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Mat. Sb., 2012, Volume 203, Number 2, Pages 3–32 (Mi msb7798)  

This article is cited in 20 scientific papers (total in 20 papers)

Asymptotic behaviour of an eigenvalue in the continuous spectrum of a narrowed waveguide

G. Cardonea, S. A. Nazarovb, K. Ruotsalainenc

a Facoltà di Ingegneria, Università degli Studi del Sannio
b Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg
c University of Oulu

Abstract: The existence of an eigenvalue embedded in the continuous spectrum is proved for the Neumann problem for Helmholtz's equation in a two-dimensional waveguide with two outlets to infinity which are half-strips of width $1$ and $1-\varepsilon$, where $\varepsilon>0$ is a small parameter. The width function of the part of the waveguide connecting these outlets is of order $\sqrt{\varepsilon}$; it is defined as a linear combination of three fairly arbitrary functions, whose coefficients are obtained from a certain nonlinear equation. The result is derived from an asymptotic analysis of an auxiliary object, the augmented scattering matrix.
Bibliography: 29 titles.

Keywords: acoustic waveguide, water waves in a channel, eigenvalues in the continuous spectrum, asymptotic behaviour, augmented scattering matrix.
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English version:
Sbornik: Mathematics, 2012, 203:2, 153–182

Bibliographic databases:

UDC: 517.956.8+517.956.227
MSC: Primary 35J05, 35P05; Secondary 35P25
Received: 11.10.2010 and 28.04.2011

Citation: G. Cardone, S. A. Nazarov, K. Ruotsalainen, “Asymptotic behaviour of an eigenvalue in the continuous spectrum of a narrowed waveguide”, Mat. Sb., 203:2 (2012), 3–32; Sb. Math., 203:2 (2012), 153–182

Citation in format AMSBIB
\by G.~Cardone, S.~A.~Nazarov, K.~Ruotsalainen
\paper Asymptotic behaviour of an eigenvalue in the continuous spectrum of a~narrowed waveguide
\jour Mat. Sb.
\yr 2012
\vol 203
\issue 2
\pages 3--32
\jour Sb. Math.
\yr 2012
\vol 203
\issue 2
\pages 153--182

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    This publication is cited in the following articles:
    1. Cardone G., Nazarov S.A., Ruotsalainen K., “Bound states of a converging quantum waveguide”, ESAIM Math. Model. Numer. Anal., 47:1 (2013), 305–315  crossref  mathscinet  zmath  isi  scopus
    2. Cardone G., Pastukhova S.E., Perugia C., “Estimates in homogenization of degenerate elliptic equations by spectral method”, Asymptotic Anal., 81:3-4 (2013), 189–209  crossref  mathscinet  zmath  isi  elib  scopus
    3. G. Cardone, “Asymptotic Analysis of an Eigenvalue in the Discrete Spectrum of a Quantum Waveguide”, 11th International Conference of Numerical Analysis and Applied Mathematics 2013, Pts 1 and 2 (ICNAAM-2013), AIP Conf. Proc., 1558, eds. Simos T., Psihoyios G., Tsitouras C., Amer. Inst. Phys., 2013, 1809–1812  crossref  adsnasa  isi  scopus
    4. T. Durante, “Asymptotic Behaviour of Eigenfunctions in a Thin Cylinder with Distorted Ends”, 11th International Conference of Numerical Analysis and Applied Mathematics 2013, Pts 1 and 2 (ICNAAM-2013), AIP Conf. Proc., 1558, eds. Simos T., Psihoyios G., Tsitouras C., Amer. Inst. Phys., 2013, 1813–1816  crossref  mathscinet  adsnasa  isi  scopus
    5. S. A. Nazarov, “Scheme for interpretation of approximately computed eigenvalues embedded in a continuous spectrum”, Comput. Math. Math. Phys., 53:6 (2013), 702–720  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    6. S. A. Nazarov, K. Ruotsalainen, P. Uusitalo, “The Y-junction of quantum waveguides”, ZAMM Z. Angew. Math. Mech., 94:6 (2014), 477–486  crossref  mathscinet  zmath  isi  elib  scopus
    7. S. A. Nazarov, K. Ruotsalainen, P. Uusitalo, “Bound states of waveguides with two right-angled bends”, J. Math. Phys., 56:2 (2015), 021505, 24 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    8. A.-S. Bonnet-Ben Dhia, L. Chesnel, S. A. Nazarov, “Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions”, Inverse Problems, 31:4 (2015), 045006  crossref  mathscinet  zmath  isi  scopus
    9. P. Exner, H. Kovařík, Quantum waveguides, Theoretical and Mathematical Physics, Springer, Cham, 2015, xxii+382 pp.  crossref  mathscinet  zmath  isi
    10. V. A. Kozlov, S. A. Nazarov, A. Orlof, “Trapped modes supported by localized potentials in the zigzag graphene ribbon”, C. R. Math. Acad. Sci. Paris, 354:1 (2016), 63–67  crossref  mathscinet  zmath  isi
    11. A. R. Bikmetov, R. R. Gadyl'shin, “On local perturbations of waveguides”, Russ. J. Math. Phys., 23:1 (2016), 1–18  crossref  mathscinet  zmath  isi  scopus
    12. 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  crossref  mathscinet  zmath  isi  scopus
    13. V. A. Kozlov, S. A. Nazarov, A. Orlof, “Trapped modes in zigzag graphene nanoribbons”, Z. Angew. Math. Phys., 68:4 (2017), 78, 31 pp.  crossref  mathscinet  zmath  isi  scopus
    14. T. Durante, “Waveguides with a box-shaped perturbation: Eigenvalues of the Neumann problem”, Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2016 (ICNAAM-2016), AIP Conf. Proc., 1863, eds. T. Simos, C. Tsitouras, Amer. Inst. Phys., Melville, NY, 2017, 510003  crossref  isi  scopus
    15. Cardone G., Durante T., Nazarov S.A., “Embedded Eigenvalues of the Neumann Problem in a Strip With a Box-Shaped Perturbation”, J. Math. Pures Appl., 112 (2018), 1–40  crossref  mathscinet  zmath  isi  scopus
    16. Bonnet-Ben Dhia A.-S., Chesnel L., Nazarov S.A., “Perfect Transmission Invisibility For Waveguides With Sound Hard Walls”, J. Math. Pures Appl., 111 (2018), 79–105  crossref  mathscinet  zmath  isi  scopus
    17. Chesnel L., Nazarov S.A., Pagneux V., “Invisibility and Perfect Reflectivity in Waveguides With Finite Length Branches”, SIAM J. Appl. Math., 78:4 (2018), 2176–2199  crossref  mathscinet  zmath  isi  scopus
    18. Durante T., “Dirichet Laplacian in a Perforated Plane With Semi-Infinite Inclusions”, AIP Conference Proceedings, 1978, Amer Inst Physics, 2018, UNSP 140004-1  crossref  isi  scopus
    19. Chesnel L., Nazarov S.A., “Non Reflection and Perfect Reflection Via Fano Resonance in Waveguides”, Commun. Math. Sci., 16:7 (2018), 1779–1800  crossref  mathscinet  isi  scopus
    20. Gaudiello A., Gomez D., Perez-Martinez M.-E., “Asymptotic Analysis of the High Frequencies For the Laplace Operator in a Thin T-Like Shaped Structure”, J. Math. Pures Appl., 134 (2020), 299–327  crossref  mathscinet  zmath  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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