Inverse scattering for waveguides in topological insulators

Abstract

This paper concerns the inverse scattering problem of a topologically non-trivial waveguide separating two-dimensional topological insulators. We consider the specific model of a Dirac system. For scalar short-range perturbations, we prove linearized uniqueness and stability and obtain local uniqueness in a finite-dimensional setting under a smallness constraint. For general Hermitian perturbations, the scattering data are invariant under a natural gauge transformation; at the linearized level, the data determine the potential modulo precisely this gauge. We then solve the problem numerically by means of a standard adjoint method and illustrate our theoretical findings with several numerical simulations.