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

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

DOI: https://doi.org/10.4213/mzm8719

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

Bibliographic databases:

UDC: 517.956.227:517.958

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
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• http://mi.mathnet.ru/eng/mz8719
• https://doi.org/10.4213/mzm8719
• http://mi.mathnet.ru/eng/mz/v87/i5/p764

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
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
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
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
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
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
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
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
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
11. Khrabustovskyi A., “Opening Up and Control of Spectral Gaps of the Laplacian in Periodic Domains”, J. Math. Phys., 55:12 (2014), 121502
12. Nazarov S.A., Taskinen J., “Spectral Gaps For Periodic Piezoelectric Waveguides”, Z. Angew. Math. Phys., 66:6 (2015), 3017–3047
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
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
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
16. Cardone G. Khrabustovskyi A., “Spectrum of a Singularly Perturbed Periodic Thin Waveguide”, J. Math. Anal. Appl., 454:2 (2017), 673–694
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
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
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
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
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
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
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
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