Weakly bound states in heterogeneous waveguides

TL;DR

This paper investigates how a localized increase in material density within a 2D waveguide creates a bound state below the continuum threshold, providing a rigorous perturbation analysis up to third order.

Contribution

It introduces a rigorous perturbation scheme to analyze weak heterogeneities in waveguides and derives an explicit formula for the bound state energy up to third order.

Findings

A denser localized heterogeneity induces a bound state below the continuum.

The derived energy expression is accurate up to third order in heterogeneity strength.

The existence of the bound state is rigorously proven for weak heterogeneities.

Abstract

We study the spectrum of the Helmholtz equation in a two-dimensional infinite waveguide, containing a weak heterogeneity localized at an internal point, and obeying Dirichlet boundary conditions at its border. We prove that, when the heterogeneity corresponds to a locally denser material, the lowest eigenvalue of the spectrum falls below the continuum threshold and a bound state appears, localized at the heterogeneity. We devise a rigorous perturbation scheme and derive the exact expression for the energy to third order in the heterogeneity.