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 ${\sigma }^{\circ }{\sigma }^{\bullet }$ due to a new non-empty spectral band caused by the semi-infinite inclusion called an open waveguide in the double-periodic medium.

25 pages, 5 figures

References in corpus (2)

• Homogenization and norm resolvent convergence for elliptic operators in a strip perforated along a curve
• Spectra of open waveguides in periodic media

Cited by in corpus (1)

• Embedded eigenvalues of the Neumann problem in a strip with a box-shaped perturbation