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

Bakharev, F.; Matveenko, S.; Nazarov, S.

doi:10.1090/spmj/1500

Rectangular lattices of cylindrical quantum waveguides. I. Spectral problems on a finite cross
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by F. L. Bakharev, S. G. Matveenko and S. A. Nazarov
Translated by: A. Plotkin

St. Petersburg Math. J. 29 (2018), 423-437

DOI: https://doi.org/10.1090/spmj/1500

Published electronically: March 30, 2018

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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.

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