Localization for Lipschitz monotone quasi-periodic Schr\"odinger operators on $\mathbb{Z}^d$ via Rellich functions analysis

TL;DR

This paper proves Anderson localization and exponential dynamical localization for certain quasi-periodic Schrödinger operators on , using Rellich function analysis and multi-scale techniques to handle Lipschitz monotone potentials.

Contribution

It introduces a novel Schur complement approach to show that Rellich functions preserve Lipschitz monotonicity across scales, advancing localization theory for these operators.

Findings

Established Anderson localization for Lipschitz monotone potentials.

Proved exponential dynamical localization in the perturbative regime.

Developed a new method to analyze Rellich functions via Schur complements.

Abstract

We establish the Anderson localization and exponential dynamical localization for a class of quasi-periodic Schr\"odinger operators on $\mathbb{Z}^d$ with bounded or unbounded Lipschitz monotone potentials via multi-scale analysis based on Rellich function analysis in the perturbative regime. We show that at each scale, the resonant Rellich function uniformly inherits the Lipschitz monotonicity property of the potential via a novel Schur complement argument.