Spectral and scattering theory for topological crystals perturbed by infinitely many new edges

TL;DR

This paper studies the spectral and scattering properties of operators on topological crystals, especially when these structures are perturbed by infinitely many edges, finite edge removals, or measure modifications.

Contribution

It provides a detailed analysis of how infinite and finite perturbations affect the spectrum and scattering theory of operators on topological crystals, extending existing results.

Findings

Spectrum characterization for perturbed operators

Existence and completeness of wave operators

Impact of infinite edge additions on spectral properties

Abstract

In this paper we investigate the spectral and scattering theory for operators acting on topological crystals and on their perturbations. A special attention is paid to perturbations obtained by the addition of an infinite number of edges, and / or by the removal of a finite number of them, but perturbations of the underlying measures and perturbations by the addition of a multiplication operator are also considered. The description of the nature of the spectrum of the resulting operators, and the existence and completeness of the wave operators are standard outcomes for these investigations.