Mathematical Results in Quantum Mechanics: International Conference in Blossin (Germany), May 17–21, 1993

345 Pages·1994·6.638 MB·English

by  M. Sh. Birman (auth.), M. Demuth, P. Exner, H. Neidhardt, V. Zagrebnov (eds.)

Operator Theory
Advances and Applications
Vol. 70
Editor
I. Gohberg
Editorial Office: T. Kailath (Stanford)
School of Mathematical H.G. Kaper (Argonne)
Sciences S.T. Kuroda (Tokyo)
Tel Aviv University P. Lancaster (Calgary)
Ramat Aviv, Israel L.E. Lerer (Haifa)
E. Meister (Darmstadt)
Editorial Board: B. Mityagin (Columbus)
J. Arazy (Haifa) V.V. Peller (Manhattan, Kansas)
A. Atzmon (Tel Aviv) J.D. Pincus (Stony Brook)
J.A. Ball (Blackburg) M. Rosenblum (Charlottesville)
A. Ben-Artzi (Tel Aviv) J. Rovnyak (Charlottesville)
H. Bercovici (Bloomington) D.E. Sarason (Berkeley)
A. Bottcher (Chemnitz) H. Upmeier (Lawrence)
L. de Branges (West Lafayette) S.M. Verduyn-Lunel (Amsterdam)
K. Clancey (Athens, USA) D. Voiculescu (Berkeley)
L.A. Coburn (Buffalo) H. Widom (Santa Cruz)
K.R. Davidson (Waterloo, Ontario) D. Xia (Nashville)
R.G. Douglas (Stony Brook) D. Yafaev (Rennes)
H. Dym (Rehovot)
A. Dynin (Columbus)
P.A. Fillmore (Halifax) Honorary and Advisory
C. Foias (Bloomington) Editorial Board:
P.A. Fuhrmann (Beer Sheva) P.R. Halmos (Santa Clara)
S. Goldberg (College Park) T. Kato (Berkeley)
B. Gramsch (Mainz) P.O. Lax (New York)
G. Heinig (Chemnitz) M.S. Livsic (Beer Sheva)
J.A. Helton (La Jolla) R. Phillips (Stanford)
M.A. Kaashoek (Amsterdam) B. Sz.-Nagy (Szeged)
Mathematical Results
in Quantum Mechanics
International Conference in Blossin (Germany), May 17-21, 1993
Edited by
M. Demuth
P. Exner
H. Neidhardt
V. Zagrebnov
Birkhauser Verlag
Basel . Boston . Berlin
Editors
M.Demuth P.Exner
Technische Universitlit Clausthal Laboratory of Theoretical Physics
Institut fiir Mathematik Joint Institute for Nuclear Research
Erzstrasse 1 Head Post Office Box 79
D-38678 Clausthal-Zellerfeld Moscow
Germany Russia
H. Neidhardt V. Zagrebnov
Fachbereich Mathematik MA 7-2 Universite d' Aix-Marseille n et
Technische Universitiit Berlin Centre de Physique Theorique
Strasse des 17. Juni 136 CNRS-Luminy-Case~
D-I0623 Berlin F-13288 Marseille Cedex 9
Germany France
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Mathematical results in quantum mechanics: international
conference in Blossin (Germany), May 17 - 21,1993/ ed. by M.
Demuth ... - Basel; Boston; Berlin: Birkhiiuser, 1994
(Operator theory; Vol. 70)
ISBN 3-7643-5025-3 (Basel ... )
ISBN 0-8176-5025-3 (Boston)
NE: Demuth, Michael [Hrsg.]
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on
microfilms or in other ways, and storage in data banks. For any kind of use the permission of the copyright holder
must be obtained.
