Asymptotics of the Eigenvalues and Eigenfunctions of a Thin Square Dirichlet Lattice with a Curved Ligament

Research output

Abstract

The spectrum of the Dirichlet problem on the planar square lattice of thin quantum waveguides has a band-gap structure with short spectral bands separated by wide spectral gaps. The curving of at least one of the ligaments of the lattice generates points of the discrete spectrum inside gaps. A complete asymptotic series for the eigenvalues and eigenfunctions are constructed and justified; those for the eigenfunctions exhibit a remarkable behavior imitating the rapid decay of the trapped modes: the terms of the series have compact supports that expand unboundedly as the number of the term increases.

Original languageEnglish
Pages (from-to)559-579
JournalMathematical Notes
Volume105
Issue number3-4
DOIs
Publication statusPublished - 1 Mar 2019

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Eigenvalues and Eigenfunctions
Dirichlet
Asymptotic series
Spectral Gap
Lattice Points
Discrete Spectrum
Compact Support
Band Gap
Term
Square Lattice
Dirichlet Problem
Expand
Waveguide
Eigenfunctions
Decay
Series

Scopus subject areas

  • Mathematics(all)

Cite this

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title = "Asymptotics of the Eigenvalues and Eigenfunctions of a Thin Square Dirichlet Lattice with a Curved Ligament",
abstract = "The spectrum of the Dirichlet problem on the planar square lattice of thin quantum waveguides has a band-gap structure with short spectral bands separated by wide spectral gaps. The curving of at least one of the ligaments of the lattice generates points of the discrete spectrum inside gaps. A complete asymptotic series for the eigenvalues and eigenfunctions are constructed and justified; those for the eigenfunctions exhibit a remarkable behavior imitating the rapid decay of the trapped modes: the terms of the series have compact supports that expand unboundedly as the number of the term increases.",
keywords = "asymptotic expansion, eigenvalues, essential and discrete spectra, gaps, lattice of thin quantum waveguides, perturbation",
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AB - The spectrum of the Dirichlet problem on the planar square lattice of thin quantum waveguides has a band-gap structure with short spectral bands separated by wide spectral gaps. The curving of at least one of the ligaments of the lattice generates points of the discrete spectrum inside gaps. A complete asymptotic series for the eigenvalues and eigenfunctions are constructed and justified; those for the eigenfunctions exhibit a remarkable behavior imitating the rapid decay of the trapped modes: the terms of the series have compact supports that expand unboundedly as the number of the term increases.

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KW - essential and discrete spectra

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KW - lattice of thin quantum waveguides

KW - perturbation

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