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Mathematical Surveys and Monographs Volume 158

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.

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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 Schrödinger Operator 71 1. Trace theorems 71 2. The free Hamiltonian 75 3. The Schrödinger 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 Schrödinger 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 Schrödinger operator 341 2. High-energy asymptotics of the SSF 353 3. Trace identities for the multidimensional Schrödinger operator 365 Chapter 10. The Schrödinger 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 Schrödinger 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 applications to differential operators, primarily to the Schrödinger 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 Schrödinger 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. Abstract results are illustrated in §3.8. However, specific properties of this function are studied by concrete methods in §4.5, §5.3 and in Chapter 9. Here perturbation determinants are also discussed and trace identities are derived. Note that Chapters 1 and 3 and large parts of Chapters 4 and 5 contain essen- tially a “necessary minimum” on scattering theory, whereas the other chapters are of a slightly more special nature. The book is mainly devoted to a study of perturbations by differential operators with short-range coefficients. Nevertheless, basic results on long-range scattering, in particular, properties of the scattering matrix, can be found in Chapter 10. We mention that the recent progress in scattering theory is to a large extent related to multiparticle systems. This very interesting and difficult problem is discussed in [16]and[61]. Similarly to [I], in working on the book the author has tried to resolve two opposite problems. The first of them is a systematic exposition of the material starting from the general background of [I]. The second problem is the exposition of a number of topics to a degree of completeness which might possibly be of interest to experts in spectral theory. We have also tried to fill in numerous gaps present in monographic literature. This pertains especially to the exposition of works of Russian and, in particular, Saint Petersburg mathematicians. Compared to [I], the author’s tastes are also more thoroughly represented here. As a whole the book is oriented toward a reader (for example, a graduate student in mathematical physics) interested in a deeper study of scattering theory. In references we use the “three-stage” enumeration of formulas and theorems and the “two-stage” enumeration of sections. However, the first number is omitted within a chapter. This book is based on the graduate courses taught by the author several times in Saint-Petersburg and Rennes Universities. The concept and structure of the entire book, as well as many specific ques- tions, were discussed with the author’s teacher M. Sh. Birman. To a large extent, mathematical tastes of the author were influenced by L. D. Faddeev. The author is deeply grateful to M. Sh. Birman and L. D. Faddeev. Numerous discussions with P. Deift, A. B. Pushnitski, G. Raikov and M. Z. Solomyak are also gratefully acknowledged. PREFACE xiii

Interdependence of chapters

Chapter 0 H H H ) ? HH X Hj - Chapter 4 XXX Chapter 3 Chapter 1 Chapter 2 XXX XXX ? XXXz ? Chapter 5 - Chapter 6 - Chapter 10 PiPP PP ? PPq Chapter 7 Chapter 11

? ? Chapter 9 Chapter 8

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Absence of positive eigenvalues, 231, 399 a generalized Fourier integral, 255 Absolutely continuous subspace, 19 relation to the wave operators, 248 Admissible functions, 28 standing waves, 250 Agmon-H¨ormander spaces, 235 the diagonalization of the Hamiltonian, Analytic Fredholm alternative, 33 245 Asymptotic completeness, see also Wave operators Free Hamiltonian, 75 Averaged scattering cross section, 331 resolvent, 78 high-energy asymptotics, 337 spectral representation, 75 semiclassical asymptotics, 336 unitary group, 76 universal upper bounds, 331 Friedrichs’ extension, 25

Birman-Kato-Kre˘ın theorem, 42 Generalized Fourier transform, 170 Birman-Kre˘ın formula, 47 in the one-dimensional problem, 214 Boundary values of the resolvent of a self- relation with wave operators, 215 adjoint operator, 44 on half-line, 170 Boundedness of integral operators, 68 relation with wave operators, 175 Break-down of completeness, 91 Hamiltonian of a relativistic spinless parti- additional channels of scattering, 91 cle, 104 Cauchy integral, 358 Hardy-Rellich inequalities, 68 Compact operators, 20 High-energy asymptotic expansion of the re- singular numbers, 20 solvent, 277 Completeness of wave operators, see also away from the spectrum, 277 Asymptotic completeness in the whole complex plane, 361 Complex interpolation, 22 High-energy asymptotics of scattering data, Hadamard three-line theorem, 22 188 three-line theorem for operator-valued in the one-dimensional problem, 219 on half-line, 188 functions in Sp,22 Conditions of self-adjointness, 24 High-energy asymptotics of the spectral Cook’s criterion, 29 shift function, see also spectral shift function Determinant, 21 High-energy estimates of the resolvent, 268 regularized determinant, 21 the free case, 268 Diagonalization of a self-adjoint operator, 20 Hilbert identity, 17 Differential operators, 53, 136 Homogeneous Schrödinger equation, 241 with constant coefficients, 53, 136 an exhaustive description of all solutions, perturbations, 54, 136 241 spectral analysis, 54, 109 Dirac operator, 106 Integral operators, 133 Direct integral, 19 from Schatten-von Neumann classes, 133 from trace class, 133 Eikonal approximation, 319 Intertwining property, 173 Exceptional set N ,33 Invariance principle, 29 Expansion theorem, 248 for perturbations of trace class type, 43

