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 Zh. Vychisl. Mat. Mat. Fiz., 2013, Volume 53, Number 6, Pages 878–897 (Mi zvmmf9838)

Scheme for interpretation of approximately computed eigenvalues embedded in a continuous spectrum

S. A. Nazarov

St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: It is assumed that a trapped mode (i.e., a function decaying at infinity that leaves small discrepancies of order $\varepsilon\ll1$ in the Helmholtz equation and the Neumann boundary condition) at some frequency $\kappa^0$ is found approximately in an acoustic waveguide $\Omega^0$. Under certain constraints, it is shows that there exists a regularly perturbed waveguide $\Omega^\varepsilon$ with the eigenfrequency $\kappa^\varepsilon=\kappa^0+O(\varepsilon)$. The corresponding eigenvalue $\lambda^\varepsilon$ of the operator belongs to the continuous spectrum and, being naturally unstable, requires “fine tuning” of the parameters of the small perturbation of the waveguide wall. The analysis is based on the concepts of the augmented scattering matrix and the enforced stability of eigenvalues in the continuous spectrum.

Key words: acoustic waveguide, approximate computation of an eigenvalue in the continuous spectrum, enforced stability, augmented scattering matrix.

DOI: https://doi.org/10.7868/S0044466913060136

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English version:
Computational Mathematics and Mathematical Physics, 2013, 53:6, 702–720

Bibliographic databases:

UDC: 519.624

Citation: S. A. Nazarov, “Scheme for interpretation of approximately computed eigenvalues embedded in a continuous spectrum”, Zh. Vychisl. Mat. Mat. Fiz., 53:6 (2013), 878–897; Comput. Math. Math. Phys., 53:6 (2013), 702–720

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Bikmetov A.R., Gadyl'shin R.R., “On Local Perturbations of Waveguides”, Russ. J. Math. Phys., 23:1 (2016), 1–18
2. Nazarov S.A., Ruotsalainen K.M., “A Rigorous Interpretation of Approximate Computations of Embedded Eigenfrequencies of Water Waves”, Z. Anal. ihre. Anwend., 35:2 (2016), 211–242
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