Overview
This paper investigates the inverse scattering problem specifically for a topologically non-trivial waveguide. This waveguide functions as a separator between two-dimensional topological insulators. The research adopts a Dirac system as its specific model for this investigation.
Research Context
The core subject of the paper is the inverse scattering problem. This problem is framed within the context of waveguides exhibiting topological non-triviality, which are positioned between two-dimensional topological insulators. The theoretical framework for this analysis is defined as a Dirac system.
Approach
The study employs both theoretical proofs and numerical methods. For the theoretical component, it focuses on scalar short-range perturbations, aiming to establish specific properties. Additionally, it considers general Hermitian perturbations, analyzing the behavior of scattering data under these conditions. The numerical solution relies on a standard adjoint method.
Findings
- For scalar short-range perturbations, the study proves linearized uniqueness and stability.
- Under a smallness constraint, local uniqueness is obtained in a finite-dimensional setting for scalar short-range perturbations.
- For general Hermitian perturbations, the scattering data exhibit invariance under a natural gauge transformation.
- At the linearized level, for general Hermitian perturbations, the data determine the potential modulo precisely this gauge transformation.
- The problem was solved numerically using a standard adjoint method.
- Numerical simulations were conducted to illustrate the theoretical findings.
Why This Matters
The research contributes to the understanding of inverse scattering problems within the specific domain of topological insulators and their separating waveguides. By proving uniqueness and stability properties and detailing the invariance of scattering data, the study provides theoretical insights into the characteristics of such systems. The numerical validation further bridges the theoretical models with practical computational approaches.