Jacobi Operators and Completely Integrable Nonlinear Lattices

This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy. Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations. Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed. Corrections and complements to this book are available from: http://www.mat.univie.ac.at/ ̃gerald/ftp/book-jac/ Typeset by AMS-LATEX, PicTEX, and Makeindex. Version: August 24, 2017. Library of Congress Cataloging-in-Publication Data Teschl, Gerald, 1970– Jacobi Operators and Completely Integrable Nonlinear Lattices / Gerald Teschl. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; V. 72) Includes bibliographical references and index. ISBN 0-8218-1940-2 1. Jacobi operators. 2. Differentiable dynamical systems. I. Title. II. Series: Mathemtical surveys and monographs ; no. 72. QA329.2.T47 1999 99-39165 515’.7242–dc21 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permissions should be addressed to the Assistant to the Publisher, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. c © 2000 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted too the United States Government.