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

TL;DR

This paper develops Green's function estimates for quasi-periodic operators on integer lattices with long-range interactions, leading to results on localization, regularity of the integrated density of states, and spectral properties.

Contribution

It provides the first quantitative Green's function bounds for such operators with power-law long-range hopping, advancing understanding of their spectral behavior.

Findings

Proves arithmetic localization for the operators.

Establishes Hölder continuity of the integrated density of states.

Shows absence of eigenvalues for Aubry dual operators.

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).