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

## TL;DR

This paper analyzes the spectral properties of the Dirichlet Laplacian in perforated planes with semi-infinite inclusions, identifying a new spectral band caused by the inclusion that affects wave propagation.

## Contribution

It introduces a novel spectral analysis framework for domains with semi-infinite inclusions where standard Floquet-Bloch methods fail, revealing a new spectral band and waveguide effects.

## Key findings

- Identification of a new spectral band due to the semi-infinite inclusion
- Demonstration that the spectrum of the perforated domain is strictly contained in that of the domain with the inclusion
- Formulation of radiation conditions leading to a Fredholm operator of index zero

## 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.