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 Mat. Sb., 2009, Volume 200, Number 12, Pages 121–156 (Mi msb7579)

The resonance spectrum of a Schrödinger operator with a rapidly decaying potential

S. A. Stepinab, A. G. Tarasova

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b University of Bialystok

Abstract: Resonances of the one-dimensional Schrödinger operator are investigated, that is, the poles of the analytic extension of the corresponding scattering matrix. For a certain class of superexponentially decreasing potentials, including the Gaussian potential, the Born approximation is substantiated for the problem of localizing the poles of the scattering matrix. This makes it possible to find an asymptotic law (a quantization rule) for the distribution of these poles. For the first time, using the method developed in the paper, asymptotic formulae for resonances are obtained in the case of potentials with noncompact support.
Bibliography: 15 titles.

Keywords: resonance, pole of a scattering matrix, asymptotic distribution, Schrödinger operator, superexponentially decreasing potential.
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DOI: https://doi.org/10.4213/sm7579

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English version:
Sbornik: Mathematics, 2009, 200:12, 1847–1880

Bibliographic databases:

UDC: 517.927
MSC: Primary 34L25, 34L40; Secondary 47E05

Citation: S. A. Stepin, A. G. Tarasov, “The resonance spectrum of a Schrödinger operator with a rapidly decaying potential”, Mat. Sb., 200:12 (2009), 121–156; Sb. Math., 200:12 (2009), 1847–1880

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb7579
• https://doi.org/10.4213/sm7579
• http://mi.mathnet.ru/eng/msb/v200/i12/p121

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This publication is cited in the following articles:
1. S. A. Stepin, “Asymptotic estimates for the kernel of the semigroup generated by a perturbation of the biharmonic operator by a potential”, Sb. Math., 203:6 (2012), 893–921
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