www.TheSolutionManual.com METHODS OF MODERN M ATHEMATICAL PH YSIC S III: SCA TTERIN G TH EO RY BARRY SIMON Department of Mathematíes Duke University Departments of Mat hematíes and Physics Princeton ¡Jniversity www.TheSolutionManual.com MICHAEL REED ACADEMIC PRESS, INC Harcourt Braca Jovanovich, Publithars San Diego New York Berkeley Boston London Sydney Tokyo Toronto C o p y r i g h t © 1979, by A c a d e m ic P r e s s , In c ALL RIGHTS RESERVED NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETR1EVAL SYSTEM, W ITHOUT PERMISSION IN WRITING FROM THE PUBLISHER ACADEMIC PRESS, INC United Kingdom Edition published by ACADEMIC PRESS, INC (LONDON) LTD 24/28 Oval Road, London NW1 7DX Library of Congress Cataloging in Publication Data Reed, Michael Methods of modern mathematical physics Vol Scattering Theory Includes bibliographical references CONTENTS: v Functional analysis.-v Fourier analysis, self-adjointness.-v Scattering theory.-v Analysis of operators Mathematical physics Simón, Barry,joint author II Title QC20.R37 1972 530.T5 -182650 ISBN -1 -5 0 -4 (v 3) AMS (MOS) 1970 Subject Classifications: -0 , -0 P R I N T E D IN T H E U N I T E D S T A T E S O F A M E R I C A 88 89 90 91 92 10 www.TheSolutionManual.com 1250 Sixth Avenue, San Diego, California 92101 www.TheSolutionManual.com T o M a rth a and Ja ckie www.TheSolutionManual.com Scattering theory is the study of an interacting system on a scale of time and/or distance which is large com pared to the scale of the interaction itself As such^it is the most effective means, sometimes the only means, to study microscopio nature To understand the im portance of scattering theory, consider the variety of ways in which it arises First, there are variou»phenom ena in nature (like the blue of the sky) which are the result of scattering In order to understand the phenomenon (and to identify it as the result of scattering) one must understand the underlying dynamics and its scattering theory Second, one often wants to use the scattering of waves or particles whose dynamics one knows to determine the structure and position of small or inaccessible objects For example, in x-ray crystallography (which led to the discovery of DNA), tom ography, and the detection of underwater objects by sonar, the underlying dynamics is well understood W hat one would like to construct are correspondences that link, via the dynamics, the position, shape, and intem al structure of the object to the scattering data Ideally, the correspondence should be an explicit formula which allows one to reconstruct, at least approximately, the object from the scattering data A third use of scattering theory is as a probe of dynamics itself In elementary particle physics, the underlying dynamics is not well understood and essentially all the experimental data are scattering data The main test of any proposed particle dynamics is whether one can construct for the dynamics a scattering theory that predicts the observed experimental data Scattering theory was not always so central to physics Even though the C oulom b cross section could have been computed by Newton, had he bothered to ask the right question, its calculation is generally attributed to Rutherford more than two hundred years later O f course, Rutherford’s calculation was in connection with the first experiment in nuclear physics Scattering theory is so im portant for atomic, condensed m atter, and high www.TheSolutionManual.com energy physics that an enorm ous physics literature has grown up Unfortunately, the development of the associated m athem atics has been much slower This is partially because the m athem atical problems are hard but also because lack of com m unication often m ade it difficult for mathematicians to appreciate the many beautiful and challenging problems in scattering theory The physics literature, on the other hand, is not entirely satisfactory because of the m any heuristic formulas and ad hoc methods Much of the physics literature deais with the “ t im e-independent ” approach to scatteringtheory because the time-independent approach provides powerful calculational tools We feel that to use the time-independent formulas one m ust understand them in terms of and derive them from the underlying dynamics Therefore, in this book we emphasize scattering theory as a time-dependent phenomenon, in particular, as a com parison between the interacting and free dynamics This approach leads to a certain im balance in our presen tation since we therefore emphasize