Asymptotics of eigenvalues in spectral gaps under regular perturbations of walls of a periodic waveguide (Q1688473)

The authors consider a two-dimensional periodic waveguide with a small localized perturbation. The waveguide is modeled by a two-dimensional cylinder with a periodically varying boundary. In this cylinder, the Dirichlet Laplacian is considered. A localized perturbation is a small localized variation of the boundary of the considered waveguide. The unperturbed operator has a pure essential spectrum, which has a band structure and can contain gaps. The perturbation keeps the essential spectrum unchanged but can generate isolated eigenvalues in the gaps. The main result of the paper describes the existence of such eigenvalues near the edges of the internal gaps and provides their asymptotics. Various cases of the structure of edges of the gaps are considered and their influence of the asymptotic behavior of the emerging eigenvalues is demonstrated.