Perturbation Theory for Linear ... - School of Mathematics

[Pages:643]C L A S S I C S I N M A T H E M AT I C S

Tosio Kato

Perturbation Theory for Linear Operators

Springer

Tosio Kato

Perturbation Theory for Linear Operators

Reprint of the 1980 Edition

Springer

Tosio Kato

Department of Mathematics, University of California Berkeley, CA 94720-3840 USA

Originally published as Vol. 132 of the Grundlehren der mathematischen Wissenschaften

Mathematics Subject Classification (1991): 46BXX, 46CXX, 47AXX, 47BXX, 47D03, 47E05, 47F05, 81Q10, 81Q15, 81UXX

ISBN 3-540-58661-X Springer-Verlag Berlin Heidelberg New York

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SPIN 10485260

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Tosio Kato

Perturbation Theory for Linear Operators

Corrected Printing of the Second Edition

Springer-Verlag

Berlin Heidelberg New York 1980

Dr. Tosio Kato Professor of Mathematics, University of California, Berkeley

AMS Subject Classification (1970): 46Bxx, 46Cxx, 47Axx, 47Bxx, 47D05, 47Exx, 47Fxx, 81A09, 81A10, 81A45 ISBN 3-540-07558-5 2. Auflage Springer-Verlag Berlin Heidelberg New York ISBN 0-387-07558-5 2nd edition Springer-Verlag New York Heidelberg Berlin

ISBN 3-540-03526-5 1. Auflage Berlin Heidelberg New York ISBN 0-387-03526-5 1st edition New York Heidelberg Berlin Library of Congress Cataloging in Publication Data. Kato, Tosio, 1917-Perturbation theory for linear operators. (Grundlehren der mathematischen Wissenschaften; 132). Bibliography: p. Includes indexes. 1. Linear operators. 2. Perturbation (Mathematics). I. Title. II. Series: Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen; Bd. 132. QA329.2.K37 1976. 515'.72. 76-4553. This work is subject to copyright. All rights are reserved, whether thewhole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under ? 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to thepublisher, the amount of the fee to be determined by agreement with the publisher. ? by Springer-Verlag Berlin Heidelberg 1966, 1976 Printed in Germany. Typesetting, printing and binding: Briihlsche Universit5tsdruckerei Giellen

To the memory of my patents

Preface to the Second Edition

In view of recent development in perturbation theory, supplementary notes and a supplementary bibliography are added at the end of the new edition. Little change has been made in the text except that the paragraphs V-? 4.5, VI-? 4.3, and VIII-? 1.4 have been completely rewritten, and a number of minor errors, mostly typographical, have been corrected. The author would like to thank many readers who brought the errors to his attention.

Due to these changes, some theorems, lemmas, and formulas of the first edition are missing from the new edition while new ones are added. The new ones have numbers different from those attached to the old ones which they may have replaced.

Despite considerable expansion, the bibliography is not intended to be complete.

Berkeley, April 1976

Tosio KATO

Preface to the First Edition

This book is intended to give a systematic presentation of perturbation theory for linear operators. It is hoped that the book will be useful to students as well as to mature scientists, both in mathematics and in the physical sciences.

Perturbation theory for linear operators is a collection of diversified results in the spectral theory of linear operators, unified more or less loosely by their common concern with the behavior of spectral properties when the operators undergo a small change. Since its creation by RAYLEIGH and SCHRODINGER, the theory has occupied an important place in

applied mathematics; during the last decades, it has grown into a

mathematical discipline with its own interest. The book aims at a mathematical treatment of the subject, with due consideration of applications.

The mathematical foundations of the theory belong to functional analysis. But since the book is partly intended for physical scientists, who might lack training in functional analysis, not even the elements of that subject are presupposed. The reader is assumed to have only a basic knowledge of linear algebra and real and complex analysis. The necessary tools in functional analysis, which are restricted to the most elementary part of the subject, are developed in the text as the need for them arises (Chapters I, III and parts of Chapters V, VI).

An introduction, containing a brief historical account of the theory, precedes the main exposition. There are ten chapters, each prefaced by a

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