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Gaps in the spectrum of a waveguide composed of domains with different limiting dimensions


We consider an acoustic waveguide (the Neumann problem for the Helmholtz equation) shaped like a periodic family of identical beads on a thin cylinder rod. Under minor restrictions on the bead and rod geometry, we use asymptotic analysis to establish the opening of spectral gaps and find their geometric characteristics. The main technical difficulties lie in the justification of asymptotic formulas for the eigenvalues of the model problem on the periodicity cell due to its arbitrary shape.

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  1. 1.

    Khrabustovskyi A., “Periodic elliptic operators with asymptotically preassigned spectrum,” Asymptot. Anal., 82, No. 1–2, 1–37 (2013).

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Khrabustovskyi A. and Khruslov E., “Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed traps,” Math. Methods Appl. Sci., DOI: 10.1002/ mma.3046 (2014).

    Google Scholar 

  3. 3.

    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  Google Scholar 

  4. 4.

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

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  6. 6.

    Kozlov V. A., Maz’ya V. G., and Movchan A. B., “Asymptotic analysis of a mixed boundary value problem in a multistructure,” Asymptot. Anal., 8, 105–143 (1994).

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Nazarov S. A., “Junctions of singularly degenerating domains with different limit dimensions. II,” J. Math. Sci., 97, No. 3, 4085–4108 (1999).

    Article  Google Scholar 

  8. 8.

    Nazarov S. A., “Asymptotic analysis and modeling of the jointing of a massive body with thin rods,” J. Math. Sci., 127, No. 127, 2192–2262 (2005).

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Gadyl’shin R. R., “On the eigenvalues of a ‘dumb-bell with a thin handle’,” Izv. Math., 69, No. 2, 265–329 (2005).

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Nazarov S. A., “Asymptotics of solutions to the spectral elasticity problem for a spatial body with a thin coupler,” Siberian Math. J., 53, No. 2, 274–290 (2012).

    Article  MATH  Google Scholar 

  11. 11.

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

    Google Scholar 

  12. 12.

    Skriganov M. M., “Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators,” Proc. Steklov Inst. Math., 171, 1–121 (1987).

    MathSciNet  Google Scholar 

  13. 13.

    Kuchment P. A., “Floquet theory for partial differential equations,” Russian Math. Surveys, 37, No. 4, 1–60 (1982).

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Ladyzhenskaya O. A., The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York etc. (1985).

    Book  MATH  Google Scholar 

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Correspondence to F. L. Bakharev.

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Original Russian Text Copyright © 2015 Bakharev F.L. and Nazarov S.A.

The authors were supported by St. Petersburg State University (Project and the Russian Foundation for Basic Research (Grant 15–01–02175).

St. Petersburg. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 4, pp. 732–751, July–August, 2015; DOI: 10.17377/smzh.2015.56.402.

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Bakharev, F.L., Nazarov, S.A. Gaps in the spectrum of a waveguide composed of domains with different limiting dimensions. Sib Math J 56, 575–592 (2015).

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  • Neumann problem
  • junction of domains with different limiting dimensions
  • periodic waveguide
  • spectral gaps
  • asymptotics