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The asymptotic analysis of gaps in the spectrum of a waveguide perturbed with a periodic family of small voids

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We study the spectrum of the Dirichlet problem for the Laplace operator in a cylindrical waveguide with periodic family of small (of diameter ε > 0) voids. Based on the asymptotic analysis of eigenvalues of the problem in a singularly perturbed periodicity cell, we show that the waveguide spectrum contains gaps of width O(ε), i.e., we provide a rigorous mathematical justification of the effect of splitting of edges of spectral bands. We consider several variants of splitting (or their absence), present asymptotic formulas for the gap edges and formulate open questions. The results are illustrated by examples. Bibliography: 38 titles. Illustrations: 13 figures.

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  1. Institute for Problems in Mechanical Engineering RAS 61, Bolshoi pr. V.O., St. Petersburg, 199178, Russia

    S. A. Nazarov

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Translated from Problems in Mathematical Analysis 66, August 2012, pp. 101–146.

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Nazarov, S.A. The asymptotic analysis of gaps in the spectrum of a waveguide perturbed with a periodic family of small voids. J Math Sci 186, 247–301 (2012). https://doi.org/10.1007/s10958-012-0985-y

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