Methods of modern mathematical physics volume 1 functional analysis michael reed, barry simon

www.TheSolutionManual.com I: FUNCTIONAL ANALYSIS Revised and Enlarged Edition MICHAEL REED BA RRY SIMON Department of Mathematíes Duke University Departments of Mal hematíes and Physics Princeton University ACADEMIC P R E SS , INC Harcourt Brace Jovanovlch, Publlshera San Diego london New York Sydney Berketey Tokyo Toronto Boston www.TheSolutionManual.com METHODS OF MODERN MATHEMATICAL PHYSICS To R S P hillip s a n d A S W ightm an, C o p y r i g h t © 1980, b y A c a d e m i c P r e s s , I n c ALL RIGHTS RESERVED NO PART OF TH IS P U B U C A T IO N MAY BE REPRO DU CED OR TRA N SM ITTED IN ANY F O R M OR BY ANY M EA N S, EL EC TR O N IC OR M ECHANICAL, INCLUDJNG PH O TO C O PY , RECORDING, OR ANY IN FO R M ATION STORAGE AND RETRIEVAL SY STE M , W IT H O U T PER M ISSIO N IN W R IT IN G FR O M TH E PUBL1SHER ACADEMIC PRESS, INC S ix t h A v e n u e , S an D i e g o , C a lif or n ia 1 United Kingdom Edition published by ACADEMIC PRESS, INC (LO N D O N ) LTD /2 O v al R o a d , L o n d o n N W 7D X Library of Congress Cataloging in Publication Data Reed, Michael Methods of modern mathematical physics Vol Functional analysis, revised and enlarged edition Includes bibliographical references CONTENTS: v Functional analysis.-v Fourier analysis, self-adjointness.-v Scattering theory.-v Analysis of operators Mathematical physics I Simón, Barry.joint author II Title QC20.R37 1972 530.1’5 75-182650 ISBN -1 - 5 -6 (v 1) AMS (MOS) 1970 Subject Classifications: -0 ,4 - , -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 8 89 91 92 10 www.TheSolutionManual.com M entors, C olleagues, F rien d s This book is the first o f a multivolume series devoted to an exposition of functional analysis methods in modem mathematical physics It describes the funda­ mental principies o f functional analysis and is essentially self-contained, although there are occasional references to later volumes We have included a few applications when we thought that they would provide motivation for the reader Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics, modem physics, and partial differential equations This revised and enlarged edition differs from the first in two major ways First, many coileagues have suggested to us that it would be helpful to include some material on the Fourier transform in Volume I so that this important topic can be conveniently included in a standard functional analysis course using this book Thus, we have included in this edition Sections IX 1, IX 2, and part of IX from Volume II and some additional material, together with relevant notes and problems Secondly, we have included a variety o f supplementary material at the end o f the book Some o f these supplementary sections provide proofs of theorems in Chapters II - I V which were omitted in the first edition W hile these proofs make Chapters I I - I V more self-contained, we still recommend that students with no previous experience with this material consult more elementary texts Other supplementary sections provide expository material to aid the in­ structor and the student (for example, “ Applications of Compact O perators” ) Still other sections introduce and develop new material (for example, “ Minimization o f Functionals ” ) It gives us pleasure to thank many individuáis: The students who took our course in 1970-1971 and especially J E Taylor for constructive comments about the lectures and lecture notes L Gross, T Kato, and especially D Ruelle for reading parts o f the manuscript and for making numerous suggestions and corrections www.TheSolutionManual.com P reface F Armstrong, E Epstein, B Farrell, and H Wertz for excellent typing M Goldberger, E Nelson, M Simón, E Stein, and A Wightman for aid and encouragement M ike R eed B arry S imón www.TheSolutionManual.com April 1980 M athem atics has its roots in num eroiogy, geometry, and physics Since the tim e o f N ew ton, the search for m athem atical m odels for physical phenom ena has been a source o f m athem atical problem s In fact, whole branches o f m athem atics have grown out o f attem pts to analyze particular physical situations An example is the developm ení o f harm onic analysis from F ourier’s work on the heat equation A lthough m athem atics and physics have grown ap art in this century, physics has continued to stim ulate m athem atical research Partially because o f this, the influence o f physics on m athem atics is well understood However, the contributions o f m athem atics to physics are not as well understood Jt is a com m on fallacy to suppose th a t m athem atics is im postant for physics only because it is a useful tool for m aking com putations Actually, m athem atics plays a m ore subtle role which in the long run is m ore im portant W hen a successful m athem atical m odel is created for a physical phenom enon, that is, a model which can be used for accurate com putations and predictions, the m athem atical structure o f the model itself provides a new way o f thinking about the phenom enon Put slightly differently, when a model is successful it is natural to think o f the physical quantities in term s o f the m athem atical objects which represent them and to interpret sim ilar o r secondary phenom ena in term s o f th e sam e model Because o f this, an investigation o f the internal m athem atical structure o f the model can alter and enlarge our understanding o f the physical phenom