Green's function estimates for long-range quasi-periodic operators on $\mathbb{Z}^d$ and applications

TL;DR

This paper develops Green's function estimates for long-range quasi-periodic operators on multidimensional integer lattices, leading to spectral localization and bounds on quantum dynamics, with novel techniques for resonant block analysis.

Contribution

It introduces a new approach using separation properties of resonant blocks to analyze quantum dynamics in long-range quasi-periodic operators.

Findings

Proves spectral localization for the class of operators.

Establishes upper bounds on quantum dynamics.

Develops a novel method for resonant block separation.

Abstract

We establish quantitative Green's function estimates for a class of quasi-periodic (QP) operators on $\mathbb{Z}^d$ with certain slowly decaying long-range hopping and analytic cosine type potentials. As applications, we prove the arithmetic spectral localization, and obtain upper bounds on quantum dynamics for all phase parameters. To deal with quantum dynamics estimates, we develop an approach employing separation property (rather than the sublinear bound) of resonant blocks in the regime of Green's function estimates.