Mathematical Scattering Theory
MathematicalSurveysandMonographsVolume 158MathematicalScattering TheoryAnalytic TheoryD. R. YafaevAmerican Mathematical Society
tering TheoryAnalytic Theory
MathematicalSurveysandMonographsVolume 158MathematicalScattering TheoryAnalytic TheoryD. R. YafaevAmerican Mathematical SocietyProvidence, Rhode Island
EDITORIAL COMMITTEEJerry L. BonaMichael G. EastwoodRalph L. Cohen, ChairJ. T. StaffordBenjamin Sudakov2000 Mathematics Subject Classification. Primary 34L25, 35-02, 35P10,35P25, 47A40, 81U05.For additional information and updates on this book, visitwww.ams.org/bookpages/surv-158Library of Congress Cataloging-in-Publication DataÍAf̀aev, D. R. (Dmitriı̆ Rauel evich), 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 2009515 .724–dc222009027382Copying and reprinting. Individual readers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Requests for suchpermission should be addressed to the Acquisitions Department, American Mathematical Society,201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made bye-mail to reprint-permission@ams.org.c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rightsexcept 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/10 9 8 7 6 5 4 3 2 115 14 13 12 11 10
To the memory of my parents
ContentsPrefacexiBasic Notation1Introduction5Chapter 0. Basic Concepts1. Classification of the spectrum2. Classes of compact operators3. The resolvent equation. Conditions for self-adjointness4. Wave operators (WO)5. The smooth method6. The stationary scheme7. The scattering operator and the scattering matrix (SM)8. The trace class method9. The spectral shift function (SSF) andthe perturbation determinant (PD)10. Differential operators11. Function spaces and embedding theorems12. Pseudodifferential operators13. Miscellaneous analytic facts1717202326293338424552565867Chapter 1. Smooth Theory. The Schrödinger Operator1. Trace theorems2. The free Hamiltonian3. The Schrödinger operator4. Existence of wave operators5. Wave operators for long-range potentials6. Completeness of wave operators7. The limiting absorption principle (LAP)8. The scattering matrix9. Absence of the singular continuous spectrum10. General differential operators of second order11. The perturbed polyharmonic operator12. The Pauli and Dirac operators71717579828693959698101103104Chapter 2. Smooth Theory. General Differential Operators1. Spectral analysis of differential operators with constant coefficients2. Scalar differential operators3. Nonelliptic differential operators4. Matrix differential operators109109116118122vii
viiiCONTENTS5. Scattering problems for perturbations of a medium6. Strongly propagative systems. Maxwell’s equations124128Chapter 3. Scattering for Perturbations of Trace Class Type1. Conditions on an integral operator to be trace class2. Perturbations of differential operators with constant coefficients3. The Schrödinger operator4. The perturbed polyharmonic operator5. General differential operators of second order6. Scattering problems for perturbations of a medium7. Wave equation8. The scattering matrix and the spectral shift function133133136139145147154157159Chapter 4. Scattering on the Half-line1. Jost solutions. Volterra equations2. Generalized Fourier transform and WO3. Low-energy asymptotics4. High-energy asymptotics5. The SSF for the radial Schrödinger operator6. Trace identities7. Perturbation by a boundary condition. Point interaction161161170178188191198203Chapter 5. One-Dimensional Scattering1. A direct approach2. Low- and high-energy asymptotics3. The SSF and trace identities4. Potentials with different limits at “ ” and “ ” infinities209209216221223Chapter 6. The Limiting Absorption Principle (LAP), the RadiationConditions and the Expansion Theorem1. Absence of positive eigenvalues and radiation conditions2. Boundary values of the resolvent3. A sharp form of the limiting absorption principle4. Nonhomogeneous Schrödinger equation5. Homogeneous Schrödinger equation6. Expansion theorem7. The wave function. The scattering amplitude8. A generalized Fourier integral9. The Mourre method231231233235239241245251255259Chapter 7. High- and Low-Energy Asymptotics1. High-energy and uniform resolvent estimates2. Asymptotic expansion of the Green function for large values of thespectral parameter3. Small time asymptotics of the heat kernel4. Low-energy behavior of the resolvent5. Low-energy behavior of the resolvent. Slowly decreasing potentials267267275280285291Chapter 8. The Scattering Matrix (SM) and the Scattering Cross Section1. Basic properties of the SM297297
