Abstract:
It is shown in the paper that, under several orthogonality and normalization conditions and a proper choice of accessory parameters, a simple eigenvalue lying between thresholds of the continuous spectrum of the Dirichlet problem in a domain with a cylindrical outlet to infinity is not taken out from the spectrum by a small compact perturbation of the Helmholtz operator. The result is obtained by means of an asymptotic analysis of the augmented
scattering matrix.
Keywords:
continuous spectrum, discrete spectrum, perturbation of eigenvalue, local perturbations of quantum waveguide.
Citation:
S. A. Nazarov, “Enforced Stability of a Simple Eigenvalue in the Continuous Spectrum of a Waveguide”, Funktsional. Anal. i Prilozhen., 47:3 (2013), 37–53; Funct. Anal. Appl., 47:3 (2013), 195–209
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\by S.~A.~Nazarov
\paper Enforced Stability of a Simple Eigenvalue in the Continuous Spectrum of a Waveguide
\jour Funktsional. Anal. i Prilozhen.
\yr 2013
\vol 47
\issue 3
\pages 37--53
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\jour Funct. Anal. Appl.
\yr 2013
\vol 47
\issue 3
\pages 195--209
\crossref{https://doi.org/10.1007/s10688-013-0026-8}
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Linking options:
https://www.mathnet.ru/eng/faa3117
https://doi.org/10.4213/faa3117
https://www.mathnet.ru/eng/faa/v47/i3/p37
This publication is cited in the following 61 articles:
S. A. Nazarov, “Raspredelenie mod sobstvennykh kolebanii v plastine, zaglublennoi v absolyutno zhëstkoe poluprostranstvo”, Matematicheskie voprosy teorii rasprostraneniya voln. 53, Zap. nauchn. sem. POMI, 521, POMI, SPb., 2023, 154–199
Lucas Chesnel, Jérémy Heleine, Sergei A. Nazarov, Jari Taskinen, “Acoustic waveguide with a dissipative inclusion”, ESAIM: M2AN, 57:6 (2023), 3585
S. A. Nazarov, “Natural Oscillations of an Elastic Half-Strip with a Different Arrangement of Fixation Areas of Its Edges”, Akusticheskii zhurnal, 69:4 (2023), 398
S. A. Nazarov, “Natural Oscillations of an Elastic Half-Strip with a Different Arrangement of Fixation Areas of Its Edges”, Acoust. Phys., 69:4 (2023), 424
S. A. Nazarov, “Asimptoticheskii analiz spektra kvantovogo volnovoda s shirokim “oknom” Neimana v svete mekhaniki treschin”, Matematicheskie voprosy teorii rasprostraneniya voln. 52, Zap. nauchn. sem. POMI, 516, POMI, SPb., 2022, 176–237
S. A. Nazarov, “Abnormal Transmission of Elastic Waves through a Thin Ligament Connecting Two Planar Isotropic Waveguides”, Mech. Solids, 57:8 (2022), 1908
Lucas Chesnel, Jérémy Heleine, Sergei A. Nazarov, “Acoustic passive cloaking using thin outer resonators”, Z. Angew. Math. Phys., 73:3 (2022)
S. A. Nazarov, “The preservation of threshold resonances and the splitting off of eigenvalues from the threshold of the continuous spectrum of quantum waveguides”, Sb. Math., 212:7 (2021), 965–1000
Nazarov S.A., Chesnel L., “Transmission and Trapping of Waves in An Acoustic Waveguide With Perforated Cross-Walls”, Fluid Dyn., 56:8 (2021), 1070–1093
S. A. Nazarov, “Scattering of Low-Frequency Elastic Waves in An Infinite Kirchhoff Plate”, J Math Sci, 252:5 (2021), 664
S. A. Nazarov, “Trapping of Waves in Semiinfinite Kirchhoff Plate with Periodically Damaged Edge”, J Math Sci, 257:5 (2021), 684
S. A. Nazarov, L. Chesnel, “Anomalies of acoustic wave propagation in two semi-infinite cylinders connected by a flattened ligament”, Comput. Math. Math. Phys., 61:4 (2021), 646–663
V. A. Kozlov, S. A. Nazarov, A. Orlof, “Trapped Modes in Armchair Graphene Nanoribbons”, J Math Sci, 252:5 (2021), 624
S. A. Nazarov, “Construction of a trapped mode with a small frequency in an elastic waveguide”, Funct. Anal. Appl., 54:1 (2020), 31–44
S. A. Nazarov, “Waveguide with double threshold resonance at a simple threshold”, Sb. Math., 211:8 (2020), 1080–1126
S. A. Nazarov, “Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides”, Izv. Math., 84:6 (2020), 1105–1160
F. L. Bakharev, S. A. Nazarov, “The discrete spectrum of an infinite kirchhoff plate in the form of a locally perturbed strip”, Siberian Math. J., 61:2 (2020), 233–247
Nazarov S.A., “Anomalies of Acoustic Wave Scattering Near the Cut-Off Points of Continuous Spectrum (a Review)”, Acoust. Phys., 66:5 (2020), 477–494
Chesnel L., Nazarov S.A., “Exact Zero Transmission During the Fano Resonance Phenomenon in Non-Symmetric Waveguides”, Z. Angew. Math. Phys., 71:3 (2020), 82
Chesnel L. Nazarov S.A. Taskinen J., “Surface Waves in a Channel With Thin Tunnels and Wells At the Bottom: Non-Reflecting Underwater Topography”, Asymptotic Anal., 118:1-2 (2020), 81–122
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