Methods of modern mathematical physics volume 2 fourier analysis, self adjointness michael reed, barry simon

www.TheSolutionManual.com METHODS OF MODERN M ATHEM ATICAL PH YSICS FOURIER ANALYSIS SELF-ADJOINTNESS M ICHAEL REED BARRY SIMON D epartm ent o f M a t hem atíes D uke U niversity D epartm ents o f M athem atics and P hysics Princeton U niversity ACADEMIC P R ESS , INC Harcourt Braca Jovanovich, Publishers San Diego London New York Sydney Berkeley Tokyo Toronto Boston www.TheSolutionManual.com II: C o p y r i g h t © 1975, b y A c a d e m ic P r e s s , I n c ALL RIGHTS RESERVED NO PART O F TH IS PU BL IC A T IO N MAY BE REPRO DU CED OR TR A N SM ITTED IN ANY F O R M OR BY ANY M EA N S, EL EC TR O N IC OR M ECH AN ICA L, IN CLU DIN G PH O TO C O PY , RECORDING, OR ANY IN FO R M A TIO N STORAGE AND RETRIEV AL S Y S T E M , W IT H O U T PERM 1SS10N IN W RIT1N G FR O M T H E P U B L ISH ER ACADEMIC PRESS INC 1250 Si.xth A venue San Diego C alifornia 92101 24/28 O val R oad, London N W I L ibrary o f C ongress C atalo g in g in P u b licatio n D ata R eed, M ichael M ethods o f m o d ern m ath e m a tic a l physics In clu d es b ib lio g rap h ical references C O N T E N T S: v F u n c tio n a l a n a ly s is -v F ourier analysis, s e lf-a d jo in tn e ss I M athem atical p h ysics I S im ó n , B a rr y jo in t a u th o r II T itle Q C 20 R 1972 'S -1 ISBN - - 0 - ( v 2) AMS (M O S) S u b je c t C lassifications: -0 , -0 P R IN T E D IN T H E U N IT E D S T A T E S O F A M E R I C A >8 89 987 www.TheSolutionManual.com U n ite d K in g d o m E d itio n p u b lis h e d b y A C A D E M IC P R E S S , I N C ( L O N D O N ) L T D T o our parents H elen and Gerald R ee d www.TheSolutionManual.com M innie and H y Simón Preface M ire R eed B arry S imón June 1975 www.TheSolutionManual.com This volume continúes our series of texts devoted to functional analysis methods in m athem atical physics In Volume I we announced a table of contents for Volume II However, in the preparation of the m aterial it became clear th at we would be unable to treat the subject m atter in sufficient depth in one volume Thus, the volume contains Chapters IX and X ; we expect that a third volume will appear in the near future containing the rest of the m aterial announced as “ Analysis of O perators.” We hope to continué this series with an additional volume on algebraic methods It gives us pleasure to thank many individuáis: E Nelson for a critical reading of C hapter X; W Beckner, H Kalf, R S Phillips, and A S W ightm an for critically reading one or more sections N um erous other colleagues for contributing valuable suggestions F A rm strong for typing m ost of the prelim inary manuscript J H agadorn, R Israel, and R W olpert for helping us with the proofreading Academic Press for its aid and patience; the N ational Science and Alfred P Sloan F oundations for financial support Jackie and M artha for their encouragement and understanding Introduction A fu n ctio n a l an a lyst is an analyst, firs i and fo re m o si, and not a degeneróte species o f topologist www.TheSolutionManual.com E H ille M ost texts in functional analysis suffer from a serious defect that is shared to an extent by Volume I of M ethods of M odern M athem atical Physics Namely, the subject is presented as an abstract, elegant Corpus generally divorced from applications Consequently, the students who learn from these texts are ignorant of the fact that alm ost all deep ideas in functional analysis have their ímmediate roots in “ applications,” either to ciassical areas of analysis such as harmonio analysis or pardal differential equations, or to another science, prim arily physics F or example, it was ciassical electromagnetic potential theory that m otivated Fredholm ’s work on integral equations and thereby the w ork o f Hilbert, Schmidt, Weyl, and Riesz on the abstractions of Hilbert space and com pact operator theory And it was the Ímpetus of quantum mechanics that led von Neum ann to his development of unbounded operators and later to his work on operator algebras M ore deleterious than historical ignorance is the fact th at students are too often misled into believing that the most profitable directions for research in functional analysis are the abstract ones In our opinión, exactly the opposite is true We not m ean to imply that abstraction has no role to play Indeed, it has the critical role of taking an idea from a concrete situation and, by elim inating the extraneous notions, m aking the idea more easily understood as well as applicable to a broader range of www.TheSolutionManual.com situations But it is the study of specific applications and the consequent generalizations th at have been the m ore im portant, rather than the consideration of abstract questions about abstract objects for their own sake This volume contains a m ixture of abstract results and applications, while the next contains mainly applications The intention is to offer the readers of the whole series a properly balanced view We hope that this volume will serve several purposes: to provide an introduction for gradúate students not previously acquainted with the material, to serve as a reference for m