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Mat. Zametki, 2010, Volume 87, Issue 5, Pages 764–786 (Mi mz8719)  

This article is cited in 26 scientific papers (total in 26 papers)

Opening of a Gap in the Continuous Spectrum of a Periodically Perturbed Waveguide

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: It is established that a small periodic singular or regular perturbation of the boundary of a cylindrical three-dimensional waveguide can open up a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator in the resulting periodic waveguide. A singular perturbation results in the formation of a periodic family of small cavities while a regular one leads to a gentle periodic bending of the boundary. If the period is short, there is no gap, while if it is long, a gap appears immediately after the first segment of the continuous spectrum. The result is obtained by asymptotic analysis of the eigenvalues of an auxiliary problem on the perturbed cell of periodicity.

Keywords: cylindrical waveguide, gap in a continuous spectrum, Laplace operator, Dirichlet problem, Helmholtz equation, cell of periodicity, Sobolev space


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English version:
Mathematical Notes, 2010, 87:5, 738–756

Bibliographic databases:

UDC: 517.956.227:517.958
Received: 18.08.2008

Citation: S. A. Nazarov, “Opening of a Gap in the Continuous Spectrum of a Periodically Perturbed Waveguide”, Mat. Zametki, 87:5 (2010), 764–786; Math. Notes, 87:5 (2010), 738–756

Citation in format AMSBIB
\by S.~A.~Nazarov
\paper Opening of a Gap in the Continuous Spectrum of a Periodically Perturbed Waveguide
\jour Mat. Zametki
\yr 2010
\vol 87
\issue 5
\pages 764--786
\jour Math. Notes
\yr 2010
\vol 87
\issue 5
\pages 738--756

