Table Of ContentTexts and
Monographs
in Physics
W. Beiglbock
J. L. Birman
R. P. Geroch
E. H. Lieb
T. Regge
W. Thirring
Series Editors
Texts and Monographs in Physics
S. Albeverio. F. Gesztesy. R. H0egh-Krohn. and H. Holden: Solvable Models in
Quantum Mechanics (1988).
R. Bass: Nuclear Reactions with Heavy Ions (1980).
A. Bohm: Quantum Mechanics: Foundations and Applications, Second Edition
(1986).
O. Bratelli and D.W. Robinson: Operator Algebras and Quantum Statistical
Mechanics. Volume I: C*- and W*-Algebras. Symmetry Groups. Decomposition
of States (1979). Volume II: Equilibrium States. Models in Quantum Statistical
Mechanics (1981).
K. Chadan and P.C. Sabatier: Inverse Problems in Quantum Scattering Theory
(1977).
M. Chaichian and N.F. Nelipa: Introduction to Gauge Field Theories (1984).
G. Gallavotti: The Elements of Mechanics (1983).
W. Glockle: The Quantum Mechanical Few-Body Problem (1983).
W. Greiner, B. MUlier. and J. Rafelski: Quantum Electrodynamics of Strong Fields
(1985).
J.M. Jauch and F. Rohrlich: The Theory of Photons and Electrons: The Relativistic
Quantum Field Theory of Charged Particles with Spin One-half, Second
Expanded Edition (1980).
J. Kessler: Polarized Electrons (1976). Out of print. (Second Edition available as
Springer Series in Atoms and Plasmas, Vol. I.)
G. Ludwig: Foundations of Quantum Mechanics I (1983).
G. Ludwig: Foundations of Quantum Mechanics II (1985).
R.G. Newton: Scattering Theory of Waves and Particles, Second Edition (1982).
A. Perelomov: Generalized Coherent States and Their Applications (1986).
H. Pilkuhn: Relativistic Particle Physics (1979).
R.D. Richtmyer: Principles of Advanced Mathematical Physics. Volume I (1978).
Volume II (1981).
W. Rindler: Essential Relativity: Special, General, and Cosmological, Revised
Second Edition (1980).
P. Ring and P. Schuck: The Nuclear Many-Body Problem (1980).
R.M. Santilli: Foundations of Theoretical Mechanics. Volume I: The Inverse
Problem in Newtonian Mechanics (1978). Volume II: Birkhoffian Generalization
of Hamiltonian Mechanics (1983).
M.D. Scadron: Advanced Quantum Theory and Its Applications Through
Feynman Diagrams (1979).
N. Straumann: General Relativity and Relativistic Astrophysics (1984).
C. Truesdell and S. Bharatha: The Concepts and Logic of Classical
Thermodynamics as a Theory of Heat Engines: Rigourously Constructed
upon the Foundation Laid by S. Carnot and F. Reech (1977).
F.J. Ynduniin: Quantum Chromodynamics: An Introduction to the Theory of
Quarks and Gluons (\983). '
Sergio Albeverio Friedrich Gesztesy
Raphael H0egh-Krohn Helge Holden
Solvable Models in
Quantum Mechanics
With 51 Illustrations
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo
Sergio Albeverio Friedrich Gesztesy
Faculty of Mathematics and Institute for Theoretical Physics
Bielefeld-Bochum Research Center University of Graz
Ruhr-Universitiit Bochum A-SOlO Graz
D-4630 Bochum 1 Austria
West Germany
Raphael H"egh-Krohn Helge Holden
Institute of Mathematics Institute of Mathematics
University of Oslo University of Trondheim
N-0316 Oslo 3 N-7034 Trondheim-NTH
Norway Norway
Series Editors
Wolf Beiglbock Joseph L. Birman Robert P. Geroch
Institut flir Angewandte Department of Physics Enrico Fermi Institute
Mathematik The City College of the University of Chicago
Universitiit Heidelberg City University of Chicago, IL 60637
D-6900 Heidelberg 1 New York U.S.A.
Federal Republic of New York, NY 10031
Germany U.S.A.
Elliott H. Lieb Tullio Regge Walter Thirring
Department of Physics Istituto de Fisica Teorica Institut fiir Theoretische Physik
Joseph Henry Laboratories Universita di Torino der Universitiit Wien
Princeton University 1-10125 Torino A-1090Wien
Princeton, NJ 08540 Italy Austria
U.S.A.
Library of Congress Cataloging in Publication Data
Solvable models in quantum mechanics.
(Texts and monographs in physics)
Bibliography: p.
Includes index.
1. Quantum theory-Mathematical models.
I. Albeverio, Sergio. II. Series.
QC174.12.S65 1988 530.1'2 87-12685
© 1988 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1988
All rights reserved. This work may not be translated or copied in whole or in part without the
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10010, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. Use
in connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
Typeset by Asco Trade Typesetting Ltd., Hong Kong.
