The analytic continuation of the scattering kernel associated with the Schrödinger operator with a penetrable wall interaction
type: Item
instance of: scholarly article
title: The analytic continuation of the scattering kernel associated with the Schrödinger operator with a penetrable wall interaction
zbMATH Open document ID: 0811.35111
author: Shin-ichi Shimada
published in: Journal of Mathematics of Kyoto University
publication date: 1994-09-05T00:00:00Z
review text: The author studies the analytical properties of the scattering matrix with respect to the wave vector \(k\) for the Schrödinger operator \(H = - \Delta + q(x)\) \(\delta (| x | - a)\) in \(L_ 2 (\mathbb{R}^ 3)\) where the function \(q(x)\) is real and smooth and \(\delta\) means the Dirac delta-function. He proves that the scattering kernel can be continued to the whole complex plane as a meromorphic function. If an imaginary part of a variable \(k\) is positive, the scattering kernel may have only simple poles and only on the imaginary axis and at most a finite number. They correspond to the negative eigenvalues of the Schrödinger operator. On the real axis the scattering kernel may have a zero resonance if any. The author examines domains of the complex plane free of poles and deals with the case where \(q(x)\) is constant.
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81U20
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 633570
zbMATH Keywords: analytical properties of the scattering matrix
zbMATH Keywords: scattering kernel
reviewed by: E. D. Belokolos
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519067
OpenAlex ID: W1488698252
DOI: 10.1215/KJM/1250519067
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description: scientific article; zbMATH DE number 633570