Integral formulation of Klein-Gordon singular waveguides

TL;DR

This paper develops an integral formulation for analyzing singular waveguides in two-dimensional insulating materials, introducing a fast numerical solver and demonstrating its effectiveness through numerical examples.

Contribution

The paper introduces a novel integral formulation for singular waveguides and implements an efficient solver with demonstrated fast convergence and practical numerical examples.

Findings

Fast convergence of the proposed numerical method

Effective modeling of waveguides with jump conditions

Numerical demonstration of scattering effects

Abstract

We consider the analysis of singular waveguides separating insulating phases in two-space dimensions. The insulating domains are modeled by a massive Schr\"odinger equation and the singular waveguide by appropriate jump conditions along the one-dimensional interface separating the insulators. We present an integral formulation of the problem and analyze its mathematical properties. We also implement a fast multipole and sweeping-accelerated iterative algorithm for solving the integral equations, and demonstrate numerically the fast convergence of this method. Several numerical examples of solutions and scattering effects illustrate our theory.