987 6 5 4 3 2 1
ISBN 978-3-642-88203-6 ISBN 978-3-642-88201-2 (eBook)
DOl 10.1007/978-3-642-88201-2
e
"La filosofia scritta in questo grandissimo libro che continuamente ci sta aperto
innanzi a gli occhi (io dico l'universo), rna non si puo intendere se prima non
e e
s'impara a intender la lingua, e conoscer i canitteri, ne' quali scritto. Egli scritto
in lingua matematica, e i canitteri son triangoli, cerchi, ed altre figure geometriche,
e e
senza i quaJi mezi impossibile a intenderne umanamente parola; senza questi
un aggirarsi vanamente per un oscuro laberinto."
Galileo Galilei, p. 38 in Il Saggiatore, Ed. L. Sosio, Feltrinelli, Milano (1965)
"Philosophy is written in this grand book-I mean the universe-which stands
continually open to our gaze, but it cannot be understood unless one first learns
to comprehend the language and to interpret the characters in which it is written.
It is written in the language of mathematics, and its characters are triangles,
circles, and other geometrical figures, without which it is humanly impossible
to understand a single word of it; without these, one is wandering about in a dark
labyrinth."
Galileo Galilei, in The Assayer (trans!. from Italian by S. Drake, pp. 106-107
in L. Geymonat, Galileo Galilei, McGraw-Hill, New York (1965»
Preface
Solvable models play an important role in the mathematical modeling of
natural phenomena. They make it possible to grasp essential features of the
phenomena and to guide the search for suitable methods of handling more
complicated and realistic situations.
In this monograph we present a detailed study of a class of solvable models
in quantum mechanics. These models describe the motion of a particle in a
potential having support at the positions of a discrete (finite or infinite) set of
point sources. We discuss both situations in which the strengths of the sources
and their locations are precisely known and the cases where these are only
known with a given probability distribution. The models are solvable in
the sense that their resolvents and associated mathematical and physical
quantities like the spectrum, the corresponding eigenfunctions, resonances,
and scattering quantities can be determined explicitly.
There is a large literature on such models which are called, because of the
interactions involved, by various names such as, e.g., "point interactions,"
"zero-range potentials," "delta interactions," "Fermi pseudopotentials,"
"contact interactions." Their main uses are in solid state physics (e.g., the
Kronig-Penney model of a crystal), atomic and nuclear physics (describing
short-range nuclear forces or low-energy phenomena), and electromagnetism
(propagation in dielectric media).
The main purpose of this monograph is to present in a systematic way the
mathematical approach to these models, developed in recent years, and to
illustrate its connections with previous heuristic derivations and computa
tions. Results obtained by different methods in disparate contexts are unified
vii
viii Preface
in this way and a systematic control on approximations to the models, in
which the point interactions are replaced by more regular ones, is provided.
There are a few happy cases in mathematical physics in which one can find
solvable models rich enough to contain essential features of the phenomena
to be studied, and to serve as a starting point for gaining control of general
situations by suitable approximations. We hope this monograph will convince
the reader that point interactions provide such basic models in quantum
mechanics which can be added to the standard ones of the harmonic oscillator
and the hydrogen atom.
Acknowledgments
Work on this monograph has extended over several years and we are grateful
to many individuals and institutions for helping us accomplish it.
We enjoyed the collaboration with many mathematicians and physicists
over topics included in the book. In particular, we would like to mention Y.
Avron, W. Bulla, J. E. Fenstad, A. Grossmann, S. Johannesen, W. Karwowski,
W. Kirsch, T. Lindstf0m, F. Martinelli, M. Mebkhout, P. Seba, L. Streit,
T. Wentzel-Larsen, and T. T. Wu.
We thank the following persons for their steady and enthusiastic support
of our project: J.-P. Antoine, J. E. Fenstad, A. Grossmann, L. Streit, and
W. Thirring. In particular, we are indebted to W. Kirsch for his generous help
in connection with Sect. III.1A and Ch. III.5.
In addition to the names listed above we would also like to thank J. Brasche,
R. Figari, and J. Shabani for stimulating discussions.
We are indebted to J. Brasche and W. Bulla, and most especially to
P. Hjorth and J. Shabani, for carefully reading parts of the manuscript and
suggesting numerous improvements.
Hearty thanks also go to M. Mebkhout, M. Sirugue-Collin, and M. Sirugue
for invitations to the Universite d'Aix-Marseille II, Universite de Provence,
and Centre de Physique Theorique, CNRS, Luminy, Marseille, respectively.
Their support has given a decisive impetus to our project.
