On the spectral properties of long-range perturbations of a class of block finite difference operators

TL;DR

This paper investigates the spectral characteristics of certain block finite difference operators with long-range perturbations, establishing key spectral properties and extending previous results to new classes of perturbations.

Contribution

It introduces a novel analysis of spectral properties for a class of long-range perturbed block finite difference operators using the conjugate operator method.

Findings

Proves the existence of limiting absorption principles.

Shows absence of singular continuous spectrum.

Extends spectral analysis to new classes of long-range perturbations.

Abstract

We analyze spectral properties of a family of self-adjoint first-order finite difference operators acting on $\ell^2(\mathbb{Z}; \mathbb{C}^2)$ or $\ell^2(\mathbb{Z}_+; \mathbb{C}^2)$. Applying the conjugate operator method, we prove the existence of limiting absorption principles and the absence of singular continuous spectrum for these operators. Our results cover classes of admissible long-range perturbations that have not been previously addressed.