Abstract
The spectra of open angular waveguides obtained by thickening or thinning the links of a thin square lattice of quantum waveguides (the Dirichlet problem for the Helmholtz equation) are investigated. Asymptotics of spectral bands and spectral gaps (i.e., zones of wave transmission and wave stopping, respectively) for waveguides with variously shaped periodicity cells are found. It is shown that there exist eigenfunctions of two types: localized around nodes of a waveguide and on its links. Points of the discrete spectrum of a perturbed lattice with eigenfunctions concentrated about corners of the waveguide are found.
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References
A.-S. Bonnet-Bendhia and F. Starling, “Guided waves by electromagnetic gratings and nonuniqueness examples for the diffraction problem,” Math. Meth. Appl. Sci. 17, 305–338 (1994).
S. A. Nazarov, “Properties of spectra of boundary value problems in cylindrical and quasicylindrical domain,” in Sobolev Spaces in Mathematics, Vol. 2, Ed. by V. Maz’ya (Springer, New York, 2008), pp. 261–309.
I. M. Gel’fand, “Eigenfunction expansion for an equation with periodic coefficients,” Dokl. Akad. Nauk SSSR 73, 1117–1120 (1950).
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).
S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries (Walter de Gruyter, Berlin, 1994).
P. Kuchment, Floquet Theory for Partial Differential Equations (Birkhäuser, Basel, 1993).
A.-S. Bonnet-Ben Dhia, G. Dakhia, C. Hazard, and L. Chorfi, “Diffraction by a defect in an open waveguide: A mathematical analysis based on a modal radiation condition,” SIAM J. Appl. Math. 70 (3), 677–693 (2009).
A.-S. Bonnet-Ben Dhia, B. Goursaud, and C. Hazard, “Mathematical analysis of the junction of two acoustic open waveguides,” SIAM J. Appl. Math. 71 (6), 2048–2071 (2001).
G. Cardone, S. A. Nazarov, and J. Taskinen, “Spectra of open waveguides in periodic media,” J. Funct. Anal. 269 (8), 2328–2364 (2015).
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).
S. A. Nazarov, “Discrete spectrum of cross-shaped quantum waveguides,” in Problems in Mathematical Analysis (Novosibirsk, 2013), Vol. 73, pp. 101–127 [in Russian].
S. A. Nazarov, Discrete spectrum of cranked, branching, and periodic waveguides," St. Petersburg Math. J. 23 (2), 351–379 (2012).
S. A. Nazarov and A. V. Shanin, “Trapped modes in angular joints of 2D waveguides,” Appl. Anal. 93 (3), 572–582 (2014).
S. A. Nazarov, “The spectrum of rectangular gratings of quantum waveguides,” Izv. Ross. Akad. Nauk, Ser. Mat. 81 (1) (2017) (in press).
D. Grieser, “Spectra of graph neighborhoods and scattering,” Proc. London Math. Soc. 97 (3), 718–752 (2008).
S. A. Nazarov, “On the spectrum of an infinite Dirichlet ladder,” St. Petersburg Math. J. 23, 1023–1045 (2012).
M. Sh. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space (Leningr. Gos. Univ., Leningrad, 1980; Reidel, Dordrecht, 1987).
S. A. Nazarov, “Elliptic boundary value problems with periodic coefficients in a cylinder,” Izv. Akad. Nauk SSSR, Ser. Mat. 45 (1), 101–112 (1981).
S. A. Nazarov, “Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold,” Sib. Math. J. 51 (5), 866–878 (2010).
S. A. Nazarov, “Trapped modes in a T-shaped waveguide,” Acoust. Phys. 56 (6), 1004–1015 (2010).
S. A. Nazarov, “Asymptotics of eigenvalues of the Dirichlet problem in a skewed T-shaped waveguide,” Comput. Math. Math. Phys. 54 (5), 811–830 (2014).
M. I. Vishik and L. A. Lyusternik, “Regular degeneration and boundary layer for linear differential equations with small parameter,” Usp. Mat. Nauk 12 (5), 3–122 (1957).
S. A. Nazarov, “Structure of the spectrum of a net of quantum waveguides and bounded solutions of a model problem at the threshold,” Dokl. Math. 90 (2), 637–641 (2014).
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, “Localization of elastic oscillations in cross-shaped planar orthotropic waveguides,” Phys.-Dokl. 59 (9), 411–415 (2014).
M. D. Van Dyke, Perturbation Methods in Fluid Mechanics (Academic, New York, 1964; Mir, Moscow, 1967).
A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Nauka, Moscow, 1989; Am. Math. Soc., RI, Providence, 1992).
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Original Russian Text © S.A. Nazarov, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 1, pp. 144–162.
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Nazarov, S.A. Open waveguides in a thin Dirichlet ladder: I. Asymptotic structure of the spectrum. Comput. Math. and Math. Phys. 57, 156–174 (2017). https://doi.org/10.1134/S0965542517010110
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DOI: https://doi.org/10.1134/S0965542517010110