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The discrete spectrum of cross-shaped waveguides


Authors: F. L. Bakharev, S. G. Matveenko and S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 28 (2016), nomer 2.
Journal: St. Petersburg Math. J. 28 (2017), 171-180
MSC (2010): Primary 81Q37
DOI: https://doi.org/10.1090/spmj/1444
Published electronically: February 15, 2017
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Abstract: The discrete spectrum of the Dirichlet problem for the Laplace operator on the union of two circular unit cylinders whose axes intersect at the right angle consists of a single eigenvalue. For the threshold value of the spectral parameter, this problem has no bounded solutions. When the angle between the axes reduces, the multiplicity of the discrete spectrum grows unboundedly.


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Additional Information

F. L. Bakharev
Affiliation: St. Petersburg State University, Universitetskii pr. 28, Petrodvorets, 198504 St. Petersburg; P. L. Chebyshev laboratory, 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 3A, office 417, 194100 St. Petersburg; P. L. 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; Peter the Great St. Petersburg Polytechnical University, Polytekhnicheskaya yl., 29, 195251 St. Petersburg; 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, s.nazarov@spbu.ru

DOI: https://doi.org/10.1090/spmj/1444
Keywords: Cross-shaped quantum waveguide, discrete spectrum multiplicity, stabilizing solution at the threshold of the continuous spectrum
Received by editor(s): October 29, 2015
Published electronically: February 15, 2017
Additional Notes: The research was done in the framework of the SPbGU project no. 0.38.237.2014
The first two authors were supported by the P. L. Chebyshev Laboratory of SPbGU (RF Governement grant no. 11.G34.31.0026)
Article copyright: © Copyright 2017 American Mathematical Society