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St. Petersburg Mathematical Journal

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


Authors: F. L. Bakharev, S. G. Matveenko and S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 29 (2017), nomer 3.
Journal: St. Petersburg Math. J. 29 (2018), 423-437
MSC (2010): Primary 81Q37
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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Additional Information

F. L. Bakharev
Affiliation: St. Petersburg State University, Universitetskii pr. 28, Petrodvorets, 198504 St. Petersburg, Russia; Chebyshev Laboratory, St. Petersburg State University, 14 Liniya V. O. 29B, 199178 St. Petersburg, Russia
Email: fbakharev@yandex.ru, f.bakharev@spbu.ru

S. G. Matveenko
Affiliation: National Research University Higher School of Economics, ul. Kantemirovskaya 3 building 1, Lit. A, 194100, St. Petersburg, Russia; Chebyshev Laboratory, St. Petersburg State University, 14 Liniya V. O. 29B, 199178 St. Petersburg, Russia
Email: matveis239@gmail.com

S. A. Nazarov
Affiliation: St. Petersburg State University, Universitetskii pr. 28, Petrodvorets, 198154 St. Petersburg, Russia; Peter the Great St. Petersburg Polytechnical University, Polytekhnicheskaya ul., 29, 195251 St. Petersburg, Russia; Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bol′shoi pr. V. O. 61, 199178 St. Petersburg, Russia
Email: srgnazarov@yahoo.co.uk

DOI: https://doi.org/10.1090/spmj/1500
Keywords: Infinite and truncated cross-shaped quantum waveguides, discrete spectrum, stable and decaying solution at the threshold of the continuous spectrum, asymptotics
Received by editor(s): November 1, 2016
Published electronically: March 30, 2018
Additional Notes: Supported by the Russian Science Foundation (grant no. 14-21-00035)
Article copyright: © Copyright 2018 American Mathematical Society

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