We are also grateful to L. Streit and ZiF, Universitat Bielefeld, for invita
tions and great hospitality at the ZiF Research Project Nr. 2 (1984/85) and to
Ph. Blanchard and L. Streit, Universitat Bielefeld, for invitations to the
Research Project Bielefeld-Bochum Stochastics (BiBoS) (V olkswagenstiftung).
We gratefully acknowledge invitations by the following persons and
institutions:
J.-P. Antoine, Institut de Physique Theorique, Universite Louvain-Ia-Neuve
(F. G., H. H.);
E. Balslev, Matematisk Institut, Aarhus Universitet (S. A., F. G.);
D. Bolle, Instituut voor Theoretische Fysica, Universiteit Leuven (F. G.);
L. Carleson, Institut Mittag-Lerner, Stockholm (H. H.);
K. Chadan, Laboratoire de Physique Theorique et Hautes Energies, CNRS,
Universite de Paris XI, Orsay (F. G.);
Preface ix
G. F. Dell' Antonio, Instituto di Matematica, Universita di Roma and SISSA,
Trieste (S. A.);
R. Dobrushin, Institute for Information Transmission, Moscow (S. A., R. H.-K.);
J. Glimm and O. McBryan, Courant Institute of Mathematical Sciences,
New York University (H. H.);
A. Jensen, Matematisk Institut, Aarhus Universitet (H. H.);
G. Lassner, Mathematisches Institut, Karl-Marx-UniversiHit, Leipzig (S. A.);
Mathematisk Seminar, NAV F, Universitetet i Oslo (S. A., F. G., H. H.);
R. Minlos, Mathematics Department, Moscow University (S. A., R. H.-K.);
Y. Rozanov, Steklov Institute of Mathematical Sciences, Moscow (S. A.);
B. Simon, Division of Physics, Mathematics and Astronomy, Caltech, Pasadena
(F. G.);
W. Wyss, Theoretical Physics, University of Colorado, Boulder (S. A.).
F. G. would like to thank the Alexander von Humboldt Stiftung, Bonn, for
a research fellowship. H. H. is grateful to the Norway-America Association
for a "Thanks to Scandinavia" Scholarship and to the U.S. Educational
Foundation in Norway for a Fulbright scholarship.
Special thanks are due to F. Bratvedt and C. Buchholz for producing all
the figures except the ones in Sect. 111.1.8.
We are indebted to B. Rasch, Matematisk Bibliotek, Universitetet i Oslo,
for her constant help in searching for original literature.
We thank I. Jansen, D. Haraldsson, and M. B. Olsen for their excellent and
patient typing of a difficult manuscript.
We gratefully acknowledge considerable help from the staff of Springer
Verlag in improving the manuscript.
Contents
Preface vii
Introduction 1
PART I
The One-Center Point Interaction 9
CHAPTER I.l
The One-Center Point Interaction in Three Dimensions 11
1.1.1 Basic Properties 11
1.1.2 Approximations by Means of Local as well as Nonlocal
Scaled Short-Range Interactions 17
1.1.3 Convergence of Eigenvalues and Resonances 28
1.1.4 Stationary Scattering Theory 37
Notes 46
CHAPTER 1.2
Coulomb Plus One-Center Point Interaction in Three Dimensions 52
1.2.1 Basic Properties 52
1.2.2 Approximations by Means of Scaled Coulomb-Type Interactions 57
1.2.3 Stationary Scattering Theory 66
Notes 74
xi
xii Contents
CHAPTER 1.3
The One-Center (j-Interaction in One Dimension 75
1.3.1 Basic Properties 75
1.3.2 Approximations by Means of Local Scaled Short-Range Interactions 79
1.3.3 Convergence of Eigenvalues and Resonances 83
1.3.4 Stationary Scattering Theory 85
Notes 89
CHAPTER 1.4
The One-Center (j'-Interaction in One Dimension 91
Notes 95
CHAPTER 1.5
The One-Center Point Interaction in Two Dimensions 97
Notes 105
PART II
Point Interactions with a Finite Number of Centers 107
CHAPTER 11.1
Finitely Many Point Interactions in Three Dimensions 109
11.1.1 Basic Properties 109
11.1.2 Approximations by Means of Local Scaled Short-Range Interactions 121
11.1.3 Convergence of Eigenvalues and Resonances 125
11.1.4 Multiple Well Problems 132
11.1.5 Stationary Scattering Theory 134
Notes 138
CHAPTER 11.2
Finitely Many (j-Interactions in One Dimension 140
11.2.1 Basic Properties 140
11.2.2 Approximations by Means of Local Scaled Short-Range Interactions 145
11.2.3 Convergence of Eigenvalues and Resonances 148
11.2.4 Stationary Scattering Theory 150
Notes 153
CHAPTER 11.3
Finitely Many (j'-Interactions in One Dimension 154
Notes 159
CHAPTER 11.4
Finitely Many Point Interactions in Two Dimensions 160
Notes 165
