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Mat. Zametki:

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

This article is cited in 14 scientific papers (total in 14 papers)

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.


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

Bibliographic databases:

UDC: 517.956
Received: 02.02.2012

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
\by D.~I.~Borisov, K.~V.~Pankrashin
\paper Gap Opening and Split Band Edges in Waveguides Coupled by a Periodic System of Small Windows
\jour Mat. Zametki
\yr 2013
\vol 93
\issue 5
\pages 665--683
\jour Math. Notes
\yr 2013
\vol 93
\issue 5
\pages 660--675

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    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.  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  crossref  mathscinet  elib  elib
    3. 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
    4. A. Khrabustovskyi, “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
    5. S. A. Nazarov, J. Taskinen, “Spectral gaps for periodic piezoelectric waveguides”, Z. Angew. Math. Phys., 66:6 (2015), 3017–3047  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    7. P. Exner, H. Kovařík, Quantum waveguides, Theoretical and Mathematical Physics, Springer-Verlag, Cham, 2015, xxii+382 pp.  crossref  mathscinet  zmath  isi
    8. G. Raikov, “Spectral asymptotics for waveguides with perturbed periodic twisting”, J. Spectr. Theory, 6:2 (2016), 331–372  crossref  mathscinet  zmath  isi  elib  scopus
    9. A. Khrabustovskyi, M. Plum, “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
    10. D. I. Borisov, “Creation of spectral bands for a periodic domain with small windows”, Russ. J. Math. Phys., 23:1 (2016), 19–34  crossref  mathscinet  zmath  isi  elib  scopus
    11. A. S. Melikhova, I. Yu. Popov, “Spectral problem for solvable model of bent nano peapod”, Appl. Anal., 96:2 (2017), 215–224  crossref  mathscinet  zmath  isi  elib  scopus
    12. G. Cardone, A. Khrabustovskyi, “Spectrum of a singularly perturbed periodic thin waveguide”, J. Math. Anal. Appl., 454:2 (2017), 673–694  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
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