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

We consider the spectral Dirichlet problem for the Laplace operator in the plane ${\mathrm{\Omega }}^{\circ }$ with double-periodic perforation but also in the domain ${\mathrm{\Omega }}^{\bullet }$ 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 ${\sigma }^{\circ }$ and ${\sigma }^{\bullet }$ of the problems in ${\mathrm{\Omega }}^{\circ }$ and ${\mathrm{\Omega }}^{\bullet },$ namely we present a concrete geometry which supports the relation ${\sigma }^{\circ }⫋{\sigma }^{\bullet }$ due to a new non-empty spectral band caused by the semi-infinite inclusion called an open waveguide in the double-periodic medium.