Table Of ContentILBERT SPACE METHODS
....................... QUANTUM MECHANICS
FuNDAMENTAL SciENCES
ILBERT SPACE METHODS
QUANTUM MECHANICS
Werner 0. An1rein
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This book is published under the editorial direction of
Professor Philippe-Andre Martin (EPFL, Lausanne).
is an in1print owned by Presses polytechniques et universitaires romandes, a
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Preface
This text is based on lectures given at the University of Geneva during the period
1994- 2005. These courses were intended for advanced undergraduate students who
had completed a first course in quantum mechanics and were thus expected to be fa
miliar with the physical aspects and the basic 1nathernatical formalism of quantum
theory. Partly due to lack of time, quantum mechanics is often taught with no or little
exposition of the mathematical questions arising through the introduction of infinite
dimensional Hilbert spaces. I hope that the present volume will prove useful, espe
cially to somewhat theoretically minded students, for deepening their knowledge and
understanding of the Hilbert space aspects of quantum mechanics, and prepare them
for reading research papers.
Mostly the lectures were organised as one-year courses (80 hours plus 25 hours of
problem sessions) and covered essentially the contents of Chapters 1 -5 and parts of
Chapters 6 or 7. To offer a few more applications, some 1naterial has been added to the
original lecture notes (Sections 5.8, 6.6, 6.7, 7.2 and 7.5). Of course a strict selection
of topics and applications to be treated had to be made from the outset. The emphasis is
placed on a certain number of basic mathematical techniques, usually without striving
for the most general results. For example there is no discussion of quadratic forms,
and we have avoided the use of techniques from stochastic analysis. However we give
essentially complete proofs for all results involving Hilbert space objects. Some of
these proofs are collected in appendices to the various chapters. Son1e acquaintance
with measure theory is required: the essential facts are explained without detailed
proofs.
Chapter 1 gives the basic properties of Hilbert space and a description of the neces
sary material from measure theory. In Chapter 2 we present various classes of bounded
linear operators and general notions on unbounded operators, including the invariance
of self-adjointness under a class of perturbations. The problem of self-adjointness is
further investigated in Chapter 3 which contains the theory of extensions of symmetric
operators, with applications to Sturm-Liouville and Schrodinger operators. Chapter 4
deals with the spectral theory of self-adjoint operators, in particular with the spectral
theorem and with the various spectral types. In Chapter 5 we prove Stone's Theo
rem and then discuss the fundamental aspects of scattering theory: scattering states,
asymptotic condition and wave operators, S-matrix, scattering cross sections. Chapter
6 is devoted to the Mourre method for controlling the resolvent of self-adjoint oper
ators near the real axis and to implications for their spectrum. In the final Chapter 7
PREFACE
we present stationary-state scattering theory and various applications of the results of
Chapter 6: asymptotic completeness, properties of the S-matrix, time delay and the
Flux-Across-Surfaces Theorem.
Each chapter ends with some bibliographical notes and a selection of problems.
The bibliography consists mostly of books. They are referred to for alternative or more
advanced presentations of some material or for certain points not treated in the present
text. A very small number of original and review papers are cited for those interested
a deeper understanding of certain topics treated in the text. In combination with
tnodern electronic means. these papers can also be useful as a basis for searching in
the vast literature.
The majority of the problems have been tested in class-room sessions. Many of
then1 are meant to help students to become familiar with the concepts discussed in the
main body; son1e require a more detailed study of technical aspects of the text. A few
of the more difficult problems are provided with a hint for the solution.
As regards notations, it should be pointed out that some symbols have more than
one 1neaning in the text. In particular: the symbol II · II is used for various norms, the
Greek letter for spectra and for scattering cross sections, the letter P for projections
CJ
and for mon1entum, and the letter R for resolvents and for S - I, where S is the
scattering operator. For the convenience of the reader we have included a Notation
Index (page 385). Constants are often denoted generically by c, but in some proofs
different constants are numbered as c c etc.
1, 2,
is a great pleasure to thank all those who helped me in one way or another during
the preparation of the lectures or of this volume. The problems in the text are mostly
due to Marius Mantoiu, Joachim Stubbe and Rafael Tiedra de Aldecoa; they assumed
the responsibility for the problem sessions with much competence and devotion. The
continual interest of my students and their constructive comments on earlier versions
of the text have influenced a considerable number of details. I received precious sup
port from Philippe Jacquet, Andreas Malaspinas, Peter Wittwer and Luis Zuleta for
coming to grips with TEX -related difficulties. I thank Philippe Martin for proposing
the publication of my lectures, the referee for pointing out some errors and for useful
suggestions, and Fred Fenter from EPFL Pre,ss for advice and his very efficient man
agement of the publishing process. am indebted to the Physics Department
Finally~'
of the University of Geneva for its kind hospitality after my retirement.
