Integral formulation of Dirac singular waveguides

Abstract

This paper concerns a boundary integral formulation for the two-dimensional massive Dirac equation. The mass term is assumed to jump across a one-dimensional interface, which models a transition between two insulating materials. This jump induces surface waves that propagate outward along the interface but decay exponentially in the transverse direction. After providing a derivation of our integral equation, we prove that it has a unique solution for almost all choices of parameters using holomorphic perturbation theory. We then extend these results to a Dirac equation with two interfaces. Finally, we implement a fast numerical method for solving our boundary integral equations and present several numerical examples of solutions and scattering effects.