large times rather than large distances However, as the reader will see, there is considerable geometry lurking in the background The scattering theories in branches of physics as different as classical mechanics, continuum mechanics, and quantum mechanics, have in common the two foundational questions of the existence and completeness of the wave operators These two questions are, therefore, our main object of study in individual systems and are the unifying theme that runs throughout the book Because we treat so m any different systems, we not carry the analysis much beyond the construction and completeness of the wave operators, except in two-body quantum scattering, which we develop in some detail However, even there, we have not been able to include such important topics as Regge theory, inverse scattering, and double dispersión relations Since quantum mechanics is a linear theory, it is not surprising that the heart of the mathematical techniques is the spectral analysis of Hamiltonians Bound States (corresponding to point spectra) of the interaction Ham iltonian not scatter, while States from the absolutely continuous spectrum The mathematical property that distinguishes these two cases (and that connects the physical intuition with the m athem atical formulation) is the decay of the Fourier transform of the corresponding spectral measures The case of singular continuous spectrum lies between and the crucial (and o fien hardest) step in most proofs of asym ptotic completeness is the proof that the interacting Ham iltonian has no singular continuous spectrum Conversely, one of the best ways of showing that a self-adjoint operator has no singular continuous spectrum is to show that it is the interaction Hamiltonian of a quantum system with com plete wave operators This deep www.TheSolutionManual.com connection between scattering theory and spectral analysis shows the artificiality of the división of m aterial into Volumes III and IV We have, therefore, preprinted at the end of this volume three sections on the absence of continuous singular spectrum from Volume IV While we were reading the galley proofs for this volume, V Enss introduced new and beautiful m ethods into the study of quantum -m echanical scattering Enss’s paper is not only of interest for what it proves, but also for the future direction that it suggests In particular, it seems likely that the m ethods will provide strong results in the theory of multiparticle scattering W ehaveadded a section at the end of this Chapter (Section XI 17) to describe Enss’s method in the two-body case We would like to thank Professor Enss for his generous attitude, which helped us to include this material The general remarks about notes and problems made in earlier introductions are applicable here with one addition: the bulk of the material presented íh this volume is from advanced research literature, so m any of the problems are quite substantial Some of the starred problems summarize the contents of research papers! Contents XI: SCATTERIN G TH EO RY An overview o f scattering phenomena Classical partióle scattering The basic principies o f scattering in Hilbert space Appendix Stationary phase methods Appendix Trace ideal properties o ff(x )g ( — iV) Appendix A general invariance principie for wave operators Quantum scattering I: Two-body case Quantum scattering II: N-body case Quantum scattering III: Eigenfunction expansions Appendix Introduction to eigenfunction expansions by the auxiliar y space method Quantum scattering IV: Dispersión relations Quantum scattering V: Central potentials A Reduction o f the S-matrix by symmetries B The partial wave expansión and its convergence C Phase shifts and their connection to the Schródinger equation D The variable phase equation E Jost functions and Levinsons theorem F Analyticity o f the partial wave amplitude for generalized Yukawa potentials G The Kohn variational principie ini ix xv www.TheSolutionManual.com Preface ¡ntroduction Contents o f Other Volumes 16 37 47 49 54 75 96 112 116 121 121 127 129 133 136 143 147 Appendix Legendre polynomials and spherical Bessel functions Jost Solutions fo r oscillatory potentials Jost Solutions and the fundamental problems o f 149 155 scattering theory Long-range potentials 10 Optical and acoustical scattering I: Schródinger operator methods Appendix Trace class properties o f Greerís functions 11 Optical and acoustical scattering II: The Lax-Phillips method Appendix The twisting trick 12 The linear Boltzmann