger operator 191 6. Trace identities 198 7. Perturbation by a boundary condition. Point interaction 203 Chapter 5. One-Dimensional Scattering 209 1. A direct approach 209 2. Low- and high-energy asymptotics 216 3. The SSF and trace identities 221 4. Potentials with different limits at “ + ” and “ − ” infinities 223 Chapter 6. The Limiting Absorption Principle (LAP), the Radiation Conditions and the Expansion Theorem 231 1. Absence of positive eigenvalues and radiation conditions 231 2. Boundary values of the resolvent 233 3. A sharp form of the limiting absorption principle 235 4. Nonhomogeneous Schr¨odinger equation 239 5. Homogeneous Schr¨odinger equation 241 6. Expansion theorem 245 7. The wave function. The scattering amplitude 251 8. A generalized Fourier integral 255 9. The Mourre method 259 Chapter 7. High- and Low-Energy Asymptotics 267 1. High-energy and uniform resolvent estimates 267 2. Asymptotic expansion of the Green function for large values of the spectral parameter 275 3. Small time asymptotics of the heat kernel 280 4. Low-energy behavior of the resolvent 285 5. Low-energy behavior of the resolvent. Slowly decreasing potentials 291 Chapter 8. The Scattering Matrix (SM) and the Scattering Cross Section 297 1. Basic properties of the SM 297 CONTENTS ix 2. The spectrum of the SM. The modified SM 302 3. The scattering cross section 306 4. High-energy asymptotics of the SM. The ray expansion 311 5. The eikonal approximation 319 6. The averaged scattering cross section. Singular potentials 328 7. The semiclassical limit 337 Chapter 9. The Spectral Shift Function and Trace Formulas 341 1. The regularized PD and SSF for the multidimensional Schr¨odinger operator 341 2. High-energy asymptotics of the SSF 353 3. Trace identities for the multidimensional Schr¨odinger operator 365 Chapter 10. The Schr¨odinger Operator with a Long-Range Potential 369 1. Propagation estimates 369 2. Long-range scattering 374 3. The eikonal and transport equations 380 4. Scattering matrix for long-range potentials 384 Chapter 11. The LAP and Radiation Estimates Revisited 399 1. The efficient form of the LAP 399 2. Absence of positive eigenvalues and uniqueness theorem 403 3. Nonhomogeneous Schr¨odinger equation with a long-range potential 408 Review of the Literature 415 Bibliography 429 Index 441 Preface This book can be considered as the second volume of the author’s monograph “Mathematical Scattering Theory (General Theory)” [I]. It is oriented to applica- tions to differential operators, primarily to the Schr¨odinger operator. A necessary background from [I] is collected (but the proofs are of course not repeated) in Chapter 0. Therefore it is presumably possible to read this book independently of [I]. Everything said in the preface to [I] pertains also to this book. In particular, we proceed again from the stationary approach. Its main advantage is that, si- multaneously with proofs of various facts, the stationary approach gives formula representations for the basic objects of the theory. Along with wave operators, we also consider properties of the scattering matrix, the spectral shift function, the scattering cross section, etc. A consistent use of the stationary approach as well as the choice of concrete material distinguishes this book from others such as the third volume of the course of M. Reed and B. Simon [43]. The latter course has become a desktop copy for many, in particular, for the author of the present book. However, in view of the broad compass of material, the course [43] was necessarily written in encyclopedic style and apparently cannot replace a systematic exposition of the theory. Hopefully, vol. 3 of [43] and this book can be considered as complementary to one another. There are two different trends in scattering theory for differential operators. The first one relies on the abstract scattering theory. The second one is almost independent of it. In this approach the abstract theory is replaced by a concrete investigation of the corresponding differential equation. In this book we present both of these trends. The first of them illustrates basic theorems of [I]. Thus, Chapters 1 and 2 are devoted to applications of the smooth method. Of course the abstract results of [I] should be supplemented by some analytic tools, such as the Sobolev trace theorem. The smooth method works well for perturbations of differential operators with constant coefficients. In Chapter 3 applications of the trace class method are discussed. The main advantage of this method is that it does not require an explicit spectral analysis of an “unperturbed” operator. Other chapters are much less dependent on [I]. Chapters 4 and 5 are devoted to the one-dimensional problem (on the half-axis and the entire axis, respectively) which is a touchstone for the multidimensional case because specific methods of ordinary differential equations can be used here. In the following chapters we return to the multidimensional problem and discuss different analytic methods appropriate to differential operators. In particular, in Chapter 6 scattering theory is formulated in terms of solutions of the Schr¨odinger equation satisfying some “boundary conditions” (radiation conditions) at infinity. xi xii PREFACE High- and low-energy asymptotics of the Green function (the resolvent kernel) and of related objects are discussed in Chapter 7. Chapter 8 is devoted to a study of the scattering matrix and of the scattering cross section. Here some asymptotic methods, such as the ray expansion and eikonal expansion, are also discussed. As an example of a useful interaction of abstract and analytic methods, we mention the theory of the spectral shift function.