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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
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.
Received: 25.04.2019 Revised: 08.10.2019
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
Linking options:
https://www.mathnet.ru/eng/im8928https://doi.org/10.1070/IM8928 https://www.mathnet.ru/eng/im/v84/i6/p73
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Abstract page: | 425 | Russian version PDF: | 70 | English version PDF: | 25 | References: | 54 | First page: | 13 |
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