enon O f course, th e outstanding exam ple o f this is N ew tonian m echanics which provided such a clear and coherent picture o f celestial m otions th at it was used to interpret practically all physical phenom ena The model itself becam e central to an understanding o f the physical world and it was difficult to give it up in the late nineteenth century, even in the face o f contradictory evidence A m ore m odern exam ple o f this influence o f m athem atics on physics is the use o f group theory to classify elem entary particles vii www.TheSolutionManual.com Introduction www.TheSolutionManual.com The analysis o f m athem atical models for physical phenom ena is p art o f the subject m atter o f m athem atical physics By analysis is m eant both the rigorous derivation o f explicit form ulas and investigations of the internal m athem atical structure o f the models In both cases the m athem atical problems which arise iead to m ore general m athem atical questions n ot associated with any particular model A lthough these general questions are som etim es problem s in puré m athem atics, they are usually classified as m athem atical physics since they arise from problem s in physics M athem atical physics has traditionally been concerned with the m athe­ m atics o f classical physics: m echanics, fluid dynam ics, acoustics, potential theory, and optics The m ain m athem atical too! for the study o f these branches o f physics is the theory of ordinary and partiai differential equations and related areas like integral equations and the calculus o f variations This classical m athem atical physics has long been part o f curricula in m athem atics and physics departm ents However, since 1926 the frontiers o f physics have been concentrated increasingly in quantum mechanics and the subjects opened up by the quantum theory: atom ic physics, nuclear physics, solid State physics, elem entary particle physics The central m athem atical discipline for the study o f these branches o f physics is functional analysis, though the theories o f group representations and several complex variables are also im portant Von N eum ann began the analysis o f the fram ew ork o f quantum mechanics in the years foliowing 1926, but there were few attem pts to study the structure o f specific quantum systems (exceptions would be some o f the work o f Friedrichs and Rellich) This situation changed in the early 1950’s when K ato proved the self-adjointness o f atom ic H am iltonians and G árding and W ightm an form ulated the axioms for quantum field theory These events dem onstrated the usefulness o f functional analysis and pointed out the m any difficult m athem atical questions arising in m odern physics Since then the range and breadth o f both the functional analysis techniques used and the subjects discussed in m odern m athem atical physics have increased enorm ously The problem s range from the concrete, for exam ple how to com pute or estím ate the point spectrum o f a particular operator, to the general, for example the rep resen taro n theory o f C “"-algebras The techniques used and the general approach to the subject have becom e m ore abstract A lthough in some areas the physics is so well understood th at the problem s are exercises in puré m athem atics, there are other areas where neither the physics ñor the m athem atical models are well understood These developm ents have had severa! serious effects not the least o f which is the difficulty o f com m unication between m athem aticians and physicists Physicists are often dismayed at the breadth o f background and increasing m athem atical sophistication which are required to understand the models M athem aticians are often frastrated by www.TheSolutionManual.com their own inability to undersíand the physics and the inability o f physicists to form úlate the problem s in a way th at m athem aticians can understand A few specific rem arks are appropriate The prerequisite for reading this volume is roughly the m athem atical sophistication acquired in a typical undergraduate m athem atics education in the U nited States C hapter I is intended as a review o f b ackground m aterial W e expect th a t the reader will have som e acquaintance w ith parts o f the m aterial covered in C hapters II—IV and have occasionally om itted proofs in these chapters when they seem uninspiring and unim portant for the reader The m aterial in this book is sufficient for a two-semester course A lthough we taught m ost o f the m aterial in a special one-semester course at Princeton which m et five days a week, we d o not recom m end a repetition o f that, either for facuity or students In order th at the m aterial may be easily adapted for lectures, we have w ritten most o f the chapters so that the earlier sections contain the basic topics while the iater sections contain more specialized and advanced topics and applications F o r example, one can give students the basic ideas about unbounded operators in nine or ten lectures from Sections 1-4 o f C hapter VIII On the other hand, by doing the details of the proofs and by adding m aterial from the notes and problem s, C hapter VIII could easily become a one-sem ester course by itself Each chapter o f this book ends with a long set o f problem s Some o f the problem s fill gaps in the text (these are m arked with a dagger) O thers develop altérnate proofs to the theorem s in the text or introduce new material We have also included harder problem