Spectrum created by line defects in periodic structures

TL;DR

This paper studies how line defects in three-dimensional periodic structures induce new spectral features within the gaps of the original spectrum of differential operators, revealing that even small defects can create additional spectral components.

Contribution

It demonstrates that line defects in periodic differential operators generate new spectrum within spectral gaps, providing insights into defect-induced spectral phenomena.

Findings

Small perturbations create spectrum in spectral gaps.

Line defects induce additional spectral components.

Properties of defect-induced spectrum are analyzed.

Abstract

The spectrum of periodic differential operators typically exhibits a band-gap structure. In this paper, we will consider perturbations to periodic differential operators and investigate the spectrum the perturbation induces in the gaps. More specifically, we consider the operator $$ L_0 =-\frac{1}{\eps_0(x,y,z)}\Delta $$ in $\R^3$ with $\eps_0$ periodic in all three directions. The perturbation is introduced by replacing $\eps_0$ by $\eps_0+\eps_1$ where we assume that $\eps_1$ is still periodic in one direction, but compactly supported in the remaining two directions, creating a line defect. We will show that even small perturbations $\eps_1$ lead to additional spectrum in the spectral gaps of the unperturbed operator $L_0$ and investigate some properties of the spectrum that is created.