Abstract:
Perturbations of an eigenvalue in the continuous spectrum of the Neumann problem for the Laplacian in a strip waveguide with an obstacle symmetric about the midline are studied. Such an eigenvalue is known to be unstable, and an arbitrarily small perturbation can cause it to leave the spectrum to become a complex resonance point. Conditions on the perturbation of the obstacle boundary are found under which the eigenvalue persists in the continuous spectrum. The result is obtained via the asymptotic analysis of an auxiliary object, namely, an augmented scattering matrix.
Key words:
waveguide with an obstacle, perturbation, eigenvalue in the continuous spectrum, enforced stability, augmented scattering matrix.
Citation:
S. A. Nazarov, “Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle”, Zh. Vychisl. Mat. Mat. Fiz., 52:3 (2012), 521–538; Comput. Math. Math. Phys., 52:3 (2012), 448–464
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\paper Enforced stability of an eigenvalue in the continuous spectrum of a~waveguide with an~obstacle
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2012
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\issue 3
\pages 521--538
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\jour Comput. Math. Math. Phys.
\yr 2012
\vol 52
\issue 3
\pages 448--464
\crossref{https://doi.org/10.1134/S096554251203013X}
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Linking options:
https://www.mathnet.ru/eng/zvmmf9675
https://www.mathnet.ru/eng/zvmmf/v52/i3/p521
This publication is cited in the following 25 articles:
S. A. Nazarov, “Abnormal Transmission of Elastic Waves through a Thin Ligament Connecting Two Planar Isotropic Waveguides”, Mech. Solids, 57:8 (2022), 1908
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
V. A. Kozlov, S. A. Nazarov, A. Orlof, “Trapped modes in armchair graphene nanoribbons”, Matematicheskie voprosy teorii rasprostraneniya voln. 49, Zap. nauchn. sem. POMI, 483, POMI, SPb., 2019, 85–115
A.-S. Bonnet-Ben Dhia, L. Chesnel, S. A. Nazarov, “Perfect transmission invisibility for waveguides with sound hard walls”, J. Math. Pures Appl., 111 (2018), 79–105
V. Ch. Piat, S. A. Nazarov, J. Taskinen, “Embedded eigenvalues forwater-waves in athree-dimensional channel with athin screen”, Q. J. Mech. Appl. Math., 71:2 (2018), 187–220
L. Chesnel, S. A. Nazarov, V. Pagneux, “Invisibility and perfect reflectivity in waveguides with finite length branches”, SIAM J. Appl. Math., 78:4 (2018), 2176–2199
S. A. Nazarov, “Various manifestations of Wood anomalies in locally distorted quantum waveguides”, Comput. Math. Math. Phys., 58:11 (2018), 1838–1855
Chesnel L., Nazarov S.A., “Non Reflection and Perfect Reflection Via Fano Resonance in Waveguides”, Commun. Math. Sci., 16:7 (2018), 1779–1800
A. Laptev, S. M. Sasane, “Perturbations of embedded eigenvalues for a magnetic Schrödinger operator on a cylinder”, J. Math. Phys., 58:1 (2017), 012105
V. A. Kozlov, S. A. Nazarov, A. Orlof, “Trapped modes in zigzag graphene nanoribbons”, Z. Angew. Math. Phys., 68:4 (2017), 78
S. A. Nazarov, “Almost standing waves in a periodic waveguide with a resonator and near-threshold eigenvalues”, St. Petersburg Math. J., 28:3 (2017), 377–410
A. R. Bikmetov, R. R. Gadyl'shin, “On local perturbations of waveguides”, Russ. J. Math. Phys., 23:1 (2016), 1–18
L. Chesnel, S. A. Nazarov, “Team organization may help swarms of flies to become invisible in closed waveguides”, Inverse Probl. Imaging, 10:4 (2016), 977–1006
S. A. Nazarov, K. M. Ruotsalainen, M. Silvola, “Trapped modes in piezoelectric and elastic waveguides”, J. Elast., 124:2 (2016), 193–223
S. A. Nazarov, K. M. Ruotsalainen, “A rigorous interpretation of approximate computations of embedded eigenfrequencies of water waves”, Z. Anal. ihre. Anwend., 35:2 (2016), 211–242
V. A. Kozlov, S. A. Nazarov, A. Orlof, “Trapped modes supported by localized potentials in the zigzag graphene ribbon”, C. R. Math., 354:1 (2016), 63–67
S. A. Nazarov, “Transmission Conditions in One-Dimensional Model of a Rectangular Lattice of Thin Quantum Waveguides”, J Math Sci, 219:6 (2016), 994
J. T. Kemppainen, S. A. Nazarov, K. M. Ruotsalainen, “Perturbation analysis of embedded eigenvalues for water-waves”, J. Math. Anal. Appl., 427:1 (2015), 399–427
A.-S. B.-B. Dhia, L. Chesnel, S. A. Nazarov, “Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions”, Inverse Probl., 31:4 (2015), 045006
L. Chesnel, N. Hyvonen, S. Staboulis, “Construction of indistinguishable conductivity perturbations for the point electrode model in electrical impedance tomography”, SIAM J. Appl. Math., 75:5 (2015), 2093–2109
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