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 Zap. Nauchn. Sem. POMI, 2018, Volume 471, Pages 168–210 (Mi znsl6632)

Asymptotics of eigenvalues in spectral gaps of periodic waveguides with small singular perturbations

S. A. Nazarovab

a St. Petersburg State University, St. Petersburg, Russia
b Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

Abstract: We study asymptotics of eigenvalues appearing near the lower edge of a spectral gap of the Dirichlet problem for the Laplace operator in $d$-dimensional periodic waveguide with the singular perturbation of the boundary by creating a hole with a small diameter $\varepsilon$ is studied. Several versions of the structure of the gap edge are considered. As usual the asymptotic formulas are different in the cases $d\geq3$ and $d=2$ where eigenvalues occur at the distances $O(\varepsilon^{2(d-2)})$ or $O(\varepsilon^{2d})$ and $O(|\ln\varepsilon|^{-2})$ or $O(\varepsilon^4)$, respectively, from the gap edge. Other types of singular perturbation of the waveguide surface and other types of boundary conditions are discussed which provide the appearance of eigenvalues near both edges of one or several gaps.

Key words and phrases: periodic waveguide, spectral problems for the Laplace operator, singular perturbation of boundaries, discrete spectrum, asymptotics of eigenvalues.

 Funding Agency Grant Number Russian Science Foundation 17-11-01003

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Citation: S. A. Nazarov, “Asymptotics of eigenvalues in spectral gaps of periodic waveguides with small singular perturbations”, Mathematical problems in the theory of wave propagation. Part 48, Zap. Nauchn. Sem. POMI, 471, POMI, St. Petersburg, 2018, 168–210

Citation in format AMSBIB
\Bibitem{Naz18} \by S.~A.~Nazarov \paper Asymptotics of eigenvalues in spectral gaps of periodic waveguides with small singular perturbations \inbook Mathematical problems in the theory of wave propagation. Part~48 \serial Zap. Nauchn. Sem. POMI \yr 2018 \vol 471 \pages 168--210 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6632}