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Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbations

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Abstract

We consider a quantum waveguide modelled by an infinite straight tube with arbitrary cross-section in n-dimensional space. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators “localized” in a certain sense. We study the asymptotic behaviour of the discrete spectrum of such system as the distance between the “supports” of localized perturbations tends to infinity. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We provide a list of the operators, which can be chosen as distant perturbations. In particular, the distant perturbations may be a potential, a second order differential operator, a magnetic Schrödinger operator, an arbitrary geometric deformation of the straight waveguide, a delta interaction, and an integral operator.

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Authors and Affiliations

  1. Nuclear Physics Institute, Academy of Sciences, Řež near Prague, 25068, Czechia

    Denis Borisov

  2. Bashkir State Pedagogical University, October rev. st. 3a, 450000, Ufa, Russia

    Denis Borisov

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  1. Denis Borisov
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Correspondence to Denis Borisov.

Additional information

The research was supported by Marie Curie International Fellowship within 6th European Community Framework Programm (MIF1-CT-2005-006254). The author is also supported by the Russian Foundation for Basic Researches (No. 07-01-00037) and by the Czech Academy of Sciences and Ministry of Education, Youth and Sports (LC06002). The author gratefully acknowledges the support from Deligne 2004 Balzan prize in mathematics and the grant of Republic Bashkortostan for young scientists and young scientific collectives.

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Borisov, D. Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbations. Math Phys Anal Geom 10, 155–196 (2007). https://doi.org/10.1007/s11040-007-9028-1

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