Eur. Phys. J. B (2016) 89: 163
https://doi.org/10.1140/epjb/e2016-70197-0

Regular Article

Weakly bound states in heterogeneous waveguides

¹ Facultad de Ciencias, CUICBAS, Universidad de Colima, Bernal Díaz del Castillo 340, 28040 Colima, Mexico
² INIFTA (UNLP, CCT La Plata-CONICET), Division Quimica Teorica, Blvd. 113 S/N, Sucursal 4, Casilla de Correo 16, 1900 La Plata, Argentina

^a e-mail: paolo.amore@gmail.com

Received: 1 April 2016
Received in final form: 25 May 2016
Published online: 6 July 2016

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 use the variational theorem to derive the condition for which 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.