CONTENTS2.3.4.5.6.7.The spectrum of the SM. The modified SMThe scattering cross sectionHigh-energy asymptotics of the SM. The ray expansionThe eikonal approximationThe averaged scattering cross section. Singular potentialsThe semiclassical limitix302306311319328337Chapter 9. The Spectral Shift Function and Trace Formulas1. The regularized PD and SSF for the multidimensional Schrödingeroperator2. High-energy asymptotics of the SSF3. Trace identities for the multidimensional Schrödinger operator341341353365Chapter 10. The Schrödinger Operator with a Long-Range Potential1. Propagation estimates2. Long-range scattering3. The eikonal and transport equations4. Scattering matrix for long-range potentials369369374380384Chapter 11. The LAP and Radiation Estimates Revisited1. The efficient form of the LAP2. Absence of positive eigenvalues and uniqueness theorem3. Nonhomogeneous Schrödinger equation with a long-range potential399399403408Review of the Literature415Bibliography429Index441
PrefaceThis 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 necessarybackground from [I] is collected (but the proofs are of course not repeated) inChapter 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, simultaneously with proofs of various facts, the stationary approach gives formularepresentations for the basic objects of the theory. Along with wave operators, wealso consider properties of the scattering matrix, the spectral shift function, thescattering cross section, etc.A consistent use of the stationary approach as well as the choice of concretematerial distinguishes this book from others such as the third volume of the course ofM. 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 broadcompass of material, the course [43] was necessarily written in encyclopedic styleand 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 almostindependent of it. In this approach the abstract theory is replaced by a concreteinvestigation of the corresponding differential equation. In this book we presentboth of these trends.The first of them illustrates basic theorems of [I]. Thus, Chapters 1 and 2are devoted to applications of the smooth method. Of course the abstract resultsof [I] should be supplemented by some analytic tools, such as the Sobolev tracetheorem. The smooth method works well for perturbations of differential operatorswith constant coefficients. In Chapter 3 applications of the trace class method arediscussed. The main advantage of this method is that it does not require an explicitspectral analysis of an “unperturbed” operator.Other chapters are much less dependent on [I]. Chapters 4 and 5 are devotedto the one-dimensional problem (on the half-axis and the entire axis, respectively)which is a touchstone for the multidimensional case because specific methods ofordinary differential equations can be used here.In the following chapters we return to the multidimensional problem and discussdifferent analytic methods appropriate to differential operators. In particular, inChapter 6 scattering theory is formulated in terms of solutions of the Schrödingerequation satisfying some “boundary conditions” (radiation conditions) at infinity.xi
xiiPREFACEHigh- and low-energy asymptotics of the Green function (the resolvent kernel) andof related objects are discussed in Chapter 7. Chapter 8 is devoted to a study ofthe scattering matrix and of the scattering cross section. Here some asymptoticmethods, such as the ray expansion and eikonal expansion, are also discussed.As an example of a useful interaction of abstract and analytic methods, wemention the theory of the spectral shift function. Abstract results are illustrated in§3.8. However, specific properties of this function are studied by concrete methodsin §4.5, §5.3 and in Chapter 9. Here perturbation determinants are also discussedand trace identities are derived.Note that Chapters 1 and 3 and large parts of Chapters 4 and 5 contain essentially a “necessary minimum” on scattering theory, whereas the other chapters areof a slightly more special nature.The book is mainly devoted to a study of perturbations by differential operatorswith 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 extentrelated to multiparticle systems. This very interesting and difficult problem isdiscussed in [16] and [61].Similarly to [I], in working on the book the author has tried to resolve twoopposite problems. The first of them is a systematic exposition of the materialstarting from the general background of [I]. The second problem is the expositionof a number of topics to a degree of completeness which might possibly be of interestto experts in spectral theory. We have also tried to fill in numerous gaps presentin monographic literature. This pertains especially to the exposition of works ofRussian and, in particular, Saint Petersburg mathematicians. Compared to [I], theauthor’s tastes are also more thoroughly represented here. As a whole the book isoriented 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 theoremsand the “two-stage” enumeration of sections. However, the first number is omittedwithin a chapter.This book is based on the graduate courses taught by the author several timesin Saint-Petersburg and Rennes Universities.The concept and structure of the entire book, as well as many specific questions, 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 authoris deeply grateful to M. Sh. Birman and L. D. Faddeev. Numerous discussionswith P. Deift, A. B. Pushnitski, G. Raikov and M. Z. Solomyak are also gratefullyacknowledged.
