Inverse problems for locally perturbed lattices -- Discrete Hamiltonian and quantum graph

TL;DR

This paper investigates inverse scattering problems on perturbed periodic lattices, showing that the S-matrix across all energies uniquely determines the graph structure and Hamiltonian coefficients for both discrete and metric graph models.

Contribution

It establishes the uniqueness of inverse scattering solutions for Schrödinger operators on perturbed lattices, extending previous results to both discrete and metric graph settings.

Findings

S-matrix for all energies determines graph structure and Hamiltonian coefficients in discrete case.

S-matrix for all energies determines graph structure in metric graph case.

Results apply to locally perturbed periodic lattices in quantum graph models.

Abstract

We consider the inverse scattering problems for two types of Schr\"odinger operators on locally perturbed periodic lattices. For the discrete Hamiltonian, the knowledge of the S-matrix for all energies determines the graph structure and the coefficients of the Hamiltonian. For locally perturbed equilateral metric graphs, the knowledge of the S-matrix for all energies determines the graph structure.