Inverse scattering for Schr\"{o}dinger operators on perturbed lattices

TL;DR

This paper develops a method to reconstruct scalar potentials and lattice structures in perturbed periodic graphs from scattering data, enabling defect detection in materials like graphene.

Contribution

It introduces a novel approach linking the scattering matrix to the Dirichlet-to-Neumann map for inverse problems on perturbed lattices.

Findings

Scattering matrix is equivalent to the Dirichlet-to-Neumann map.

Reconstruction of scalar potentials and graph structures from scattering data.

Procedure for probing defects in hexagonal lattices like graphene.

Abstract

We study the inverse scattering for Schr{\"o}dinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph, and reconstruct scalar potentials as well as the graph structure from the knowledge of the S-matrix. In particular, we give a procedure for probing defects in hexagonal lattices (graphene).