The Schrödinger Equation

572 Pages·1991·55.98 MB·English

by  M. A. Shubin (auth.), F. A. Berezin

The Schrodinger Equation
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV. MGU, Moscow, U.S.S.R.
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
N. N. MOISEEV, Computing Centre, Academy ofSciences, Moscow, U.S.S.R.
S. P. NOVIKOV, Landau Institute ofTheoretical Physics, Moscow, U.S.S.R.
M. C. POLYVANOV. Steklov Institute of Mathematics, Moscow, U.S.S.R.
Yu. A. ROZANOV, Steklol' Institute ofMathematics, Moscow, U.S.S.R.
Volume 66
The Schr6dinger Equation
by
F. A. Berezint
and
M.A. Shubin
Center for Optimization and Mathematical Modelling,
Institute ofN ew Technologies,
Moscow, U.S.S.R.
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Berezin. F. A. (Feliks Aleksanarovich)
The Schroainger equation I by F.A. Berezin ana M.A. Shubin with
the assistance of G.L. Litvinov and O.A. Leites.
p. cm. -- (Mathematics ana its applications (Soviet series)
v. 66)
Incluaes bibliographical references and index.
ISBN 978-94-010-5391-4 ISBN 978-94-011-3154-4 (eBook)
DOI 10.1007/978-94-011-3154-4
1. Schrodinger equation. 1. Shubin. M. A. (Mikhail
Aleksanarovichl. 1944- II. Title. III. Series: Mathematics and
its applications (Kluwer Academie Publishers). Soviet series ; 66.
QCI74.26.W28B45 1991
530. 1 '24--dc20 91-11946
ISBN 978-94-010-5391-4
Printed on acid-free paper
This English edition is a revised, expanded version of the original Soviet publication.
This is the translation of the work
YPABHEHI1E lllPE.nI1HfEPA
Published by the Moscow State University, Moscow, © 1983.
Translated from the Russian by Yu. Rajabov, D. A. Leites and N. A. Sakharova
AII Rights Reserved
© 1991 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1991
Softcover reprint ofthe hardcover Ist edition 1991
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
inc\uding photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
SERIES EDITOR'S PREFACE
4Et moi, ..., si j'avait su comment en revenir, One service mathematics has rendered the
je n'y serais point alle.' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labelled 'discarded non·
The series is divergent; therefore we may be sense'.
able to do something with it. Eric T. Bell
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non-
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com-
puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And
all statements obtainable this way form part of the raison d'€tre of this series.
This series, Mathematics and Its ApplicatiOns, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu-
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
vi SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non-
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci-
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com-
puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre-
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub-
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis-
cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
.. influences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
That the SchrOdinger equation is central to all of quantum mechanics is nothing new. That it is
mathematically a rich and complicated equation also not. In fact it better be, as, in a way, it is the
equation of everything microphysical. That makes writing about it, and about quantum mechanics,
difficult, and virtually all books on the topic sacrifice mathematical rigour and, especially, precise
statements. This is distressing to mathematicians and puts them off; it makes it difficult for
mathematicians to feel at home in the quantum world. This book by the late FA Berezin, the origi-
nator of a great many ideas in supersymmetry and second quantization and top analyst M.A. Shu-
bin is an exception. Wherever possible the utmost mathematical precision is used. I have to say
'wherever possible' because there still are parts where more research is needed to make things
rigorous and mathematically satisfactory. A foremost example of that is the theory of one of the
central tools, the Feynman path integral (to which a large chapter is devoted). Here, it is nice to
note that there has been a great deal of progress recently based on T. Hida's white noise analysis
(infinite-dimensional stochastic calculus). A great deal of sophisticated mathematics gets involved
when one takes the Scbrodinger equation seriously, and, as the book starts at the graduate student
level and ends with the most modem developments such as supersymmetry and supermanifolds, it
has become a rather large volume. It will take time to study it completely but for those who desire
to feel comfortable in the quantum world that time will be an optimal investment.
The shortest path between two truths in the Never lend books, for no one ever returns
real domain passes through the complex them; the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole FlllJ1ce
La physique ne nous donne pas seulement The function of an expert is not to be more
l'occasion de r.:soudre des problemes ... eIle right than other people, but to be wrong for
nous fOO t pressentir la solution. more sophisticated reasons.
