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 Mat. Zametki, 2019, Volume 105, Issue 4, Pages 564–588 (Mi mz11712)

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

S. A. Nazarov

Saint Petersburg State University

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: lattice of thin quantum waveguides, perturbation, essential and discrete spectra, gaps, eigenvalues, asymptotic expansion.

 Funding Agency Grant Number Russian Science Foundation 17-11-01003 This work was supported by the Russian Science Foundation under grant 17-11-01003.

DOI: https://doi.org/10.4213/mzm11712

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English version:
Mathematical Notes, 2019, 105:4, 559–579

Bibliographic databases:

UDC: 517

Citation: S. A. Nazarov, “Asymptotics of the Eigenvalues and Eigenfunctions of a Thin Square Dirichlet Lattice with a Curved Ligament”, Mat. Zametki, 105:4 (2019), 564–588; Math. Notes, 105:4 (2019), 559–579

Citation in format AMSBIB
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