Edge states for second order elliptic operators in a channel

TL;DR

This paper develops a framework to analyze edge states in second order elliptic operators within a half channel, establishing a link between bulk material indices and boundary localized states.

Contribution

It introduces a novel index-based approach to predict edge states at material junctions for second order elliptic operators.

Findings

Edge states occur at material boundaries when indices differ.

An integer-valued index is associated with bulk materials.

The framework applies to general second order elliptic operators.

Abstract

We present a general framework to study edge states for second order elliptic operators in a half channel. We associate an integer valued index to some bulk materials, and we prove that for any junction between two such materials, localised states must appear at the boundary whenever the indices differ.