Spectral and scattering theory for Schr\"odinger operators on perturbed topological crystals

TL;DR

This paper studies the spectral and scattering properties of Schrödinger operators on perturbed periodic graphs, using Mourre theory to analyze spectrum and wave operators for various perturbations.

Contribution

It introduces a comprehensive analysis of spectral and scattering theory for Schrödinger operators on perturbed topological crystals, including new results on wave operators and spectrum characterization.

Findings

Mourre theory effectively describes the spectrum of these operators.

Existence and completeness of local wave operators are established for short-range perturbations.

The paper characterizes spectral types under different perturbation conditions.

Abstract

In this paper we investigate the spectral and the scattering theory of Schr\"odinger operators acting on perturbed periodic discrete graphs. The perturbations considered are of two types: either a multiplication operator by a short-range or a long-range function, or a short-range type modification of the measure defined on the vertices and on the edges of the graph. Mourre theory is used for describing the nature of the spectrum of the underlying operators. For short-range perturbations, existence and completeness of local wave operators are also proved.