441 442 INDEX

Jost function, 165, 210 wave operators, 145 in the one-dimensional problem, 210 completeness, 145 regularity properties, 165 existence, 145 Jost solution, 165 Phase shift, 166 regularity properties, 163 Point interaction, 203 with the vacuum, 207 Kato-Rosenblum theorem, 42 Propagation of classical waves, 129 Propagation or microlocal estimates, 372 Laplace transform, 18 Propagative systems, 109 Large time local decay of solutions of the of first order, 129 Schrödinger equation, 290 strongly, 109 Levinson’s formulas, 198, 222, 348 uniformly, 109 Limit amplitude, 166 Pseudodifferential operators, 58 Limit phase, 166 action on the exponential function, 61 Limiting absorption principle, 236, 400 boundedness and compactness, 58 efficient form, 400 elementary calculus, 59 commutator method, 400 of negative order, 65 in Schatten-von Neumann classes, 271 asymptotics of eigenvalues, 66 the sharp form, 238 on manifolds, 63 the free case, 236 oscillating symbols, 62 Lippmann-Schwinger equation, 252 essential spectrum, 62 Local asymptotic expansion of the parabolic principal symbol, 61 Green function, 280 strongly Carleman, 54 Laplace transform, 283 symbols and amplitudes, 60 Local theory of trace class perturbations, 43 with constant coefficients, 53 Long-range potentials, 380 perturbations, 54 diagonal singularity of the scattering am- spectral analysis, 54, 109 plitude, 395 with homogeneous symbols, 115 eikonal and transport equations, 380 scattering matrix, 394 Quadratic forms, 25 spectrum, 394 Long-range scattering, 86 Radiation conditions, 233, 399 existence of modified wave operators, 86 Radiation estimates, 373 Low-energy asymptotics, 178 Ray expansion, 312 in the one-dimensional problem, 216 Reflection coefficients, 166 on half-line Regular points, 17 for slowly decaying potentials, 186 Regular solution, 165 on the half-line, 178 Regularized perturbation determinant, 341 Low-energy behavior of the resolvent, 285 Resolvent, 17 Cauchy-Stieltjes integral, 17 Maxwell’s equations, 129 on half-line, 170 Mourre estimate, 261 Resolvent identity, 23 Mourre method, 259 modified, 275 limiting absorption principle, 259 propagation or microlocal estimates, 372 Scattering amplitude, 253 for long-range potentials, 385 Nonhomogeneous Schrödinger equation, 408 high-energy and smoothness asymp- existence and uniqueness of solutions, 239 totics, 385 with a long-range potential, 408 for potentials of compact support, 314 existence and unicity of solutions, 408 high-energy asymptotics, 314 for potentials with power-like decay, 318 Pauli operator, 104 high-energy and smoothness asymp- Pearson theorem, 43 totics, 318 Perturbation determinant, 45 Scattering cross section, 307 for the radial Schrödinger operator, 192 universal upper bounds, 310 in the one-dimensional problem, 221 Scattering matrix, 38, 159, 297 modified, 51 asymptotics of scattering phases, 302 regularized, 52 Born series, 301 Perturbed polyharmonic operator, 103, 145 eigenvalues, 39 INDEX 443

