Spectral Geometry: Direct and Inverse Problems With an Appendix by G
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1207 Pierre H. Berard
Spectral Geometry: Direct and Inverse Problems With an Appendix by G. Besson
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Pierre H. Berard Departement de Mathematiques, Unversite de Savoie B. P. 1104, 73011 Charnbery Cedex, France
This book is being published in a parallel edition by the Instituto de Maternatica Pura e Aplicada, Rio de Janeiro as volume 41 of the series .Monoqraflas de Matematica", Mathematics Subject Classification (1980): Primary: 58G 25, 35P 15, 52A40 Secondary: 58G 11, 58C40, 58G30 ISBN 3-540-16788-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-16788-9 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data. Berard. Pierre H. Spectral geometry. (Lecture notes in mathematics; 1207) Bibliography: p. Includes index. 1. Geometry, Riemannian. 2. Eigenvalues. 3. Operator theory. I. Title. II. Series: Lecture notes in mathematics (SpringerVerlag); 1207. 0A3.L28 no. 1207 [0A649] 510 s [516.3'73J 86-20323 ISBN 0-387-16788-9 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole 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 "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
To Rachel, Philippe, Izabel
INTRODUCTION
The purpose of these notes is to describe some aspects of direct problems in spectral geometry. Eigenvalue problems were motivated by questions in mathematical physics.
In these notes, we deal with eigenvalue problems for the
Laplace-Beltrami operator on a compact Riemannian manifold. a manifold numbers
(M,g),
To such
we can associate a sequence of non-negative real
fA}
the eigenvalues of the Laplace-Beltrami operator i:t 1 ' acting on C=(M). One can think of a Riemannian manifold as a i
musical instrument together with the musician who plays it.
In this
picture, the eigenvalues of the Laplace operator correspond to the harmonics of the instrument; they may depend on the music player, i.e. on the Riemannian metric: think of a kettledrum, or better of a Brazilian "cu{ca". Spectral geometry aims at describing the relationships between the musical instrument and the sounds it is capable of sending out. The problems which arise in spectral geometry are of two kinds: direct problems and inverse problems.
In a direct problem,
we want information on the sounds produced by the instrument, in terms of its geometry.
For example, we know that the bigger the
tension of the parchment head of a kettledrum, the higher the pitch. In an inverse problem, we investigate what geometric information on the instrument can be recovered from the sounds it sends out. Both types of problem
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