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

TL;DR

The paper studies long-range perturbations of a class of block finite difference operators, focusing on self-adjoint first-order operators on $\ell^2(\mathbb{Z}; \mathbb{C}^2)$ or $\ell^2(\mathbb{Z}_+; \mathbb{C}^2)$. Using Mourre theory with a constructed family of conjugate operators, it proves a Limiting Absorption Principle and absence of singular continuous spectrum for perturbed operators $H^{(\mathbb{G})}=H_0^{(\mathbb{G})}+V$, under carefully specified decay conditions on the matrix-valued perturbations. The authors establish stability of the essential spectrum and provide precise descriptions of possible eigenvalue accumulation points, including gapless-conical-point scenarios, with applications to SSH models and one-dimensional Dirac operators (massive and massless). The results extend known LAP for discrete Dirac and block Jacobi settings to a broader class of long-range perturbations and contribute to the spectral stability theory for block finite difference operators. These findings have potential impact on the mathematical understanding of transport phenomena in lattice models and related quantum systems.

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.