SOLVABLE MODELS IN QUANTUM MECHANICS SECOND EDITION S. HOLDEN WITH AN APPENDIX BY PAVEL EXNER AMS CHELSEA PUBLISHING American Mathematical Society Providence, Rhode Island www.com 2000 Mathematics Subject Classification. For additional information and updates on this book, visit www.org/bookpages/chel-350 Library of Congress Cataloging-in-Publication Data Solvable models in quantum mechanics with appendix written by Pavel Exner / S. Solvable models in quantum mechanics.
Includes bibliographical references and index. Quantum theory-Mathematical models. Al- beverio, Sergio. Solvable models in quantum mechanics.12-dc22 2004057452 Copying and reprinting.
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® The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.com "La filosofia 6 scritta in questo grandissimo libro the continuamente ci sta aperto innanzi a gli occhi (io dico l'universo), ma non si pud intendere se prima non s'impara a intender la lingua, e conoscer i caratteri, ne' quali a scritto. Egli 6 scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi a impossibile a intenderne umanamente parola; senza questi b un aggirarsi vanamente per un oscuro laberinto. 38 in Il Saggiatore, Ed.
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 (transl. from Italian by S. Geymonat, Galileo Galilei, McGraw-Hill, New York (1965)) www.com Preface to the Second Edition The original edition of this monograph generated continued interest as evidenced by a steady number of citations since its publication by Springer-Verlag in 1988.
Hence, we were particularly pleased that the American Mathematical Society offered to publish a second edition in its Chelsea series, and we hope this slightly expanded and corrected reprint of our book will continue to be a useful resource for researchers in the area of exactly solvable models in quantum mechanics. The Springer edition was translated into Russian by V. Ku- perin, and K. Makarov, and published by Mir, Moscow, in 1991.
The Russian edition contains an additional appendix by K. Makarov as well as further ref- erences. The field of point interactions and their applications to quantum mechanical systems has undergone considerable development since 1988. We were partic- ularly fortunate to attract Pavel Exner, one of the most prolific and energetic representatives of this area, to prepare a summary of the progress made in this field since 1988.
His summary, which centers around two-body point interaction problems, now appears as the new Appendix K in this edition; it is followed by a bibliography which focuses on some of the essential developments since 1988. A list of errata and addenda for the first Springer-Verlag edition appears at the end of this edition. We are particularly grateful to G. Panati for generously supplying us with lists of corrections.
Apart from the new Appendix K, its bibliography, and the list of errata, this second AMS-Chelsea edition is a reprint of the original 1988 Springer-Verlag edition. We thank Sergei Gelfand and the staff at AMS for their help in preparing this second edition. Due to Raphael Hoegh-Krohn's unexpected passing on January 24, 1988, he never witnessed the publication of this monograph. He was one of the principal creators of this field, and we take the opportunity to dedicate this second edition to his dear memory.com 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., "point interactions," "zero-range potentials," "delta interactions," "Fermi pseudopotentials," "contact interactions." Their main uses are in solid state physics (e., 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 www.com 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. Wentzel-Larsen, and T. We thank the following persons for their steady and enthusiastic support of our project: J. In particular, we are indebted to W.
Kirsch for his generous help in connection with Sect. In addition to the names listed above we would also like to thank J. Shabani for stimulating discussions. We are indebted to J.
Bulla, and most especially to P. Shahani, for carefully reading parts of the manuscript and suggesting numerous improvements. Hearty thanks also go to 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. Streit, Universitat Bielefeld, for invitations to the Research Project Bielefeld-Bochum Stochastics (BiBoS) (Volkswagenstiftung). We gratefully acknowledge invitations by the following persons and institutions: J. Antoine, Institut de Physique Theorique, Universite Louvain-la-Neuve (F.
Balslev, Matematisk Institut, Aarhus Universitet (S. Bolle, Instituut voor Theoretische Fysica, Universiteit Leuven (F. Carleson, Institut Mittag-Leffler, Stockholm (H. Chadan, Laboratoire de Physique Theorique et Hautes Energies, CNRS, Universite de Paris XI, Orsay (F.com Preface ix G.