Werner Amrein
Geneva, Switzerland
December 2008
Contents
v
1 Hilbert Spaces 1
1.1 Definition and elementary properties 1
1.2 Vector-valued functions . . . . . . . . 6
1.3 Subsets and dual of a Hilbert space . . 8
1.4 Measures, integrals and LP spaces 14
Problems ............ . 25
2 Linear Operators 27
2.1 The algebra B(H) ..... . 27
2.2 Projections and isometries 37
2.3 Compact operators . . . . 41
2.4 Unbounded operators .... 47
2.5 Multiplication operators . . . . . . 56
2.6 Resolvent and spectrum of an operator .. 67
2. 7 Perturbations of self-adjoint operators 7
Appendix ... . 80
Problems ................ . 82
3 Symmetric Operators and their Extensions 87
3.1 The method of the Cay ley transform . . 87
8 ....
3.2 Differential operators with constant coefficients 96
3.3 Schrodinger operators . . . . . . . 111
Appendix 126
Problems ...... . 131
4 Spectral Theory of Self-Adjoint Operators 133
4.1 Stieltjes measures . . . . . . . . . . . . . 133
4.2 Spectral measures . . . . . . . . . . . . . 141
4.3 Spectral parts of a self-adjoint operator . 157
4.4 The spectral theorem. The resolvent near the spectrum . . . . 170
0 •••
Appendix: Proof of the Spectral Theorem .... 179
Problems ........................ . 190
CONTENTS
Vlll
5 Evolution Groups and Scattering Theory 193
5.1 Evolution groups 193
5.2 Characterisation of the scattering states .. 204
5.3 Asymptotic condition. Wave operators . 213
5.4 Simple scattering systems. Scattering operator . . 218
5.5 Scattering operator and S-matrix . 226
5.6 Scattering cross sections 235
5.7 Bounds on scattering cross sections 243
5.8 Coulomb scattering .. 252
Problems .. . .. 260
6 The Conjugate Operator Method 263
6.1 A simple example . 264
6.2 The method of differential inequalities . 271
6.3 The Mourre inequality 283
6.4 Application to Schrodinger operators .. 286
6.5 Relatively smooth operators 292
6.6 Higher order resolvent estimates 300
6.7 Some comtnutators .. 306
Appendix: Interpolation of operators 310
Problems 315
7 Further Topics in Scattering Theory 317
7.1 Asymptotic completeness . 317
7.2 Flux and scattering into cones 324
7.3 Time-independent scattering theory ... 343
7.4 The scattering matrix 352
7.5 Time delay .. 364
Appendix 374
Problems 379
References I 381
Notation Index 385
Subject Index 389
CHAPTER!
Hilbert Spaces
Hilbert space sets the stage for standard quantum theory: the pure states of a physical
system are identified with the unit rays of a Hilbert space 1{ and observables with self
adjoint operators acting in H. In this initial chapter we present the essential concepts
and prove the basic results concerning separable Hilbert spaces (Sections 1.1 - 1.3 ). In
Section 1.4 we then introduce £2 spaces, which are of special importance for quantum
mechanics. This requires some familiarity with measure theory, and we include a short
description of the necessary concepts from this theory.
1.1 Definition and elementary properties
Throughout this text a Hilbert space 1neans a co1nplex linear vector space,
equipped with a Hermitian scalar product, which is complete and admits a countable
basis. More precisely a (separable) Hilbert space His defined by the four postulates
(HI)- (H4) stated below:
(Hl) H is a linear vector space over the field CC o.f co1nplex nu1nbers:
With each couple {f, g} of elements of H there is associated another element of H,
denoted f + g, and with each couple {a, f}, a E CC, f E H, there is associated an ele
ment af ofH, and these associations have the following properties (where f, hE 1{
and a, {3 E CC):
f+g==g+f f + (g + h) == (f +g) + h (1.1)
a(f+g)==af+ag (a+(3)f-af+(3f (1.2)
a({3f) == (a{3)f If- f. (1.3)
Furthermore there exists a unique eletnent 0 E H (called the zero vector) such that 1
O+f==f Of== 0 \If E H. (1.4)
1 Here 0 denotes the complex number a = 0.
