Scattering Theory

Scattering Theory Revised Edition This is Volume 26 in PURE AND APPLIED MATHEMATICS H. Bass, A. Borel, J. Moser, and S.-T. Yau, editors Paul A. Smith and Samuel Eilenberg, founding editors W list of titles in this series appears at the end of this volume. In memory of Franz Rellich This Page Intentionally Left Blank SCATTERING THEORY REVISED EDITION PETER D. LAX COURANT INSTITUTE OF MATHEMATICAL SCIENCES NEW YORK UNIVERSITY NEW YORK,NEW YORK RALPH S. PHPLLIPS DEPARTMENT OF MATHEMATICS STANFORD UNIVERSITY STANFORD, CALIFORNIA ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers Boston San Diego New York Berkley London Sydney Tokyo Toronto This is Volume 26 in PURE AND APPLIED MATHEMATICS H. Bass, A. Borel, J. Moser, S.-T. Yau, editors Paul A. Smith and Samuel Eilenberg, founding editors COPYRIGHT 0 1989 BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA 92101 United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. 24-28 Oval Road, London NW17DX Library of Congress Cataloging-in-Publication Data Lax,Peter D. Scattering theory I Peter D. Lax, Ralph S. Phillips. -Rev. ed. p. cm.-(Pure and applied mathematics ; v. 26) Includes bibliographical references. ISBN 0-12-440051-5 (alk.paper) 1. Wave-motion, Theory of. 2. Scattering (Mathematics) I. Phillips, Ralph S. (Ralph Saul), Date- . 11. Title. 111. Series: Pure and applied mathematics (Academic Press) ; 26. QA3.P8 vol. 26 1989 [QA9271 510 s-dc20 89-17819 [530.1’24] CIP PRINTED IN THE UNITED STATES OF AMERICA 89909192 9 8 7 6 5 4 3 2 1 Contents Preface to the Revised Editim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER . 1 2. 3. 4. 5. 6 7. . I . Introduction . . . . . . . . . . . . . . .... ... ix x 1 TheDynamic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Scattering Theory Formulated in Terms of Representation Theory . . . . . 6 A Semigroup of Operators Related to the Scattering Matrix . . . . . . . . . . . . . . . . 11 The Form of the Scattering Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A Simple Example ....................... ...................... 18 Scattering Theory for Transport Phenomena ... . . . . . . . . . . . . . 27 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 CHAPTER I1. Representation Theory and the Scattering Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1. The Discrete Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Scattering Operator in the Discrete Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . The Continuous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .