Arithmetic version of Anderson localization via reducibility

TL;DR

This paper extends the arithmetic version of Anderson localization to multi-dimensional quasi-periodic long-range operators using a novel approach based on arithmetic Aubry duality and reducibility.

Contribution

It introduces a new method that generalizes previous one-dimensional results to all dimensions for a broad class of operators.

Findings

Proves Anderson localization in all dimensions for quasi-periodic long-range operators.

Develops an arithmetic Aubry duality approach.

Achieves a unified framework encompassing previous special cases.

Abstract

The arithmetic version of Anderson localization (AL), i.e., AL with explicit arithmetic description on both the localization frequency and the localization phase, was first given by Jitomirskaya \cite{J} for the almost Mathieu operators (AMO). Later, the result was generalized by Bourgain and Jitomirskaya \cite{bj02} to a class of {\it one dimensional} quasi-periodic long-range operators. In this paper, we propose a novel approach based on an arithmetic version of Aubry duality and quantitative reducibility. Our method enables us to prove the same result for the class of quasi-periodic long-range operators in {\it all dimensions}, which includes \cite{J, bj02} as special cases.