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Asymptotic formulas for trapped modes and for eigenvalues below the threshold of the continuous spectrum of a waveguide with a thin screening barrier


Author: S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 23 (2011), nomer 3.
Journal: St. Petersburg Math. J. 23 (2012), 571-601
MSC (2010): Primary 35J05
DOI: https://doi.org/10.1090/S1061-0022-2012-01209-4
Published electronically: March 2, 2012
MathSciNet review: 2896170
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Abstract: Asymptotic formulas are found for the eigenvalues and eigenfunctions (trapped modes) of the mixed boundary value problem for the Laplace operator in an $ n$-dimensional cylindrical waveguide with a thin screening barrier obtained by a regular perturbation of a part $ \theta $ of the hyperplane orthogonal to the axis of the cylinder; the boundary $ \partial \theta $ is smooth and $ (n-1)$-dimensional. These asymptotic formulas agree with the sufficient conditions for the discrete spectrum to be nonempty, deduced via the variational method. For an unbounded waveguide, both the algorithm for obtaining asymptotic formulas, and even the orders themselves of the main correction terms turn out to be different from those for a bounded domain. The same refers to the justification procedure for asymptotic expansions, which employs substantially the spectral theory machinery.


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Additional Information

S. A. Nazarov
Affiliation: Institute of Mechanical Engineering Problems, Bol’shoĭ Prospekt V.O. 61, St. Petersburg 199178, Russia
Email: srgnazarov@yahoo.co.uk

DOI: https://doi.org/10.1090/S1061-0022-2012-01209-4
Keywords: Waveguide with barrier, trapped modes, discrete and continuous spectra, asymptotic formulas for eigenvalues, boundary perturbation, spectral measure
Received by editor(s): January 25, 2010
Published electronically: March 2, 2012
Additional Notes: Supported by RFBR (grant no. 09-01-00759)
Article copyright: © Copyright 2012 American Mathematical Society