© 1994 Birkhiiuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland
Printed on acid-free paper produced from chlorine-free pulp
Cover design: Heinz Hiltbrunner, Basel
ISBN 3-7643-5025-3
ISBN 0-8176-5025-3
Contents
Preface................................................................. ix
1 Schrodinger and Dirac operators
M.Sh. Birman: Discrete spectrum of the periodic Schrodinger
operator for non-negative perturbations ......................... 3
M.Sh. Birman, T. Weidl: The discrete spectrum in a gap of the
continuous one for compact supported perturbations ............. 9
R. Hempel, J. Laitenberger: Schrodinger operators with strong local
magnetic perturbations: Existence of eigenvalues in gaps of the
essential spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
M. Hoffmann-Ostenhof, T.Hoffmann-Ostenhoff, N. Nadirashvili:
Regularity of the nodal sets of solutions to Schrodinger
equations ...................................................... 19
G. Stolz: Results in the spectral theory of Schrodinger operators with
wide potential barriers ......................................... 27
V. Grecchi, M. Maioli, A. Sacchetti: Stark ladders and
perturbation theory ............................................. 33
M. Znojil: Singular potentials: algebraization ......................... 37
Y. Saito: Asymptotic behavior of the resolvent of
the Dirac operator ............................................. 45
M.Sh. Birman, A. Laptev: Discrete spectrum of the perturbed
Dirac operator ................................................. 55
2 Generalized Schrodinger operators
R. Hempel, J. Voigt: The spectrum of Schrodinger operators
in Lp(Rd) and in Co(Rd) ...................................... 63
J.F. Brasche: On spectral properties of generalized
Schrodinger operators .......................................... 73
J-P. Antoine, P. Exner, P. 20 Seba, J. Shabani: A Fermi-type rule
for contact embedded-eigenvalue perturbations .................. 79
VI Contents
P. Duclos, B. Meller: A simple model for predissociation ............. 89
P. Stovicek: Scattering on several solenoids .......................... 107
M. Klein: Hall conductance of Riemann surfaces..................... 113
3 Stochastic spectral analysis
M. Demuth, J. van Casteren: Framework and results of stochastic
spectral analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. 123
M. Demuth, W. Kirsch, I. McGillivray: Occupation time asymptotics
to the decay of eigenfunctions .................................. 133
E.M. Ouhabaz: Holomorphic semigroups and Schrodinger
equations ...................................................... 137
V.A. Liskevich, Yu.A. Semenov: Some problems on submarkovian
semigroups ..................................................... 143
V.A. Liskevich: Smoothness estimates and uniqueness for
the Dirichlet operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 149
P. Stollmann: Trace ideal properties of perturbed Dirichlet
semigroups ............... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 153
4 Many-body problems and statistical physics
K.B. Sinha: Quantum dynamical semigroups ......................... 161
A.M. Khorunzhy, L.A. Pastur: Limits of infinite order,
dimensionality or number of components for random
finite-difference operators ...................................... 171
N. Macris, Ph. Martin: Atoms at finite density and temperature and
the spectra of reduced density matrices. . . . . . . . . . . . . . . . . . . . . . . . .. 191
A. Verbeure, V.A. Zagrebnov: Quantum fluctuations in the
many-body problem .......................................... . .. 207
D. Va. Petrina: General Hamiltonians and model Hamiltonians....... 213
R. Gielerak, A.L. Rebenko: Poisson fields representations in the
statistical mechanics of conitinuous systems .................... 219
M. Fannes: Exact ground states for quantum spin chains ............. 227
M. Hiibner, H. Spohn: The spectrum of the spin-boson model. . . . . . . .. 233
H. Lange, B.V. Toomire, P.F. Zweifel: A survey of Wigner-Poisson
problems ....................................................... 239
Contents VB
5 Chaos
J. Dittrich, P. Duclos, P. Seba: Classical d'Alembert field in an
one-dimensional pulsating region ............................... 259
P. Seba: Irregular scattering in one-dimensional periodically
driven systems ................................................. 263
K. Zyczkowski: Relatively random unitary operators . . . . . . . . . . . . . . . . .. 277
6 Operator theory and its application
D. Robert: A trace formula for obstacles problems and
applications .................................................... 283
F. Bentosela: Propagation in irregular optic fibres .................... 293
H. Neidhardt, V.A. Zagrebnov: Singular perturbations, regularization
