Scattering by a collection of $δ$-function point and parallel line defects in two dimensions

Abstract

Interaction of waves with point and line defects are usually described by δ-function potentials supported on points or lines. In two dimensions, the scattering problem for a finite collection of point defects or parallel line defects is exactly solvable. This is not true when both point and parallel line defects are present. We offer a detailed treatment of the scattering problem for finite collections of point and parallel line defects in two dimensions. In particular, we perform the necessary renormalization of the coupling constants of the point defects, introduce an approximation scheme which allows for an analytic calculation of the scattering amplitude and Green's function for the corresponding singular potential, investigate the consequences of perturbing this potential, and comment on the application of our results in the study of the geometric scattering of a particle moving on a curved surface containing point and line defects. Our results provide a basic framework for the study of spectral singularities and the corresponding lasing and antilasing phenomena in two-dimensional optical systems involving lossy and/or active thin wires and parallel thin plates.