Localization for Almost-Periodic Operators with Power-law Long-range Hopping: A Nash-Moser Iteration Type Reducibility Approach

TL;DR

This paper introduces a Nash-Moser iteration method to establish localization in high-dimensional almost-periodic operators with power-law long-range hopping, extending previous results to this new setting.

Contribution

It develops a novel reducibility approach for localization in almost-periodic operators with power-law long-range interactions, including regularity bounds.

Findings

Proves inverse localization for certain operators

Provides quantitative bounds on hopping regularity

Generalizes classical results to power-law hopping case

Abstract

In this paper we develop a Nash-Moser iteration type reducibility approach to prove the (inverse) localization for some $d$-dimensional discrete almost-periodic operators with power-law long-range hopping. We also provide a quantitative lower bound on the regularity of the hopping. As an application, some results of \cite{Sar82, Pos83, Cra83, BLS83} are generalized to the power-law hopping case.