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

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.