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 Mat. Zametki, 2013, Volume 93, Issue 5, Pages 665–683 (Mi mz9284)

Gap Opening and Split Band Edges in Waveguides Coupled by a Periodic System of Small Windows

D. I. Borisovab, K. V. Pankrashinc

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa
b Bashkir State Pedagogical University, Ufa
c Unit\'e mixte de recherche 8628, CNRS, University Paris-Sud 11, Orsay, France

Abstract: As an example of two coupled waveguides, we construct a periodic second-order differential operator acting in a Euclidean domain and having spectral gaps whose edges are attained strictly inside the Brillouin zone. The waveguides are modeled by the Laplacian in two infinite strips of different width that have a common interior boundary. On this common boundary, we impose the Neumann boundary condition, but cut out a periodic system of small holes, while on the remaining exterior boundary we impose the Dirichlet boundary condition. It is shown that, by varying the widths of the strips and the distance between the holes, one can control the location of the extrema of the band functions as well as the number of the open gaps. We calculate the leading terms in the asymptotics for the gap lengths and the location of the extrema.

Keywords: Laplacian, periodic operator, waveguide, band spectrum, spectral gap, dispersion laws, matching of asymptotic expansions, boundary conditions.

DOI: https://doi.org/10.4213/mzm9284

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English version:
Mathematical Notes, 2013, 93:5, 660–675

Bibliographic databases:

UDC: 517.956

Citation: D. I. Borisov, K. V. Pankrashin, “Gap Opening and Split Band Edges in Waveguides Coupled by a Periodic System of Small Windows”, Mat. Zametki, 93:5 (2013), 665–683; Math. Notes, 93:5 (2013), 660–675

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz9284
• https://doi.org/10.4213/mzm9284
• http://mi.mathnet.ru/eng/mz/v93/i5/p665

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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. D. Borisov, K. Pankrashkin, “Quantum waveguides with small periodic perturbations: gaps and edges of Brillouin zones”, J. Phys. A, 46:23 (2013), 235203, 18 pp.
2. 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
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4. A. Khrabustovskyi, “Opening up and control of spectral gaps of the Laplacian in periodic domains”, J. Math. Phys., 55:12 (2014), 121502
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6. D. I. Borisov, “On the band spectrum of a Schrödinger operator in a periodic system of domains coupled by small windows”, Russ. J. Math. Phys., 22:2 (2015), 153–160
7. P. Exner, H. Kovařík, Quantum waveguides, Theoretical and Mathematical Physics, Springer-Verlag, Cham, 2015, xxii+382 pp.
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10. D. I. Borisov, “Creation of spectral bands for a periodic domain with small windows”, Russ. J. Math. Phys., 23:1 (2016), 19–34
11. A. S. Melikhova, I. Yu. Popov, “Spectral problem for solvable model of bent nano peapod”, Appl. Anal., 96:2 (2017), 215–224
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13. F. L. Bakharev, M. E. Perez, “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
14. S. A. Nazarov, J. Taskinen, “Essential spectrum of a periodic waveguide with non-periodic perturbation”, J. Math. Anal. Appl., 463:2 (2018), 922–933
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