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

Nazarov, S.

doi:10.1090/S1061-0022-2012-01209-4

Asymptotic formulas for trapped modes and for eigenvalues below the threshold of the continuous spectrum of a waveguide with a thin screening barrier
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by S. A. Nazarov
Translated by: A. Plotkin

St. Petersburg Math. J. 23 (2012), 571-601

DOI: https://doi.org/10.1090/S1061-0022-2012-01209-4

Published electronically: March 2, 2012

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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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