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.
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References
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).
P. A. Kuchment, “Floquet theory for partial differential equations,” Russ. Math. Surv. 37 (4), 1–60 (1982).
M. M. Skriganov, “Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators,” Proc. Steklov Inst. Math. 171 (2), 1–121 (1987).
P. Kuchment, Floquet Theory for Partial Differential Equations (Birkhäuser, Basel, 1993).
D. Grieser, “Spectra of graph neighborhoods and scattering,” Proc. London Math. Soc. 97 (3), 718–752 (2008).
S. A. Nazarov, “Discrete spectrum of cross-shaped quantum waveguides,” J. Math. Sci. 196 (3), 346–376 (2014).
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).
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).
G. Cardone, S. A. Nazarov, and J. Taskinen, “Spectra of open waveguides in periodic media,” J. Funct. Anal. 269 (8), 2328–2364 (2015).
I. M. Gel’fand, “Eigenfunction expansion for an equation with periodic coefficients,” Dokl. Akad. Nauk SSSR 73, 1117–1120 (1950).
S. A. Nazarov, “Elliptic boundary value problems with periodic coefficients in a cylinder,” Math. USSR Izv. 18 (1), 89–98 (1982).
S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries (Walter de Gruyter, Berlin, 1994).
S. A. Nazarov, “Discrete spectrum of cranked, branching, and periodic waveguides,” St. Petersburg Math. J. 23 (2), 351–379 (2012).
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).
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.
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).
N. A. Umov, Equations of Energy Transfer in Bodies (Tipogr. Ul’rikha i Shul’tse, Odessa, 1874) [in Russian].
L. I. Mandelstam, Lectures on Optics, Relativity Theory, and Quantum Mechanics (Akad. Nauk SSSR, Moscow, 1947), Vol. 2 [in Russian].
I. I. Vorovich and V. A. Babeshko, Mixed Dynamic Problems of Elasticity Theory for Nonclassical Domains (Nauka, Moscow, 1979) [in Russian].
S. A. Nazarov, “Umov–Mandelstam radiation conditions in elastic periodic waveguides,” Sb. Math. 205 (7), 953–982 (2014).
S. A. Nazarov and B. A. Plamenevskii, “On radiation conditions for self-adjoint elliptic problems,” Sov. Math. Dokl. 41 (2), 274–277 (1990).
S. A. Nazarov and B. A. Plamenevskii, “Radiation principles for self-adjoint elliptic problems,” Probl. Mat. Fiz. Leningr. Gos. Univ. 13, 192–244 (1991).
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).
J. H. Poynting, “On the transfer of energy in the electromagnetic field,” Phil. Trans. R. Soc. London 175, 343–361 (1884).
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Original Russian Text © S.A. Nazarov, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 2, pp. 237–254.
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Nazarov, S.A. Open waveguides in a thin Dirichlet lattice: II. localized waves and radiation conditions. Comput. Math. and Math. Phys. 57, 236–252 (2017). https://doi.org/10.1134/S0965542517020129
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DOI: https://doi.org/10.1134/S0965542517020129