Dell'Antonio, Instituto di Matcmatica, University di Roma and SISSA, Trieste (S. Institute for Information Transmission, Moscow (S. McBryan, Courant Institute of Mathematical Sciences. New York University (H.
Jensen, Matematisk Institut, Aarhus Universitet (H. Lassner, Mathematisches Institut, Karl-Marx-Universitat, Leipzig (S.); Mathematisk Seminar, NAVF, Universitetet i Oslo (S. Minlos, Mathematics Department, Moscow University (S. Rozanov, Steklov Institute of Mathematical Sciences.
Simon, Division of Physics, Mathematics and Astronomy, Caltech, Pasadena (F. Wyss, Theoretical Physics, University of Colorado, Boulder (S. would like to thank the Alexander von Humboldt Stiftung, Bonn, for a research fellowship. is grateful to the Norway-America Association for a "Thanks to Scandinavia" Scholarship and to the U.
Educational Foundation in Norway for a Fulbright scholarship. Special thanks are due to F. Buchholz for producing all the figures except the ones in Sect. We arc indebted to B.
Rasch, Matematisk Bibliotek, Universitetet i Oslo, for her constant help in searching for original literature. 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.com Contents Preface to second edition v Preface vii Introduction 1 PART I The One-Center Point Interaction 9 CHAPTER I.1 The One-Center Point Interaction in Three Dimensions 11 1.2 Approximations by Means of Local as well as Nonlocal Scaled Short-Range Interactions 17 1.3 Convergence of Eigenvalues and Resonances 28 1.4 Stationary Scattering Theory 37 Notes 46 CHAPTER 1.2 Coulomb Plus One-Center Point Interaction in Three Dimensions 52 1.2 Approximations by Means of Scaled Coulomb-Type Interactions 57 1.3 Stationary Scattering Theory 66 Notes 74 xi www.com xii Contents CHAPTER 1.3 The One-Center d-Interaction in One Dimension 75 1.2 Approximations by Means of Local Scaled Short-Range Interactions 79 1.3 Convergence of Eigenvalues and Resonances 83 1.4 Stationary Scattering Theory 85 Notes 89 CHAPTER 1.4 The One-Center b'-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 11.2 Approximations by Means of Local Scaled Short-Range Interactions 121 11.3 Convergence of Eigenvalues and Resonances 125 11.4 Multiple Well Problems 132 11.5 Stationary Scattering Theory 134 Notes 138 CHAPTER 11.2 Finitely Many b-Interactions in One Dimension 140 11.2 Approximations by Means of Local Scaled Short-Range Interactions 145 11.3 Convergence of Eigenvalues and Resonances 148 11.4 Stationary Scattering Theory 150 Notes 153 CHAPTER 11.3 Finitely Many 8'-Interactions in One Dimension 154 Notes 159 CHAPTER 11.4 Finitely Many Point Interactions in Two Dimensions 160 Notes 165 www.com Contents xiii PART III Point Interactions with Infinitely Many Centers 167 CHAPTER I1I.1 Infinitely Many Point Interactions in Three Dimensions 169 III.2 Approximations by Means of Local Scaled Short-Range Interactions 173 111.3 Periodic Point Interactions 176 111.9 Crystals with Defects and Impurities 239 Notes 250 CHAPTER 111.2 Infinitely Many 6-Interactions in One Dimension 253 111.2 Approximations by Means of Local Scaled Short-Range Interactions 261 111.5 Quasi-periodic b-Interactions 288 111.6 Crystals with Defects and Impurity Scattering 290 Notes 303 CHAPTER 111.3 Infinitely Many b'-Interactions in One Dimension 307 Notes 323 CHAPTER 111.4 Infinitely Many Point Interactions in Two Dimensions 324 Notes 333 CHAPTER 111.5 Random Hamiltonians with Point Interactions 334 111.2 Random Point Interactions in Three Dimensions 341 111.3 Random Point Interactions in One Dimension 349 Notes 353 APPENDICES A Self-Adjoint Extensions of Symmetric Operators 357 B Spectral Properties of Hamiltonians Defined as Quadratic Forms 360 www.