Title: 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 $\Omega^{\circ}$ with double-periodic perforation but also in the domain
$\Omega^{\bullet}$ 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 $\Omega^{\circ}$ and
$\Omega^{\bullet},$ namely we present a concrete geometry which supports the
relation $\sigma^{\circ}\varsubsetneqq\sigma^{\bullet}$ due to a new non-empty
spectral band caused by the semi-infinite inclusion called an open waveguide in
the double-periodic medium.