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Izvestiya: Mathematics, 2020, Volume 84, Issue 6, Pages 1105–1160
DOI: https://doi.org/10.1070/IM8928
(Mi im8928)
 

This article is cited in 20 scientific papers (total in 20 papers)

Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides

S. A. Nazarov

St. Petersburg State University, Mathematics and Mechanics Faculty
References:
Abstract: We describe and classify the thresholds of the continuous spectrum and the resulting resonances for general formally self-adjoint elliptic systems of second-order differential equations with Dirichlet or Neumann boundary conditions in domains with cylindrical and periodic outlets to infinity (in waveguides). These resonances arise because there are “almost standing” waves, that is, non-trivial solutions of the homogeneous problem which do not transmit energy. We consider quantum, acoustic, and elastic waveguides as examples. Our main focus is on degenerate thresholds which are characterized by the presence of standing waves with polynomial growth at infinity and produce effects lacking for ordinary thresholds. In particular, we describe the effect of lifting an eigenvalue from the degenerate zero threshold of the spectrum. This effect occurs for elastic waveguides of a vector nature and is absent from the scalar problems for cylindrical acoustic and quantum waveguides. Using the technique of self-adjoint extensions of differential operators in weighted spaces, we interpret the almost standing waves as eigenvectors of certain operators and the threshold as the corresponding eigenvalue. Here the threshold eigenvalues and the corresponding vector-valued functions not decaying at infinity can be obtained by approaching the threshold (the virtual level) either from below or from above. Hence their properties differ essentially from the customary ones. We state some open problems.
Keywords: elliptic systems, Dirichlet or Neumann boundary conditions, thresholds of continuous spectrum, virtual levels, threshold resonances, almost standing waves, spaces with separated asymptotic conditions, self-adjoint extensions of differential operators.
Funding agency Grant number
Russian Science Foundation 17-11-01003
This investigation was supported by the Russian Science Foundation (grant no. 17-11-01003).
Received: 25.04.2019
Revised: 08.10.2019
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2020, Volume 84, Issue 6, Pages 73–130
DOI: https://doi.org/10.4213/im8928
Bibliographic databases:
Document Type: Article
UDC: 517.956.8+517.956.328
Language: English
Original paper language: Russian
Citation: S. A. Nazarov, “Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides”, Izv. RAN. Ser. Mat., 84:6 (2020), 73–130; Izv. Math., 84:6 (2020), 1105–1160
Citation in format AMSBIB
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\paper Threshold resonances and virtual levels in the spectrum of~cylindrical and periodic waveguides
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\pages 73--130
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  • https://www.mathnet.ru/eng/im8928
  • https://doi.org/10.1070/IM8928
  • https://www.mathnet.ru/eng/im/v84/i6/p73
  • This publication is cited in the following 20 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Abstract page:425
    Russian version PDF:70
    English version PDF:25
    References:54
    First page:13
     
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