Rectangular lattices of cylindrical quantum waveguides. I. spectral problems on a finite cross

F. L. Bakharev, S. G. Matveenko, S. A. Nazarov

Research output

Abstract

The spectrum of truncated cross-shaped waveguides is studied under the Dirichlet conditions on the lateral surface and various boundary conditions on the ends of the column and the cross bar. The monotonicity and asymptotics of the eigenvalues are discussed in dependence on the size of a cross whose section may be fairly arbitrary. In the case of a round section, the estimates found for the second eigenvalue agree with the asymptotic formulas obtained. Such information is needed for the spectral analysis of thin periodic lattices of quantum waveguides.

Original languageEnglish
Pages (from-to)423-437
Number of pages15
JournalSt. Petersburg Mathematical Journal
Volume29
Issue number3
DOIs
Publication statusPublished - 1 Jan 2018
Externally publishedYes

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Spectral Problem
Waveguide
Waveguides
Eigenvalue
Dirichlet conditions
Spectral Analysis
Asymptotic Formula
Spectrum analysis
Monotonicity
Lateral
Cross section
Boundary conditions
Arbitrary
Estimate

Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

Cite this

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AU - Matveenko, S. G.

AU - Nazarov, S. A.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - The spectrum of truncated cross-shaped waveguides is studied under the Dirichlet conditions on the lateral surface and various boundary conditions on the ends of the column and the cross bar. The monotonicity and asymptotics of the eigenvalues are discussed in dependence on the size of a cross whose section may be fairly arbitrary. In the case of a round section, the estimates found for the second eigenvalue agree with the asymptotic formulas obtained. Such information is needed for the spectral analysis of thin periodic lattices of quantum waveguides.

AB - The spectrum of truncated cross-shaped waveguides is studied under the Dirichlet conditions on the lateral surface and various boundary conditions on the ends of the column and the cross bar. The monotonicity and asymptotics of the eigenvalues are discussed in dependence on the size of a cross whose section may be fairly arbitrary. In the case of a round section, the estimates found for the second eigenvalue agree with the asymptotic formulas obtained. Such information is needed for the spectral analysis of thin periodic lattices of quantum waveguides.

KW - Asymptotics

KW - Discrete spectrum

KW - Infinite and truncated cross-shaped quantum waveguides

KW - Stable and decaying solution at the threshold of the continuous spectrum

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JO - St. Petersburg Mathematical Journal

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SN - 1061-0022

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