Green's function estimates for quasi-periodic operators on $\mathbb{Z}^d$ with power-law long-range hopping

Abstract

We establish quantitative Green's function estimates for a class of quasi-periodic (QP) operators on $\mathbb{Z}^d$ with power-law long-range hopping and analytic cosine type potentials. As applications, we prove the arithmetic version of localization, the finite volume version of $(\frac12-)$-H\"older continuity of the IDS, and the absence of eigenvalues (for Aubry dual operators).