for differential operators, 159 continuity, 351 for long-range potentials, 385 for a trace class perturbation, 46 for perturbations of a definite sign, 301 for differential operators, 160 in the one-dimensional problem, 213 for general perturbations of trace class modified, 304 type, 47 on the half-line, 176 for perturbations of definite sign, 51 spectral properties, 297 for semibounded operators, 50 stationary representation, 38, 297 for the radial Schrödinger operator, 194 with identifications, 40 high-energy expansion, 356 Scattering operator, 38 asymptotic coefficients, 362, 364 for long-range potentials, 385 in the one-dimensional problem, 222 Scattering solutions, wave functions or Spectrum, 17 eigenfunctions of the continuous spec- absolutely continuous, 19 trum, 251 essential, 24 Schatten-von Neumann classes, 20 singular continuous, 19 Hilbert-Schmidt class, 21 conditions for its absence, 36 trace class, 21 Spherical waves, 242 Schrödinger operator, 79, 139 outgoing and incoming, 242 absence of the singular continuous spec- Strong smoothness, 31 trum, 98 Subordination of operators, 43 complex conjugation, 81 essential spectrum, 81 Time reversal invariance, 97 limiting absorption principle, 95 Time-delay, 346 magnetic, 101 Total scattering cross section, 255 perturbations of second order, 102 Trace, 21 scattering matrix, 96 Trace formula, 45 modified, 97 Trace identities, 199 spectrum, 97 for the radial Schrödinger operator, 199 stationary representations, 97 in the multidimensional problem, 365 self-adjointness, 80 in the one-dimensional problem, 222 wave operators, 82, 139 Transmission coefficients, 213 completeness, 93, 139 Uniform estimates of the spectral family, 270 completeness for anisotropic potentials, Uniqueness theorem under radiation condi- 94 tions, 233 existence, 82, 139 existence for anisotropic potentials, 84 Volterra equations, 162 zero-energy resonance, 287 the resolvent singularity, 288 Wave equation in inhomogeneous media, 157 Slowly decreasing positive potentials, 291 scattering theory, 157 a virtual shift of the continuous spectrum, Wave function, 166 291 asymptotic behavior, 253 quasiregularity of the spectral point zero, in the one-dimensional problem, 213 293 pointwise asymptotics, 257 superpower local decay of solutions of the Wave operators, 27 time-dependent Schrödinger equation, completeness, 27 295 for long-range potentials, 379 Smoothness in the Kato sense, 29 for magnetic potentials, 152 local, 30 for matrix differential operators, 122 sufficient conditions of a commutator for nonelliptic differential operators, 118 type, 32 for perturbations of a medium, 124, 154 Sobolev spaces, 56 for scalar differential operators, 116 embedding theorems, 57 for second order differential operators, 147 invariance with respect to diffeomor- for singular potentials, 154 phisms, 57 for strongly propagative systems, 128 trace theorem, 73 for the Maxwell operators, 129 Hölder continuity of traces, 74 in different spaces, 27 Spectral measure, 17 intertwining property, 27 Spectral shift function, 45, 160 local, 28 444 INDEX

multiplication theorem, 28 perturbations of boundary conditions, 154 stationary representations, 36 with identiﬁcations, 41 weak, 27 Abelian, 27 Weyl’s theorem on preservation of power asymptotics of eigenvalues, 22 Weyl’s theorem on preservation of the essential spectrum, 25

Zero-energy resonance, 287 in multidimensional problem, 287 in the one-dimensional problem, 217 on half-line, 180 Titles in This Series

158 D. R. Yafaev, Mathematical scattering theory: Analytic theory, 2010 157 Xia Chen, Random walk intersections: Large deviations and related topics, 2010 156 Jaime Angulo Pava, Nonlinear dispersive equations: Existence and stability of solitary and periodic travelling wave solutions, 2009 155 Yiannis N. Moschovakis, Descriptive set theory, 2009 154 Andreas Capˇ and Jan Slovák, Parabolic geometries I: Background and general theory, 2009 153 Habib Ammari, Hyeonbae Kang, and Hyundae Lee, Layer potential techniques in spectral analysis, 2009 152 János Pach and Micha Sharir, Combinatorial geometry and its algorithmic applications: The Alcála lectures, 2009 151 Ernst Binz and Sonja Pods, The geometry of Heisenberg groups: With applications in signal theory, optics, quantization, and field quantization, 2008 150 Bangming Deng, Jie Du, Brian Parshall, and Jianpan Wang, Finite dimensional algebras and quantum groups, 2008 149 Gerald B. Folland, Quantum field theory: A tourist guide for mathematicians, 2008 148 Patrick Dehornoy with Ivan Dynnikov, Dale Rolfsen, and Bert Wiest, Ordering braids, 2008 147 David J. Benson and Stephen D. Smith, Classifying spaces of sporadic groups, 2008 146 Murray Marshall, Positive polynomials and sums of squares, 2008 145 Tuna Altinel, Alexandre V. Borovik, and Gregory Cherlin, Simple groups of finite Morley rank, 2008 144 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: Techniques and applications, Part II: Analytic aspects, 2008 143 Alexander Molev, Yangians and classical Lie algebras, 2007 142 Joseph A. Wolf, Harmonic analysis on commutative spaces, 2007 141 Vladimir Mazya and Gunther Schmidt, Approximate approximations, 2007 140 Elisabetta Barletta, Sorin Dragomir, and Krishan L. Duggal, Foliations in Cauchy-Riemann geometry, 2007 139 Michael Tsfasman, Serge Vlˇadut¸, and Dmitry Nogin, Algebraic geometric codes: Basic notions, 2007 138 Kehe Zhu, Operator theory in function spaces, 2007 137 MikhailG.Katz, Systolic geometry and topology, 2007 136 Jean-Michel Coron, Control and nonlinearity, 2007 135 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: Techniques and applications, Part I: Geometric aspects, 2007 134 Dana P. Williams, Crossed products of C∗-algebras, 2007 133 Andrew Knightly and Charles Li, Traces of Hecke operators, 2006 132 J. P. May and J. Sigurdsson, Parametrized homotopy theory, 2006 131 Jin Feng and Thomas G. Kurtz, Large deviations for stochastic processes, 2006 130 Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, 2006 129 William M. Singer, Steenrod squares in spectral sequences, 2006 128 Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov, Painlevé transcendents, 2006 127 Nikolai Chernov and Roberto Markarian, Chaotic billiards, 2006 126 Sen-Zhong Huang, Gradient inequalities, 2006 125 Joseph A. Cima, Alec L. Matheson, and William T. Ross, The Cauchy Transform, 2006 124 Ido Efrat, Editor, Valuations, orderings, and Milnor K-Theory, 2006 TITLES IN THIS SERIES