s (indicated by a star) in order to challenge the reader W e strongly encourage students to the problem s It is trite but true th a t m athem atics is learned by doing it, not by w atching other people it We hope th at these volumes will provide physicists with an access to m odern abstract techniques and th at m athem aticians will benefit by learning the advanced techniques side by side with their applications www.TheSolutionManual.com There is a versión of the first monotone convergence theorem even in the case where Q{t^) is not dense: see Sinion's two papers quoted on page 385 e.g., under the hypotheses of Theorem S.20, The proofs in this section are such that it is easy to verify the uniformity of convergence in the i variable as t runs through compact subsets of [0 ,oo) Kato s strong versión of the product formula is proven in T Kato “ Trotter's Product Formula for an Arbitrary Pair of Self-Adjoint Contraction Semigroups,” in Topics in Functional Analysis (G C Rota, ed), Academic Press, New York, 1978, pp 185-195 Extensión to more than two factors and some nonlinear operators can be found in T Kato and K Masuda, “Trotter's Product Formula for Nonlinear Semigroups Generated by the Subdifferentials of Convex Functionals,” J Math Soc Japan 30 (1978), 169-178 Extensions to include generators of holomorphic semigroups can be found in Kato's paper and a &mí/ofTrotter product formula for unitary groups under the hypotheses of Theorem S.21 is proven in T Ichinose, “ A Product Formula and Its Application to the Schródinger Equation,” to appear Kato actually proves a stronger result than we prove in Theorem S.2I Namely, he proves that whenever A and B are positive self-adjoint operators on a Hilbert space then s-limíe' A>"e~ B/")n = e ‘ P where P is the projection onto Q{A) n Q(B) and C is the obvious operator on Ran P There has been a considerable amount of work on nonlinear product formulas See, for example: H Brezis and A Pazy, “ Semigroups of Nonlinear Contractions on Convex Sets,” J Futu tamal Ana! 5(1970) 237 281; H Brezis and A Pazy, “ Convergence and Approximation of Semigroups of Nonlinear Operators in Banach Spaces,” J Functional Anal (1972), 63-74; Paul R ChernofT, “ Product Formulas, Nonlinear Semigroups, and Addition of Unbounded Operators,” Memoirs of the American Mathematical Society, Number 140; A J Chorin, T J R Hughes, M F McCracken, and J E Marsden, “ Product Formulas and Numérica! Algorithms,” Comm Puré Appl Math.XXXl (1978), 205-256; M G Crandall and T M Liggett, “ Generation of Semi-Groups of Nonlinear Transformations on Genera! Banach Spaces,” Amer J Math 93(1971), 265-298; J Marsden, “ On Product Formulas for Nonlinear Semigroups,” J Functional Anal 13 (1973), 51-72; Eric Schechter, “ Well-Behaved Evolutions and Trotter Products,” Thesis, University of Chicago, 1978; G F Webb, “ Exponential Representation of Solutions to an Abstract Semi-Linear Differential Equation,” Pacific J Math 70 (1977), 269-279; and Fred B Weissler, “ Construction of Non-Linear Semi-Groups Using Product Formulas.” Israel J Math 29 (1978), 265-275 These product formulas play an important role in the existence theorems for certain classes of nonlinear partial differential equations www.TheSolutionManual.com Supplement VI 11.8 Chernoff’s theorem was proven in his J Functional Anal, paper quoted in the Notes to Section VIII.8 He proves it when A is the generator of any contraction senngroup on any Banach space and f(t) any famiiy of contractions (operators with ¡| /'(t)¡| < I) One advantage of the form of his theorem ts that it implies that for many real-valued functions F and G PRO BLEM S Use Theorem S.l to prove Theorems 1.19 and 1.20 in the general a-finite case (a) Let Si = {zj|r| < 1} and define Tr: L2(Sl) L 2(Sl) by (Trf) (z) = f(zr) for < r < Prove that Trf -> / as r -* for any / e L (Hint: Prove it for j e C(Q) and use an s/3 argument.) (b) I f /is analytic in {z||zj < + fijforsomee > ,prove that the Taylor series for/about z = converges to / in L2(0 ) (c) Conclude that {z"}®s=0 's a basis for jtf(Sl) Let (a) (b) (c) K be the Bergmann reproducing kernel for some set Si Prove that |/í(z, vv)| ^ K(z, z)*K{w, w)* Prove that K(z, z) -f K(w, w) — K(z, w) — K(w, z) > Fix z e Si Let d üijiz) = d I +i Xz' yi + Prove that a — {a,j} is a positive definite matrix www.TheSolutionManual.com Additional Suppiement The results of this section and Problem 25 are often called Phragmén -Lindelóf theorems Some of the original papers are: J Hadamard, “Sur les fonctions entiéres,” Bull Soc Math Franee 24 (1896), 186-187; E Phragmén, “Sur une extensión d’un théoréme ciassique de la théorie des fonctions,” Acta Math 28 (1904), 351-368; E Lindelóf and E Phragmén, “ Sur une extensión d’un principe ciassique de l’analyse et sur quelque propriétés des fonctions monogénes dans le voisinage d’un point singuiier.” Acta Math 31 (1908), 381-406; E Lindelóf, “Sur un principe général d’analyscel sesapplications la theoriede la représenlation conforme,” Acta Soc Sci Finn 46(4) (1915), 1-35 For simple proofs see L Ahlfors, “On Phragmén-Lindelóf’s principie,” Trans Amer Math Soc 41 (1937), 1-8 Remark {a,/z)} defines a metric in the sense of Riemann geometry and introduces a natural geometric structure into ÍI For example, if Í2 is the unit disk, the resuiting geometry is one of the standard non-Euclidean geometries (a) Prove that equality holds in (S.8) only if a — b9~ (b) Suppose that H/H, = j|^||4 = Prove that equality in (S.7) holds if and onlyif |