athem atical physicists already working in the field, and to provide an introduction to various advanced topics which are difficult to understand in the literature N ot all the techniques and applications are treated in the same depth In general, we give a very thorough discussion of the m athem atical techniques and applications in quantum mechanics, but provide only an introduction to the problems arising in quantum field theory, classical mechanics, and partial differential equations Finally, some of the material developed in this volume will not find application until Volume III F or all these reasons, this volume contains a great variety of subject matter To help the reader select which material is im portant for him, we have provided a “ Reader’s G uide” at the end of each chapter As in Volume I, each chapter contains a section o f notes The notes give references to the literature and sometimes extend the discussion in the text Historical comments are always limited by the knowledge and prejudices of authors, but in m athem atics that arises directly from applications, the problem of assigning credit is especialiy difficult Typically, the history is in two stages: first a specific m ethod (typically difficult, com putational, and sometimes nonrigorous) is developed to handle a small class of problems Later it is recognized th at the m ethod contains ideas which can be used to treat other problems, so the study of the m ethod itself becomes im portant The ideas are then abstracted, studied on the abstract level, and the techniques systematized W ith the newly developed m achinery the original problem becomes an easy special case In such a situation, it is often not completely clear how m any o f the m athem atical ideas were already contained in the original work Further, how one assigns credit may depend on whether one first learned the technique in the oíd com putational way or in the new easier but m ore abstract way In such situations, we hope that the reader will treat the notes as an introduction to the literature and not as a judgm ent of the historical valué o f the contributions in the papers cited Each chapter ends with a set of problems As in Volume I, parts of proofs are occasionally left to the problems to encourage the reader to www.TheSolutionManual.com particípate in the development of the mathematics Problems that fill gaps in the text are m arked with a dagger Difficult problems are m arked with an asterisk W e strongly urge students to work the problems since that is the best way to learn mathematics Contents IX : T H E F O U R IE R T R A N S F O R M The Fourier transform on SP(Un) and £P'(Un), convolutions The range o f the Fourier transform: Classical spaces The range o f the Fourier transform: Analyticity I I Estimates Appendix Abstract interpolation Fundamental Solutions o f partial dijferential equations with constant coefficients Elliptic regularity The free Hamiltonianfor nonrelativistic quantum mechanics The Gdrding-Wightman axioms Appendix Lorentz invariant measures Restriction to submanifolds 10 Products o f distributions, wave front sets, and oscillatory integráis Notes Problems Reader's Guide vii ix xv www.TheSolutionManual.com Preface Introduction Contents o f Other Volumes 15 21 32 45 49 54 61 12 16 81 108 120 133 10 Let A b e th e o p e to r I A I w ith d o m ain co n sistin g o f all pairs < / j , / 2> w ith (a) S how th at iA is sym m etric (b) S how th at iA is essentially self-adjoint by show ing th a t its deficiency índices are (c) S how th a t D(£í3)) = ( < / i , / 2> | A /, e L2(R 3) , / e X,(IR3)} (d) Let U {t) = e‘(“4>l = e ~ n a n d for < / i , / 2> D (Á ) set ( ¡^ ( x , t), u2(x, t)> = U ( t ) < / i , / 2> P ro v e th a t and IS - u , ( x , r ) - / 2(x) -♦ as t -* ¿?(R J) 61 Let {T (f)},i b e a Family o f o p e to rs on a reflexive B anach space X so th at (i) T ( t + s) = T (t) T (s ) (") U í > o R aofT 'ÍO ) *s dense in X (iii) í(T(t) there is a n so th a t (X.109) holds (c) U se sim ilar m eth o d s to prove (X l 10) www.TheSolutionManual.com | « i ( x , t ) - / i ( x ) l i ? ( R 3) - » = 0, 63 (a) P ro v e th a t the w eak to pology o n th e u n it ball in a se p ara b le H ilb ert space is m etrizable (b) C o n c lu d e th a t the balls in a se p ara b le H ilb ert space are w eakly sequentially com plete 64 Let p(x, y; t) = (4 n D t)~ n,2e ~ I*- >1í/4í)i for so m e D w ith Re D ;> 0, D # S u p p o se th a t there is a signed m easu re /i on the p a th space O o f Section X l l so th a t J

0, p ro v e th a t n x{a> | 0 u(0, x) = / ( x ) u,(0, x) = g (x) we rew rite it w ith a lin ear term added to b o th sides, u„ - Au + m 2u = - |u |2u + m 2u, a n d th e n fo rm ú late th e p ro b lem as a first-o rd er e q u a tio n www.TheSolutionManual.com 71 S how th a t if / an d g are real-valued, then the co m p o n en ts o f W ( t ) ( f g ) are realvalued for each t w h ere W (i) is the u n itary g ro u p defined in S ection X.13 U se th is to sh o w th a t th e so lu tio n o f (X 138) is real-valued if th e in itial d a ta a re real-valued m > in t as in Section X.13: X=i be a n o rth o n o rm a l basis for JÉ"+ , th e deficiency space for A D efine J : jé " + -* JÉ" + by J ( £ a„