Bound States Eigenfunction Expansions
The bound states correspond to the solutions of (I.1.6) which satisfy the boundary conditions (I.2.1) and are square-integrable on the whole positive r-axis. We are going to study them in detail because, as is well known, they are necessary in the complet
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w.
Beiglbock
series editor
K.Chadan and P. C. Sabatier
Inverse Problems
in Quantum Scattering Theory
With a Foreword by R. G. Newton
[I]
Springer Science+Business Media, LLC
Khosrow Chad an
Pierre C. Sabatier
Directeur de recherches CNRS Laboratoire de Physique Theorique Universiti: de Paris-Sud 914050rsay France
Universite des Sciences et Techniques du Languedoc 34060 Montpellier Cedex France
Professor Wolf Beiglbock
Professor R. G. Newton
Springer-Verlag Neuenheimer Landstrasse 28-30 6900 Heidelberg Federal Republic of Germany
Professeur de physique mathematique Department of Physics Indiana University Bloomington, Indiana 47401 USA
Library of Congress Cataloging in Publication Data Chadan, Khosrow. Inverse problems in quantum scattering theory. (Texts and monographs in physics) Includes bibliographical references. 1. Scattering (Physics) 2. Quantum theory. 3. Inverse problems (Differential equations) I. Sabatier, Pierre Celestin, 1935joint author. II. Title. QC20.7.S3C45 539.7'54 76-51250 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC.
© 1977 by Springer Science+Business Media New York Originally published by Springer-Verlag New York in 1977
ISBN 978-3-662-12125-2 (eBook) ISBN 978-3-662-12127-6 DOI 10.1007/978-3-662-12125-2
Foreword
The normal business of physicists may be schematically thought of as predicting the motions of particles on the basis of known forces, or the propagation of radiation on the basis of a known constitution of matter. The inverse problem is to conclude what the forces or constitutions are on the basis of the observed motion. A large part of our sensory contact with the world around us depends on an intuitive solution of such an inverse problem: We infer the shape, size, and surface texture of external objects from their scattering and absorption of light as detected by our eyes. When we use scattering experiments to learn the size or shape of particles, or the forces they exert upon each other, the nature of the problem is similar, if more refined. The kinematics, the equations of motion, are usually assumed to be known. It is the forces that are sought, and how they vary from point to point. As with so many other physical ideas, the first one we know to have touched upon the kind of inverse problem discussed in this book was Lord Rayleigh (1877). In the course of describing the vibrations of strings of variable density he briefly discusses the possibility of inferring the density distribution from the frequencies of vibration. This passage may be regarded as a precursor of the mathematical study of the inverse spectral problem some seventy years later. Its modern analogue and generalization was given in a famous lecture by Marc Kac (1966), entitled" Can one hear the shape of a drum?" With the invention of the Schr dinger equation the physical scope of mathematical notions connected with the spectra of differential equations
vi Foreword with prescribed bo