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    This publication is cited in the following articles:
    1. Nazarov S.A., “Gaps and eigenfrequencies in the spectrum of a periodic acoustic waveguide”, Acoustical Physics, 59:3 (2013), 272–280  crossref  crossref  mathscinet  elib  elib  scopus
    2. Nazarov S.A., Ruotsalainen K., Taskinen J., “Gaps in the spectrum of the Neumann problem on a perforated plane”, Dokl. Math., 86:1 (2012), 574–578  crossref  mathscinet  zmath  isi  elib  elib  scopus
    3. Nazarov S.A., “Asymptotics of eigenfrequencies in the spectral gaps caused by a perturbation of a periodic waveguide”, Dokl. Math., 86:3 (2012), 871–875  crossref  mathscinet  zmath  isi  elib  scopus
    4. Piat V.Ch., Nazarov S.A., Ruotsalainen K., “Spectral gaps for water waves above a corrugated bottom”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 469:2149 (2013), 20120545  crossref  mathscinet  zmath  isi  elib  scopus
    5. D. I. Borisov, K. V. Pankrashin, “Gap Opening and Split Band Edges in Waveguides Coupled by a Periodic System of Small Windows”, Math. Notes, 93:5 (2013), 660–675  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. Bakharev F.L., Nazarov S.A., Ruotsalainen K.M., “A Gap in the Spectrum of the Neumann-Laplacian on a Periodic Waveguide”, Appl. Anal., 92:9 (2013), 1889–1915  crossref  mathscinet  zmath  isi  elib  scopus
    7. Borisov D., Pankrashkin K., “Quantum Waveguides with Small Periodic Perturbations: Gaps and Edges of Brillouin Zones”, J. Phys. A-Math. Theor., 46:23 (2013), 235203  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Fliss S., “A Dirichlet-to-Neumann Approach for the Exact Computation of Guided Modes in Photonic Crystal Waveguides”, SIAM J. Sci. Comput., 35:2 (2013), B438–B461  crossref  mathscinet  zmath  isi  scopus
    9. Nazarov S.A., “Asymptotic behavior of spectral gaps in a regularly perturbed periodic waveguide”, Vestnik of the St. Petersburg University: Mathematics, 46:2 (2013), 89–97  crossref  mathscinet  elib  elib  scopus
    10. S. A. Nazarov, “Gap opening around a given point of the spectrum of a cylindrical waveguide by means of gentle periodic perturbation of walls”, J. Math. Sci. (N. Y.), 206:3 (2015), 288–314  mathnet  crossref
    11. Khrabustovskyi A., “Opening Up and Control of Spectral Gaps of the Laplacian in Periodic Domains”, J. Math. Phys., 55:12 (2014), 121502  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. Nazarov S.A., Taskinen J., “Spectral Gaps For Periodic Piezoelectric Waveguides”, Z. Angew. Math. Phys., 66:6 (2015), 3017–3047  crossref  mathscinet  zmath  isi  elib  scopus
    13. Chesnel L., Nazarov S.A., “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  elib  scopus
    14. Khrabustovskyi A. Plum M., “Spectral properties of an elliptic operator with double-contrast coefficients near a hyperplane”, Asymptotic Anal., 98:1-2 (2016), 91–130  crossref  mathscinet  zmath  isi  elib  scopus
    15. Nazarov S.A., “Wave Scattering in the Joint of a Straight and a Periodic Waveguide”, Pmm-J. Appl. Math. Mech., 81:2 (2017), 129–147  crossref  mathscinet  isi  scopus
    16. Cardone G. Khrabustovskyi A., “Spectrum of a Singularly Perturbed Periodic Thin Waveguide”, J. Math. Anal. Appl., 454:2 (2017), 673–694  crossref  mathscinet  zmath  isi  scopus
    17. Delourme B., Fliss S., Joly P., Vasilevskaya E., “Trapped Modes in Thin and Infinite Ladder Like Domains. Part 1: Existence Results”, Asymptotic Anal., 103:3 (2017), 103–134  crossref  mathscinet  zmath  isi  scopus
    18. Bakharev F.L. Eugenia Perez M., “Spectral Gaps For the Dirichlet-Laplacian in a 3-D Waveguide Periodically Perturbed By a Family of Concentrated Masses”, Math. Nachr., 291:4 (2018), 556–575  crossref  mathscinet  zmath  isi  scopus
    19. Nazarov S.A. Taskinen J., “Essential Spectrum of a Periodic Waveguide With Non-Periodic Perturbation”, J. Math. Anal. Appl., 463:2 (2018), 922–933  crossref  mathscinet  zmath  isi  scopus
    20. Bakharev F.L., Exner P., “Geometrically Induced Spectral Effects in Tubes With a Mixed Dirichlet-Neumann Boundary”, Rep. Math. Phys., 81:2 (2018), 213–231  crossref  mathscinet  isi  scopus
    21. Piat V.Ch., Nazarov S.A., Ruotsalainen K.M., “Spectral Gaps and Non-Bragg Resonances in a Water Channel”, Rend. Lincei-Mat. Appl., 29:2 (2018), 321–342  crossref  mathscinet  zmath  isi  scopus
    22. S. A. Nazarov, “Breakdown of cycles and the possibility of opening spectral gaps in a square lattice of thin acoustic waveguides”, Izv. Math., 82:6 (2018), 1148–1195  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    23. S. A. Nazarov, “Asymptotics of eigenvalues in spectral gaps of periodic waveguides with small singular perturbations”, J. Math. Sci. (N. Y.), 243:5 (2019), 746–773  mathnet  crossref
    24. Nazarov S.A. Orive-Illera R. Perez-Martinez M.-E., “Asymptotic Structure of the Spectrum in a Dirichlet-Strip With Double Periodic Perforations”, Netw. Heterog. Media, 14:4 (2019), 733–757  crossref  mathscinet  isi
    25. S. A. Nazarov, “Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides”, Izv. Math., 84:6 (2020), 1105–1160  mathnet  crossref  crossref  mathscinet  isi  elib
    26. 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  crossref  isi
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