H. Poincare David Butler
Amsterdam, 6 March 1991 Michiel Hazewinkel
Contents
Series Editor's Preface . v
Foreword . . . . . . . Xlll
Notational Conventions . xviii
CHAPTER 1. General Concepts of Quantum Mechanics 1
Introduction . . . . . . . . . . . . . . 1
1.1. Formulation of Basic Postulates . . . . 4
1.2. Some Corollaries of the Basic Postulates 8
1.3. Time Differentiation of Observables . . 14
1.4. Quantization . . . . . . . . . . . . 17
1.5. The Uncertainty Relations and Simultaneous Measurability of
Physical Quantities . . . . . . . . . . . . 23
1.6. The Free Particle in Three-Dimensional Space 27
1.7. Particles with Spin 30
1.8. Harmonic Oscillator . 33
1.9. Identical Particles . . 40
1.10. Second Quantization . 44
CHAPTER 2. The One-Dimensional Schrodinger Equation . 50
2.1. Self-Adjointness . . . . . . . . . . . . . . . . . . . 50
2.2. An Estimate of the Growth of Generalized Eigenfunctions 55
2.3. The Schrodinger Operator with Increasing Potential . . . 57
1. Discreteness of spectrum (57). 2. Comparison theorems and the
behaviour of eigenfunctions as x - 00. (59). 3. Theorems on zeros of
eigenfunctions (64).
2.4. On the Asymptotic Behaviour of Solutions of Certain Second-Order
Differential Equations as x - 00 .. . . . . . . . . . . . .. 69
1. The case of integrable potential (70). 2. Liouville's transformation
and operators with non-integrable potential (81).
Vill CONTENTS
2.5. On Discrete Energy Levels of an Operator with Semi-Bounded
Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
1 The operator in a half-axis with Dirichlet's boundary condition (87).
2. The case of an operator on the half-axis with the Neumann bound-
ary condition (94). 3. The case of an operator on the whole axis (97).
2.6. Eigenfunction Expansion for Operators with Decaying Potentials 99
1. Preliminary remarks (99). 2. Formulation of the main theorem
(102). 3. Two proofs of Theorem 6.1. (102). 4. One-dimensional oper-
ator obtained from the radially symmetric three-dimensional operator
(116). 5. The case of an operator on the whole axis (122).
2.7. The Inverse Problem of Scattering Theory . . . . . . . . . . . 126
1. Inverse problem on the half-axis (127). 2. Inverse problem on the
whole axis (131).
2.8. Operator with Periodic Potential . . . . . . . . . . . . . . . 134
1. Bloch functions and the band structure of the spectrum (134).
2. Expansion into Bloch eigenfunctions (141). 3. The density of states
(145).
CHAPTER 3. The Multidimensional Schrodinger Equation 150
3.1. Self-Adjointness. . . . . . . . . . . . . . 150
3.2. An Estimate of the Generalized Eigenfunctions . . . . . . 160
3.3. Discrete Spectrum and Decay of Eigenfunctions . . . . . . 164
1. Discreteness of spectrum (165). 2. Decay of eigenfunctions (167).
3. Non-degeneracy of the ground state and positiveness of the first
eigenfunction (177). 4. On the zeros of eigenfunctions (180).
3.4. The Schrodinger Operator with Decaying Potential: Essential Spec-
trum and Eigenvalues . . . . . . . . . . . . . . . . . . . . 181
1. Essential spectrum (182). 2. Separation of variables in the case of
spherically symmetric potential and the Laplace-Beltrami operator on
a sphere (183). 3. Estimation of the number of negative eigenvalues
(189).4. Absence of positive eigenvalues (191).
3.5. The Schrodinger Operator with Periodic Potential . . . . . . . . 198
1. Lattices (198). 2. Bloch functions (200). 3. Expansion in Bloch func-
tions (204). 4. Band functions and the band structure of the spectrum
(208). 5. Theorem on eigenfunction expansion (213). 6. Non-triviality
of band functions and the absence of a point spectrum (216). 7. Den-
sity of states (220).
CONTENTS ix
CHAPTER 4. Scattering Theory. . . . . . . . . . . . . . . . . 223
4.1. The Wave Operators and the Scattering Operator . . . . . . . . 223
1. The basic definitions and the statement of the problem (223).
2. Physical interpretation (225). 3. Properties of the wave operators
(226). 4. The invariance principle and the abstract conditions for the
existence and completeness of the wave operators (230).
