SOLVABLE MODELS IN QUANTUM MECHANICS SECOND EDITION S. ALBEVERIO F. GESZTESY R. HQEGH-KROHN H. HOLDEN WITH AN APPENDIX BY PAVEL EXNER AMS CHELSEA PUBLISHING American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 81Q05; Secondary 03H10, 81-02, 81Q10, 81V70. For additional information and updates on this book, visit www.ams.org/bookpages/chel-350 Library of Congress Cataloging-in-Publication Data Solvable models in quantum mechanics with appendix written by Pavel Exner / S. Albeverio...[et al.J.- 2nd ed. p. cm. Rev. ed. of. Solvable models in quantum mechanics. c1988. Includes bibliographical references and index. ISBN 0-8218-3624-2 (alk. paper) 1. Quantum theory-Mathematical models. I. Exner, Pavel, 1946-. II. Al- beverio, Sergio. III. Solvable models in quantum mechanics. QC174.12.S65 2004 530.12-dc22 2004057452 Copying and reprinting. 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Copyright © 1988 held by the American Mathematical Society. Reprinted by the American Mathematical Society, 2005 Printed in the United States of America. ® 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.ams.org/ 10987654321 100908070605 "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." 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 (transl. from Italian by S. Drake, pp. 106-107 in L. Geymonat, Galileo Galilei, McGraw-Hill, New York (1965)) 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. A. Geiler, Yu. A. Ku- perin, and K. A. Makarov, and published by Mir, Moscow, in 1991. The Russian edition contains an additional appendix by K. A. 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. F. Dell'Antonio, P. Exner, W. Karwowski, P. Kurasov, K. A. Makarov, K. Nemcova, and 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. July 2004 S. Albeverio F. Gesztesy H. Holden 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. Lindstrom, 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. 111.1.4 and Ch. 111.5. 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. Shahani, 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) (Volkswagenstiftung). We gratefully acknowledge invitations by the following persons and institutions: J.-P. Antoine, Institut de Physique Theorique, Universite Louvain-la-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-Leffler, 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 Matcmatica, University 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-Universitat, Leipzig (S. A.); Mathematisk Seminar, NAVF, 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 arc 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.