Essential spectrum of a periodic waveguide with non-periodic perturbation (Q1747077)

The authors consider a periodic waveguide with a non-compact non-periodic perturbation. The waveguide is modelled by a quasi-cylinder infinite along an one-dimensional axis with periodically varying boundary. The operator in question is the Dirichlet Laplacian in such domain. The spectrum of such operator has a band gap structure described by the corresponding band functions. The mentioned perturbation is described by an infinite series of identical holes in the considered cylinder such that the distances between two neighbouring holes goes to infinity as the distance from the considered pair of holes to zero increases. The main result of the paper states that the essential spectrum of such perturbed operator is the union of the aforementioned unperturbed periodic operator and a discrete spectrum of a model operator. The latter corresponds to the same periodic waveguide but with only one hole.