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Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbations

  • Denis Borisov
Article

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.

Keywords

Distant perturbation Waveguide Asymptotics Eigenvalue Eigenfunction 

Mathematics Subject Classifications (2000)

35P05 35B20 35B40 

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© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Nuclear Physics InstituteAcademy of SciencesŘež near PragueCzechia
  2. 2.Bashkir State Pedagogical UniversityUfaRussia

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