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Open waveguides in doubly periodic junctions of domains with different limit dimensions. (English. Russian original) Zbl 1365.35099

Sib. Math. J. 57, No. 6, 943-956 (2016); translation from Sib. Mat. Zh. 57, No. 6, 1208-1223 (2016).
Summary: Considering the spectral Neumann problem for the Laplace operator on a doubly periodic square grid of thin circular cylinders (of diameter \(\varepsilon\ll1\)) 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.

MSC:

35P25 Scattering theory for PDEs
78A50 Antennas, waveguides in optics and electromagnetic theory
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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