and extension theory ........................................... 299
G. Nenciu: Adiabatic reduction theory. Semiclassical S-matrix for
N-state one-dimensional systems ............................... 307
V. Buslaev, A. Fedotov: The functional structure of the
monodromy matrix for Harper's equation....................... 321
A.Yu. Konstantinov: Eigenfunction expansion of right definite
multiparameter problems ....................................... 343
V. Koshmanenko: Singularly perturbed operators..................... 347
A.N. Kochubei: Some problems of p-adic quantum theory ............ 353
Preface
The last decades have demonstrated that quantum mechanics is an inexhaustible
source of inspiration for contemporary mathematical physics. Of course, it seems
to be hardly surprising if one casts a glance toward the history of the subject;
recall the pioneering works of von Neumann, Weyl, Kato and their followers which
pushed forward some of the classical mathematical disciplines: functional analysis,
differential equations, group theory, etc. On the other hand, the evident powerful
feedback changed the face of the "naive" quantum physics. It created a contem
porary quantum mechanics, the mathematical problems of which now constitute
the backbone of mathematical physics. The mathematical and physical aspects
of these problems cannot be separated, even if one may not share the opinion of
Hilbert who rigorously denied differences between pure and applied mathemat
ics, and the fruitful oscilllation between the two creates a powerful stimulus for
development of mathematical physics.
The International Conference on Mathematical Results in Quantum Mechan
ics, held in Blossin (near Berlin), May 17-21, 1993, was the fifth in the series of
meetings started in Dubna (in the former USSR) in 1987, which were dedicated
to mathematical problems of quantum mechanics. A primary motivation of any
meeting is certainly to facilitate an exchange of ideas, but there also other goals.
The first meeting and those that followed (Dubna, 1988; Dubna, 1989; Liblice (in
the Czech Republic), 1990) were aimed, in particular, at paving ways to East-West
contacts. The most recent conference in Blossin was organized after a three year
period during which the old barriers were removed completely. There are, how
ever, other challenges which have nothing to do with the vagaries of politics and
geography: in a period of high specialization in scientific thought scientists with
a different bent of mind should be gathered under the same "roof" emphasizing
similarities, convergences and analogies between ideas they are advocating in their
fields of research. For us this "roof" was the Mathematical Results in Quantum
Mechanics conference.
The proceedings start with lectures devoted to the traditional core of the
Quantum Mechanics - "Schrodinger and Dirac Operators". They touch on spec
tral problems, asymptotic behaviour of the resolvents, singular potentials and
other topics. Following naturally from these subjects is the section "Generalized
Schrodinger Operators" wherein lectures from less traditional fields have been col-
x Preface
lected: the quantum Hall effect, contact perturbations, predissociation, etc. In the
section "Stochastic Spectral Analysis" readers will find results strongly motivated
by Quantum Mechanics: decay of eigenfunctions, Dirichlet operators and semi
groups and others. The section "Many-Body-Problems and Statistical Physics"
contains lectures related to the problems of quantum statistical mechanics, in
cluding the spectra of reduced density matrices, macroscopic quantum fluctua
tions, ground-states of the quantum spin chains and spectrum of the spin-boson
model etc. Lectures on delicate problems of the quantum evolution irregularities
are collected in the section "Chaos". The last section, "Operator Theory and Its
Applications", was reserved for lectures motivated by different types of mathe
matical observations with roots in Quantum Mechanics such as trace formulas for
obstacle problems, self-adjoint extensions and singular perturbations, adiabatic
reduction theory or p-adic quantum theory.
We hope that the broad areas covered by these proceedings may give read
ers an impression of the contemporary situation at the intersection of quantum
mechanics and mathematical physics at least from the point of view of a not in
considerable part of the community working in this field.
We would like to thank the Deutsche Forschungsgemeinschaft, Sonderfor
schungsbereich 288 and the Max-Planck-Gesellschaft for their financial support
which made the conference possible. We want to stress, in particular, that it was
this support that allowed the scientists of the former USSR to participate.
November 1993. The Editors
Chapter 1
Schrodinger and Dirac
operators