Inverse scattering on half-line
type: Item
instance of: scholarly article
title: Inverse scattering on half-line
zbMATH Open document ID: 0631.35085
DOI: 10.1016/0022-247X(88)90422-2
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: The author gives a new approach to the theory. This approach uses a Wiener-Hopf type equation (which is Marchenko's equation at \(x=0)\). The normalizing constants are expressed in terms of the parameters which define the general solution to this equation. A detailed analysis of invertibility of the steps in the various inversion procedures is given. In particular several ways are given to prove that the potential reconstructed from the scattering data generates the original scattering data; that the scattering data, corresponding to a potential, generate via the inversion procedure the original potential. Homeomorphism between a class of potentials and a class of scattering data is established. The class of potentials q is \(\{\) \(q: q=\bar q,\int ^{\infty}_{0} (1+x)| q(x)| dx<\infty \}\). Applications are given. In particular, an example of \(q\in C_ 0^{\infty}(0,R)\) is given for which there exist infinitely many purely imaginary resonances; a necessary and sufficient condition on the scattering data is given for \(q\in L^ 2(0,\infty)\), and some other results are obtained.
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35P25
zbMATH DE Number: 4026012
zbMATH Keywords: Wiener-Hopf type equation
zbMATH Keywords: parameters
zbMATH Keywords: scattering data
zbMATH Keywords: potential
zbMATH Keywords: imaginary resonances
author: A. G. Ramm
MaRDI profile type: Publication
cites work: Q5726845
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cites work: Inverse scattering on the line
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cites work: Sturm-Liouville operators and applications. Transl. from the Russian by A. Iacob
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cites work: Sufficient conditions for zero not to be an eigenvalue of the Schrödinger operator
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full work available at URL: 0022-247x(88)90422-2
OpenAlex ID: W2019053782
description: scientific article; zbMATH DE number 4026012