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

Weakly bound states in heterogeneous waveguides