Resonance multiplicity of a perturbed periodic Schrödinger operator
type: Item
instance of: scholarly article
title: Resonance multiplicity of a perturbed periodic Schrödinger operator
zbMATH Open document ID: 0952.47012
DOI: 10.1007/BF02557127
author: Yu. P. Chuburin
published in: Theoretical and Mathematical Physics
publication date: 2001-01-15T00:00:00Z
review text: Let \(V,W:\mathbb{R}^3\to \mathbb{R}\) be functions such that \(V\) is periodic, \(W\) is periodic in \(x_1, x_2\) and \(W\) decays exponentially as \(x_3\to\infty\). The author studies the structure of the resonances of the Schrödinger operator \(-\Delta+V+W\). It is shown that the set of resonances coincides with the set of zeros of a certain analytic function \(\alpha-1\) and that the resonance multiplicity coincides with the multiplicity of zeros of \(\alpha-1\). Under some hypothesis on \(\alpha\) it is also shown that (an analogue of) the forward scattering amplitude vanishes on the surface given by the zeros of the derivative of the eigenvalue of the periodic operator in a cell with respect to the third quasi-momentum component.
reviewed by: André Noll
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 1465991
zbMATH Keywords: structure of the resonances
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: periodic operator
zbMATH Keywords: quasi-momentum component
MaRDI profile type: Publication
cites work: Q4185363
cites work: Scattering from infinite sheets
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cites work: Q5515607
cites work: On small perturbations of the Schrödinger operator with a periodic potential
cites work: Q3994511
cites work: On the Schrödinger operator with a small potential in the case of a crystal film
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description: scientific article; zbMATH DE number 1465991