The spectrum, radiation conditions and the Fredholm property for the Dirichlet Laplacian in a perforated plane with semi-infinite inclusions

Abstract

We consider the spectral Dirichlet problem for the Laplace operator in the plane $Ω^{\circ}$ with double-periodic perforation but also in the domain $Ω^{\bullet}$ with a semi-infinite foreign inclusion so that the Floquet-Bloch technique and the Gelfand transform do not apply directly. We describe waves which are localized near the inclusion and propagate along it. We give a formulation of the problem with radiation conditions that provides a Fredholm operator of index zero. The main conclusion concerns the spectra $σ^{\circ}$ and $σ^{\bullet}$ of the problems in $Ω^{\circ}$ and $Ω^{\bullet},$ namely we present a concrete geometry which supports the relation $σ^{\circ}\varsubsetneqqσ^{\bullet}$ due to a new non-empty spectral band caused by the semi-infinite inclusion called an open waveguide in the double-periodic medium.