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Open waveguides in a thin Dirichlet lattice: II. localized waves and radiation conditions

  • S. A. Nazarov
Article

Abstract

Wave processes localized near an angular open waveguide obtained by thickening two perpendicular semi-infinite rows of ligaments in a thin square lattice of quantum waveguides (Dirichlet problem for the Helmholtz equation) are investigated. Waves of two types are discovered: the first are observed near the lattice nodes and almost do not affect the ligaments, while the second, on the contrary, excite oscillations in the ligaments, whereas the nodes stay relatively at rest. Asymptotic representations of the wave fields are derived, and radiation conditions are imposed on the basis of the Umov–Mandelstam energy principle.

Keywords

open waveguides square lattice Dirichlet problem for the Helmholtz equation asymptotic representations of wave fields Umov–Mandelstam principle 

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References

  1. 1.
    J. P. Carini, J. T. Londergan, and D. P. Murdock, Binding and Scattering in Two-Dimensional Systems: Applications to Quantum Wires, Waveguides, and Photonic Crystals (Springer-Verlag, Berlin, 1999).MATHGoogle Scholar
  2. 2.
    P. A. Kuchment, “Floquet theory for partial differential equations,” Russ. Math. Surv. 37 (4), 1–60 (1982).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    M. M. Skriganov, “Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators,” Proc. Steklov Inst. Math. 171 (2), 1–121 (1987).MathSciNetMATHGoogle Scholar
  4. 4.
    P. Kuchment, Floquet Theory for Partial Differential Equations (Birkhäuser, Basel, 1993).CrossRefMATHGoogle Scholar
  5. 5.
    D. Grieser, “Spectra of graph neighborhoods and scattering,” Proc. London Math. Soc. 97 (3), 718–752 (2008).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    S. A. Nazarov, “Discrete spectrum of cross-shaped quantum waveguides,” J. Math. Sci. 196 (3), 346–376 (2014).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    S. A. Nazarov, “Asymptotics of the scattering matrix near the edges of a spectral gap,” Izv. Akad. Nauk, Ser. Mat. 81 (1), 3–64 (2017).MathSciNetGoogle Scholar
  8. 8.
    S. A. Nazarov, “Open waveguides in a thin Dirichlet lattice: I. Asymptotic structure of the spectrum,” Comput. Math. Math. Phys. 57 (1), 156–174 (2017).MathSciNetCrossRefGoogle Scholar
  9. 9.
    G. Cardone, S. A. Nazarov, and J. Taskinen, “Spectra of open waveguides in periodic media,” J. Funct. Anal. 269 (8), 2328–2364 (2015).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    I. M. Gel’fand, “Eigenfunction expansion for an equation with periodic coefficients,” Dokl. Akad. Nauk SSSR 73, 1117–1120 (1950).Google Scholar
  11. 11.
    S. A. Nazarov, “Elliptic boundary value problems with periodic coefficients in a cylinder,” Math. USSR Izv. 18 (1), 89–98 (1982).CrossRefMATHGoogle Scholar
  12. 12.
    S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries (Walter de Gruyter, Berlin, 1994).CrossRefMATHGoogle Scholar
  13. 13.
    S. A. Nazarov, “Discrete spectrum of cranked, branching, and periodic waveguides,” St. Petersburg Math. J. 23 (2), 351–379 (2012).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    S. A. Nazarov, “Bounded solutions in a T-shaped waveguide and the spectral properties of the Dirichlet ladder,” Comput. Math. Math. Phys. 54 (8), 1261–1279 (2014).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    S. A. Nazarov, “Properties of spectra of boundary value problems in cylindrical and quasicylindrical domain,” in Sobolev Spaces in Mathematics, Ed. by V. Maz’ya (Springer, New York, 2008), Vol. 9, pp. 261–309.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators (Moscow, Nauka, 1965; Am. Math. Soc., Providence, R.I., 1969).Google Scholar
  17. 17.
    N. A. Umov, Equations of Energy Transfer in Bodies (Tipogr. Ul’rikha i Shul’tse, Odessa, 1874) [in Russian].Google Scholar
  18. 18.
    L. I. Mandelstam, Lectures on Optics, Relativity Theory, and Quantum Mechanics (Akad. Nauk SSSR, Moscow, 1947), Vol. 2 [in Russian].Google Scholar
  19. 19.
    I. I. Vorovich and V. A. Babeshko, Mixed Dynamic Problems of Elasticity Theory for Nonclassical Domains (Nauka, Moscow, 1979) [in Russian].MATHGoogle Scholar
  20. 20.
    S. A. Nazarov, “Umov–Mandelstam radiation conditions in elastic periodic waveguides,” Sb. Math. 205 (7), 953–982 (2014).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    S. A. Nazarov and B. A. Plamenevskii, “On radiation conditions for self-adjoint elliptic problems,” Sov. Math. Dokl. 41 (2), 274–277 (1990).Google Scholar
  22. 22.
    S. A. Nazarov and B. A. Plamenevskii, “Radiation principles for self-adjoint elliptic problems,” Probl. Mat. Fiz. Leningr. Gos. Univ. 13, 192–244 (1991).Google Scholar
  23. 23.
    S. A. Nazarov, “Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide,” Theor. Math. Phys. 167 (2), 606–627 (2011).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    J. H. Poynting, “On the transfer of energy in the electromagnetic field,” Phil. Trans. R. Soc. London 175, 343–361 (1884).CrossRefMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Peter the Great St. Petersburg State Polytechnical UniversitySt. PetersburgRussia
  3. 3.Institute of Problems of Mechanical EngineeringRussian Academy of SciencesSt. PetersburgRussia

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