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Open waveguides in doubly periodic junctions of domains with different limit dimensions


Considering the spectral Neumann problem for the Laplace operator on a doubly periodic square grid of thin circular cylinders (of diameter ε ≪ 1) with nodes, which are sets of unit size, we show that by changing or removing one or several semi-infinite chains of nodes we can form additional spectral segments, the wave passage bands, in the essential spectrum of the original grid. The corresponding waveguide processes are localized in a neighborhood of the said chains, forming I-shaped, V-shaped, and L-shaped open waveguides. To derive the result, we use the asymptotic analysis of the eigenvalues of model problems on various periodicity cells.

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  1. 1.

    Cardone G., Nazarov S. A., and Taskinen J., “Spectra of open waveguides in periodic media,” J. Funct. Anal., 269, No. 8, 2328–2364 (2015).

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bakharev F. L. and Nazarov S. A., “Gaps in the spectrum of a waveguide composed of domains with different limiting dimensions,” Sib. Math. J., 56, No. 4, 575–592 (2015).

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Carini J. P., Londergan J. T., and Murdock D. P., Binding and Scattering in Two-Dimensional Systems: Applications to Quantum Wires, Waveguides, and Photonic Crystals, Springer-Verlag, Berlin (1999) (Lect. Notes Phys.).

    MATH  Google Scholar 

  4. 4.

    Birman M. Sh. and Solomyak M. Z., Spectral Theory of Selfadjoint Operators in Hilbert Space, D. Reidel Publ. Co., Dordrecht (1987).

    MATH  Google Scholar 

  5. 5.

    Gelfand I. M., “Expansion in eigenfunctions of an equation with periodic coefficients,” Dokl. Akad. Nauk SSSR, 73, 1117–1120 (1950).

    Google Scholar 

  6. 6.

    Nazarov S. A., “Elliptic boundary value problems with periodic coefficients in a cylinder,” Math. USSR-Izv., 18, No. 1, 89–98 (1982).

    Article  MATH  Google Scholar 

  7. 7.

    Nazarov S. A. and Plamenevsky B. A., Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin and New York (1994).

    Book  MATH  Google Scholar 

  8. 8.

    Arsen’ev A. A., “The existence of resonance poles and scattering resonances in the case of boundary conditions of the second and third kind,” USSR Comput. Math. Math. Phys., 16, No. 3, 171–177 (1976).

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Beale J. T., “Scattering frequencies of resonators,” Comm. Pure Appl. Math., 26, No. 4, 549–563 (1973).

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Gadyl’shin R. R., “On eigenfrequencies of bodies with thin branches. II. Asymptotics,” Math. Notes, 55, No. 1, 14–23 (1994).

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Van Dyke M., Perturbation Methods in Fluid Mechanics, Academic Press, New York and London (1964).

    MATH  Google Scholar 

  12. 12.

    Il’in A. M., Matching Asymptotic Expansions for Solutions of Boundary Value Problems [Russian], Nauka, Moscow (1989).

    MATH  Google Scholar 

  13. 13.

    Nazarov S. A., “Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold,” Sib. Math. J., 51, No. 5, 866–878 (2010).

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Agmon S., Douglis A., and Nirenberg L., “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,” Comm. Pure Appl. Math., 12, 623–727 (1959).

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to F. L. Bakharev.

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Original Russian Text Copyright © 2016 Bakharev F.L. and Nazarov S.A.

The authors were supported by St. Petersburg State University (Project The first author was also supported by the Chebyshev Laboratory of St. Petersburg State University, the Government of the Russian Federation (Agreement No. 11.G34.31.0026), and Gazprom Neft PJSC.

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Bakharev, F.L., Nazarov, S.A. Open waveguides in doubly periodic junctions of domains with different limit dimensions. Sib Math J 57, 943–956 (2016).

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  • spectral Neumann problem
  • doubly periodic grid
  • localized waves
  • open waveguides