Spectral and scattering theory for Gauss-Bonnet operators on perturbed topological crystals

TL;DR

This paper studies the spectral and scattering properties of Gauss-Bonnet operators on perturbed periodic graphs, establishing results for wave operators under short-range and long-range perturbations, with implications for related Laplacian operators.

Contribution

It introduces new spectral and scattering results for Gauss-Bonnet operators on perturbed topological crystals, including wave operator existence and completeness for various perturbations.

Findings

Proved existence and completeness of local wave operators for short-range perturbations.

Extended results to Laplacian operators on edges.

Analyzed spectral properties of Gauss-Bonnet operators on perturbed graphs.

Abstract

In this paper we investigate the spectral and the scattering theory of Gauss--Bonnet operators acting on perturbed periodic combinatorial graphs. Two types of perturbation are considered: either a multiplication operator by a short-range or a long-range potential, or a short-range type modification of the graph. For short-range perturbations, existence and completeness of local wave operators are proved. In addition, similar results are provided for the Laplacian acting on edges.