Spectra of open waveguides in periodic media

TL;DR

This paper investigates the essential spectra of elliptic systems on doubly periodic media with semi-infinite perturbations, revealing how the spectrum combines the unperturbed case with localized spectral contributions from the perturbation.

Contribution

It introduces a novel analysis of the essential spectrum for elliptic systems with semi-infinite periodic perturbations using Floquet-Bloch-Gelfand transform techniques.

Findings

Essential spectrum includes the unperturbed spectrum and additional parts from the perturbation.

Model problems in the perturbation strip determine the new spectral components.

The approach clarifies spectral behavior in complex periodic media with defects.

Abstract

We study the essential spectra of formally self-adjoint elliptic systems on doubly periodic planar domains perturbed by a semi-infinite periodic row of foreign inclusions. We show that the essential spectrum of the problem consists of the essential spectrum of the purely periodic problem and another component, which is the union of the discrete spectra of model problems in the infinite perturbation strip; these model problems arise by an application of the partial Floquet-Bloch-Gelfand transform.