Abstract
The generalized Marchenko and Gel’fand-Levitan methods are based on the solution of a Riemann-Hilbert problem. This paper relates the solution of this problem to the standard Wiener-Hopf factorization of an operator-valued function. It also states a new theorem that equates the number of bound states of the underlying potential for an admissible S matrix to the dimensions of the eigenspaces, at the eigenvalues ±1, of a relevant compact operator obtained from the Fourier transform of the scattering amplitude.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. G. Newton. Scattering Theory of Waves and Particles. Springer Verlag, New York, 2nd edition, 1982.
R. G. Newton. Inverse Schrôdinger Scattering in Three Dimensions. Springer Verlag, New York, 1989. To be published.
K. Chadan and P. C. Sabatier. Inverse Problems in Quantum Scattering Theory. Springer Verlag, New York, second edition edition, 1989.
I. C. Gohberg. The factorization problem for operator functions. Izv. Akad. Nauk SSSR Ser. Mat., 28:1055-1082, 1964. English translation: American Mathematical Society Translations 49, 130 - 161 (1966).
K. Clancey and 1. C. Gohberg. Factorization of Matrix Functions and Singular integral Operators. Birkhiiuser Verlag, Basel, 1981.
I. C. Gohberg and M. A. Kaashoek. Constructive Methods in Wiener-Hopf Factorization. Birkhiiuser Verlag, Basel, 1986.
W. Greenberg, C. van der Mee, and V. Protopopescu. Boundary Value Problems in Abstract Kinetic Theory. Birkhiiuser Verlag, Basel, 1987.
T. Aktosun and C. van der Mee. Solution of the inverse scattering problem for the 3-d Schrödinger equation by Wiener-Hopf factorization of the scattering operator. Preprint, University of Delaware, 1989.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Newton, R.G. (1990). New Results on 3D Schrödinger Inverse Scattering. In: Sabatier, P.C. (eds) Inverse Methods in Action. Inverse Problems and Theoretical Imaging. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75298-8_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-75298-8_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-75300-8
Online ISBN: 978-3-642-75298-8
eBook Packages: Springer Book Archive