4.2. Existence and Completeness of the Wave Operators . . . . . . . 233
1. The abstract scheme of Enss (233). 2. The case of the Schrodinger
operator (242). 3. The scattering matrix (249). 4. One-dimensional
case (252). 5. Spherically symmetric case (256).
4.3. The Lippman-Schwinger Equations and the Asymptotics of Eigen-
functions . . . . . . . . . . . . . . . . . . . . . . . . . . 259
1. A derivation of the Lippman-Schwinger equations (259). 2. Another
derivation of the Lippman-Schwinger equations (262). 3. An outline
of the proof of the completeness of wave operators by the station-
ary method (265). 4. Discussion on the Lippman-Schwinger equation
(271). 5. Asymptotics of eigenfunctions (279).
CHAPTER 5. Symbols of Operators and Feynman Path Integrals 282
5.1. Symbols of Operators and Quantization: qp- and pq-Symbols and
Weyl Symbols 282
1. The general concept of symbol and its connection with quantization
(282).2. The qp- and pq-symbols (285). 3. Symmetric or Weyl symbols
(294). 4. Weyl symbols and linear canonical transformations (300).
5. Weyl symbols and reflections (302).
5.2. Wick and Anti-Wick Symbols. Covariant and Contravariant Symbols 304
1. Annihilation and creation operators. Fock space (304). 2. Definition
and elementary properties of Wick and Anti-Wick symbols (307).
3. Covariant and contravariant symbols (316). 4. Convexity inequali-
ties and Feynman-type inequalities (321).
5.3. The General Concept of Feynman Path Integral in Phase Space.
Symbols of the Evolution Operator . . . . . . . . . . . . . . 324
1. The method of Feynman Path integrals (324). 2. Weyl symbol of
the evolution operator (328). 3. The Wick symbol of the evolution
operator (345). 4. pq- and qp-symbols of the evolution operator and
the path integral for matrix elements (357).
5.4. Path Integrals for the Symbol of the Scattering Operator and for the
Partition Function 361
x CONTENTS
1. Path integral for the symbol of the scattering operator (361).
2. The path integral for the partition function (370).
5.5. The Connection between Quantum and Classical Mechanics. Semi-
classical Asymptotics . . . . . . . . . . . . . . . . . . . . 374
1. The concept of a semiclassical asymptotic (374). 2. The operator
initial-value problem (374).3. Asymptotics of the Green's function (377).
4. Asymptotic behaviour of eigenvalues (381). 5. Bohr's formula (383).
SUPPLEMENT 1. Spectral Theory of Operators in Hilbert Space 386
S1.1. Operators in Hilbert Space. The Spectral Theorem . . . . . . . 386
1. Preliminaries (386). 2. Theorem on the spectral decomposition of a
self-adjoint operator in a separable Hilbert space (392). 3. Examples
and exercises (406). 4. Commuting self-adjoint operators in Hilbert
space, operators with simple spectrum (407). 5. Functions of self-
adjoint operators (411). 6. One-parameter groups of unitary operators
(414).7. Operators with simple spectrum (415). 8. The classification
of spectra (416). 9. Problems and exercises (418).
S1.2. Generalized Eigenfunctions . . . . . . . . . . . . . . . . . 419
1. Preliminary remarks (419). 2. Hilbert-Schmidt operators (420).
3. Rigged Hilbert spaces (423). 4. Generalized eigenfunctions (426).
5. Statement and proof of main theorem (429). 6. Appendix to the
main theorem (430). 7. Generalized eigenfunctions of differential op-
erators (431).
S1.3. Variational Principles and Perturbation Theory for a Discrete
Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 434
S1.4. Trace Class Operators and the Trace . . . . . . . . . . . . . 448
1. Definition and main properties (448). 2. Polar decomposition of an
operator (451). 3. Trace norm (453).4. Expressing the trace in terms
of the kernel of the operator (457).
S1.5. Tensor Products of Hilbert Spaces . . . . . . . . . 462
SUPPLEMENT 2. Sobolev Spaces and Elliptic Equations 466
S2.1. Sobolev Spaces and Embedding Theorems. . . . . . 466
S2.2. Regularity of Solutions of Elliptic Equations and a priori Estimates 475
S2.3. Singularities of Green's Functions . . . . . . . 480
SUPPLEMENT 3. Quantization and Supermanifolds 483
S3.1. Supermanifolds: Recapitulations . . . . . . . . 486