Absence of bound states for waveguides in 2D periodic structures

TL;DR

This paper proves that in a 2D periodic medium with a line defect acting as a waveguide, there are no bound states, meaning the spectrum contains no eigenvalues, which impacts wave localization understanding.

Contribution

It establishes the absence of bound states in a 2D periodic waveguide with a line defect, a novel spectral property for such structures.

Findings

Spectrum contains no point spectrum (no bound states)

Wave localization does not occur in this setting

Results apply to soft-wall waveguide models

Abstract

We study a Helmholtz-type spectral problem in a two-dimensional medium consisting of a fully periodic background structure and a perturbation in form of a line defect. The defect is aligned along one of the coordinate axes, periodic in that direction (with the same periodicity as the background), and bounded in the other direction. This setting models a so-called "soft-wall" waveguide problem. We show that there are no bound states, i.e., the spectrum of the operator under study contains no point spectrum.