PREFACEInterdependence of chaptersChapter 0 HH HH ) ?HHj Chapter 1 - Chapter 2Chapter 4 XXXX Chapter 3XXXXXXXXX?z- ChapterChapter 56 - Chapter 10iPPPPPPPq?Chapter 7Chapter 11?Chapter 9 ?Chapter 8xiii
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Mathematical Surveys and Monographs Volume 158 American Mathematical Society Mathematical Scattering Theory Analytic Theory D. R. Yafaev surv-158-yafaev-cov.indd 1 2/4/10 3:47 PM
Lecture 34 Rayleigh Scattering, Mie Scattering 34.1 Rayleigh Scattering Rayleigh scattering is a solution to the scattering of light by small particles. These particles . The quasi-static analysis may not be valid for when the conductivity of the
scattering theory. As preparation for the quantum mechanical scattering problem, let us first consider the classical problem. This will allow us to develop (hopefully a revision!) some elementary concepts of scattering theory, and to introduce some notation. In a classical scattering experiment, one considers particles of energy E 1 2 mv 2
1. Weak scattering: Single‐scattering tomography and broken ray transform (BRT) 2. Strong scattering regime: Optical diffusion tomography (ODT) 3. Intermediate scattering regime: Inverting the radiative transport equation (RTE) 4. Nonlinear problem of inverse scattering
Scattering theory: outline Notations and definitions; lessons from classical scattering Low energy scattering: method of partial waves High energy scattering: Born perturbation series expansion
2.3.4 Solubility Parameter 107 2.3.5 Problems 108 2.4 Static Light Scattering 108 2.4.1 Sample Geometry in Light-Scattering Measurements 108 2.4.2 Scattering by a Small Particle 110 2.4.3 Scattering by a Polymer Chain 112 2.4.4 Scattering by Many Polymer Chains 115 2.4.5 Correlation Function and Structure Factor 117 2.4.5.1 Correlation Function 117
Computational Scattering Science 2010 Table of Contents Executive Summary 1 1. Introduction and Scope 3 1.A. Trends in Scattering Research and Computing 3 1.B. Roles for Computing in Scattering Science Today 3 2. Strategic Plan for Computational Scattering Science 7 2.A. Where We Are Today 7 2.B. Goal State 8 2.C. Path Forward 11 3. Topic .
SCATTERING AND INVERSE SCATTERING ON THE LINE FOR A . via the so-called inverse scattering transform method. The direct and inverse problems for the corresponding first-order linear sys-tem with energy-dependent potentials are investigated. In the direct problem, when . In quantum mechanics, ei .
group of employees at his work. Derogatory homophobic : comments have been posted on the staff noticeboard about him by people from this group. Steve was recently physically pushed to the floor by one member of the group but is too scared to take action. Steve is not gay but heterosexual; furthermore the group know he isn’t gay. This is harassment related to sexual orientation. Harassment at .
