Weakly bound states in heterogeneous waveguides

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 Dirichletboundary conditions at its border. We use the variational theorem to derive the conditionfor which the lowest eigenvalue of the spectrum falls below the continuum threshold and abound state appears, localized at the heterogeneity. We devise a rigorous perturbationscheme and derive the exact expression for the energy to third order in theheterogeneity.