www.TheSolutionManual.com M ETHODS OF MODERN M ATH EM ATICAL P H Y S IC S IV : ANALYSIS OF OPERATORS BARRY SIMON Department o f Mathematíes Duke University Departments o f M at hematíes and Physics Princeton University www.TheSolutionManual.com M ICHAEL REED ACADEMIC PRESS, INC Harcourt Brace Jovanovich, Publishers San Diego London New York Sydney Berkeley Tokyo Toronto Boston C o p y r i g h t © 1978, b y A c a d e m i c P r e s s , I n c ALL RIGHTS RESERVED NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY F O R M OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, W IT H O U T PERMISSION IN WRITING FRO M TH E PUBLISHER ACADEMIC PRESS, S i x t h A v e n u e , San INC D i e g o California 1 U n ited K in g d o m E d itio n p u b lish e d b y ACADEMIC PRESS, INC (LONDON) LTD 7DX Library oí Congress Cataloging in Publication Data Reed, Michael Methods o f m odem mathematical physics Vol Analysis o f Operators Includes bibliographical references CONTENTS: v Functional analysis.-v Fourier analysis, se lf-a d jo in tn e ss.-v Scattering th e o r y -v Analysis o f operators Mathematical physics I Simón, Barry,joint author II Title QC20.R37 1972 ’5 75-182650 ISBN - - 0 - (v 4) AMS (MOS) 1970 Subject Classifications: - , - PRINTED IN THE UNITED STATES OF AMERICA 87 88 89 www.TheSolutionManual.com 24/28 Oval R o a d , L o n d o n N W T o David www.TheSolutionManual.com T o Rivka and Benny Preface With the publication of Volumes III and IV we have completed our presentation of the material which we originally planned as “ Volume 11” at the time of publication of Volume I We originally promised the publisher that the entire series would be completed nine months after we submitted Volume I Well! We have Usted the contents of future volumes below We are not foolhardy enough to make any predictions We were very fortúnate to have had T Kato and R Lavine read and cnticize Chapters XII and XIII, respectively In addition, we received valuable comments from J Avron, P Deift, H Epstein, J Ginibre, I Herbst, and E Trubowitz We are grateful to these individuáis and others whose comments made this book better We would also like to th a n k : J Avron, G Battle, C Berning, P Deift, G Hagedorn, E Harrell, II, L Smith, and A Sokol for proofreading the galley and/or page proofs G Anderson, F Armstrong, and B Farrell for excellent typing The National Science Foundation, the Duke Research Council, and the Alfred P Sloan Foundation for financia! support Academic Press, without whose care and assistance these volumes would have been impossible Martha and Jackie for their encouragement and understanding vil www.TheSolutionManual.com of making books there is no end, and much study is a weariness o f the flesh Koheleth (Ecclesiastes) 12:12 Introduction Galileo Galilei The first step in the mathematical elucidation of a physical theory must be the solution of the existence problem for the basic dynamical and kinematical equations of the theory Once that is accomplished, one would like to find general qualitative features of these Solutions and also to study in detail specific special systems of physical interest Having discussed the general question of the existence of dynamics in Chapter X, we present methods for the study of general qualitative features of Solutions in this volume and its companion (Volume III) on scattering theory We concéntrate on the Hamiltonians of nonrelativistic quantum mechanics although other systems are also treated In Volume III, the main theme is the long-time behavior of dynamics, especially of Solutions which are “ asymptotically free.” In this volume, the main theme involves the five kinds of spectra defined in Sections VII.2 and VII.3: the essential spectrum, A and that A has com pact resolvent Let EH(a) be the nth eigenvalue o f A% Prove that E„(a) -* oo uniformly in a as n oo 145 Prove that the density o f states measure given by (176) is absolutely continuous with respect to Lebesgue measure (Hint: Prove that it is a sum of spectral measures for H.) www.TheSolutionManual.com In particular, conclude that infinitely many gaps occur 146 In the one-dimensional case prove that at a point E with E„(k) = E the density of states is given by 1147 Fill in the details of the proof of Theorem XIII 101 *148 Prove the analogue o f Theorem XIII 101 when periodic boundary conditions are replaced by Dirichlet or Neum ann boundary conditions 149 U sing (185), prove (186) 150 Let j | a f | < oo Let f N(z) * I lJ * i + a j z )m P r o v e Í n converges as N formly on compact sets as follows: (a) Prove uniform bounds on f N(z) by using 11 + x | oo (c) Com plete the proof o f convergence -*> oo uni 151 (a) Let An -*>A in trace class norm and suppose that p £ a(A) Prove directly (without use o f Theorem XIII 107) that m norm (b) Prove (196) knowing it is true when A is finite rank with p $ dS) Let I* ( * * ) ! Z C \ x - y \ - ‘ with a < n Suppose that L is the kernel o f an integral operator , with |¿ (x , y )| ¿ D \ x - y \ ~ f (a) If a 4■p < n , prove that A B has a bounded kernel (b) If a 4- p > n, prove that AB has a kernel M with |M (x , y )| ¿ E ( x - y ) - y y = a 4- p — n (c) Let a < n[l »- (2k ) ~ J], for k equal a positive integer Prove that A e J 2k *(d) Let