123 Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli, Fundamental algebraic geometry: Grothendieck’s FGA explained, 2005 122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities and asymptotic methods, 2005 121 Anton Zettl, Sturm-Liouville theory, 2005 120 Barry Simon, Trace ideals and their applications, 2005 119 Tian Ma and Shouhong Wang, Geometric theory of incompressible flows with applications to fluid dynamics, 2005 118 Alexandru Buium, Arithmetic differential equations, 2005 117 Volodymyr Nekrashevych, Self-similar groups, 2005 116 Alexander Koldobsky, Fourier analysis in convex geometry, 2005 115 Carlos Julio Moreno, Advanced analytic number theory: L-functions, 2005 114 Gregory F. Lawler, Conformally invariant processes in the plane, 2005 113 William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith, Homotopy limit functors on model categories and homotopical categories, 2004 112 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups II. Main theorems: The classification of simple QTKE-groups, 2004 111 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups I. Structure of strongly quasithin K-groups, 2004 110 Bennett Chow and Dan Knopf, The Ricci flow: An introduction, 2004 109 Goro Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups, 2004 108 Michael Farber, Topology of closed one-forms, 2004 107 Jens Carsten Jantzen, Representations of algebraic groups, 2003 106 Hiroyuki Yoshida, Absolute CM-periods, 2003 105 Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, second edition, 2003 104 Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence sequences, 2003 103 Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanré, Lusternik-Schnirelmann category, 2003 102 Linda Rass and John Radcliffe, Spatial deterministic epidemics, 2003 101 Eli Glasner, Ergodic theory via joinings, 2003 100 Peter Duren and Alexander Schuster, Bergman spaces, 2004 99 Philip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, 2002 97 V. A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D. Neusel and Larry Smith, Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2: Model operators and systems, 2002 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1: Hardy, Hankel, and Toeplitz, 2002

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/.

The main subject of this book is applications of methods of scattering theory to differential operators, primarily the Schrödinger operator. 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 both of these trends are presented. The first half of this book begins with the summary of the main results of the general scattering theory of the previous book by the author, Mathematical Scattering Theory: General Theory, American Mathematical Society, 1992. The next three chapters illustrate basic theorems of abstract scattering theory, presenting, in particular, their applications to scattering theory of perturbations of differential operators with constant coefficients and to the analysis of the trace class method. In the second half of the book direct methods of scattering theory for differential operators are presented. After considering the one-dimensional case, the author returns to the multi-dimensional problem and discusses various analytical methods and tools appropriate for the analysis of differential operators, including, among others, high- and low-energy asymptotics of the Green function, the scattering matrix, ray and eikonal expansions. The book is based on graduate courses taught by the author at Saint-Petersburg (Russia) and Rennes (France) Universities and is oriented towards a reader interested in studying deep aspects of scattering theory (for example, a graduate student in mathematical physics).

For additional information and updates on this book, visit AMS on the Web www.ams.org/bookpages/surv-158 www.ams.org

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