equation 13 Nonlinear wave equations Appendix Conserved currents 14 Spin wave scattering 15 Quantum field scattering I: The external field 16 Quantum field scattering II: The Haag-Ruelle theory 17 Phase space analysis o f scattering and spectral theory Appendix The RAG E theorem Notes Notes on scattering theory on C*-algebras Problems 164 169 MATERIAL PREPRINTED FROM VO LU M E IV X III.6 X I I 1.7 X III.8 The absence o f singular continuous spectrum I: General theory The absence o f singular continuous spectrum II : Smooth perturbations A Weakly coupled quantum systems B Positive commutators and repulsive potentials C Local smoothness and wave operators fo r repulsive potentials The absence o f singular continuous spectrum I I I: Weighted l i spaces Notes Problems List o f Symbols Index 184 203 www.TheSolutionManual.com Appendix Appendix 210 241 243 252 278 285 293 317 331 340 344 382 385 406 411 421 427 433 438 447 450 455 457 O u r ap p ro a c h is based on a series of lectures by A gm on to g eth er w ith helpful rem arks by H E pstein, J G in ib re, an d R Lavine T h eo rem X III.33d was proven p rio r to A gm on in T K a to an d S K u ro d a, “ T h eo ry o f sim ple scatterin g and eigenfunctions e x p a n s io n s” in Functional Analysis and Related Fields, S pringer-V erlag, Berlín an d N ew Y ork, 1970, 99-131 T h a t (d) follow s from A g m o n ’s a p rio ri estim ates an d the th eo ry o f local sm o o th n ess is a rem ark of R Lavine in the p ap er q u o te d below A gm on's w ork rep resen ts th e cu lm in atio n o f several lines o f developm ent T he first involves p roving th a t an d th a t th e associated o p e to r H h as a(H) c a(H0) an d ob ey s su p || |C , | l/2( // - z)~1 | C , | I/2|| < x for all u j (continued in th e next problem ) *54 (co n tin u ed from P ro b le m 53) Let R(//) be th e resolvent o f H0 an d R (//; X) the resolvent of H(X) = H0 + X I ? , , C D efine W±(X) by ¿ { W±(Xyi,) = (V,4,)+ \ ( ' (C¡nR(tt ¿ni jat i • - , ± iO)v>, \c,\ulR(,l + ,0 X)>P) 4) = {0} P ro v e th at, for any positive integers n ¿ m, at m o st one o f A" an d A~mis H0 sm o o th 56 Let V e R, th e R ollnik class, w ith ||F ||* < n P ro v e th a t th e w ave o p e to rs pro v id e u n itary eq uivalences o f - A a n d - A + V an d in p a rtic u la r th at scatterin g is com plete A*Bn w here 57 (a) L et //„ = //- - Bn is H 0-sm o o th an d An is //„ -sm ooth S uppose th at ||„0 s u p ,|M ||„ < oo an d l i m , ||B = P rove Ihat Clf s s - l i m , , , converges to in norm In p a rtic u la r verify the n o rm co n tin u ity of £>*(2) = s - l i m , ^ ^ ¿'Wo +xQg-itHo fo r i | jn t he context of T h eo rem XIII.26 (b) Let VH-+V in R ollnik norm P rove th a t the co rre sp o n d in g Sm atrices converge strongly (H in t: W rite Vn - WH+ Yn, V = W + y ,s o th at Y„ -* Y in 12 n R,WH-+ W in R, an d s u p , || Wm¡|* < 4«.) 58 (a) L et H0 be th e o p e to r on L2[0, oo) th a t is th e closu re of - d 2/dx2 on {w e C® [0, o o )|u (0 ) « 0} L et E £ o(H0) an d let K £(x, y) = E~ 1/2 sin f^ /E min{x, y } ]e x p [i\/E max{x, y}] w here -JE is th e sq u a re ro ot with Im -JE > Prove th at [(H0 - £)" V](.v) = | **(*< y)v(y) Jy 'o (b) \K¿x.y)\ < Jxy (c) Let V be a m easu rab le function on [0, oc) with J j su p || | x | K (x) | dx < ce T hen, V\ll2(H0 - £ ) " 1F | ,/2 || < oc Eé R (d) If x | F (x )| dx < 1, then H0 an d o p e to rs are u n itary equivalences H be b o u n d ed self-adjoint o p e to rs a n d let R(p) = {H - p) ~ P r o v e th at M I M ¿ + >«)«> an d use this to prove th a t (/[//, /4 [H >4]R(á + Íe )v )| S M il M I ])l/2 is //-s m o o th if i[ //, A] > www.TheSolutionManual.com 59 Let A an d H0 + V are unitarily equivalent an d the w ave 60 Let A an d B be b o u n d ed self-adjoint o p e to rs an d c a strictly positive real num ber P rove th at i[/4, B] > el is im possible by : (a) using th e th eo ry o f sm o o th p e rtu rb a tio n s ; (b) d irect c o m p u ta tio n (look at eiAtBe~iAt) 61 E xtend the K a to - P u tn a m th eorem to the case w here H is u n b o u n d ed a n d /[ //, A] > m eans th at i(A inductively in [0, 1], (1, 2] ) (b) C o m p lete the p ro o f of L em m a in Section (c) P rove th a t ( - A + 1)~ d/dx¡ is a b o u n d ed m ap from L] to L for any +67 Verify th e b o u n d (62) 68 Let ó > n + P ro v e th at, for any b > a > 0, there is a co n stan t C so th a t ||(-A - A- /0)">||_, «^Tppt l 53 175 175 102, 103 102, 103 206 206 295 39 2301 85 85 219 321 Jt se* ^ sing jr JC\ ** out H0 H (free H a m ilto n ia n ) H* H(C