Localization and unique continuation for non-stationary Schr\"odinger operators on the 2D lattice

TL;DR

This paper extends localization results for non-stationary Schrödinger operators on a 2D lattice by relaxing distribution assumptions, using unique continuation principles and Wegner estimates to prove localization at the spectrum's bottom.

Contribution

It introduces a new approach that replaces identical distribution assumptions with bounds on potential range and variance, enabling localization proofs for more general random potentials.

Findings

Established unique continuation principles for non-stationary potentials.

Proved Wegner estimates under relaxed distribution conditions.

Achieved localization at the bottom of the spectrum for broader classes of potentials.

Abstract

We extend methods of Ding and Smart from their breakthrough paper in 2020 which showed Anderson localization for certain random Schr\"odinger operators on $\ell^2(\mathbb{Z}^2)$ via a quantitative unique continuation principle and Wegner estimate. We replace the requirement of identical distribution with the requirement of a uniform bound on the essential range of potential and a uniform positive lower bound on the variance of the variables giving the potential. Under those assumptions, we recover the unique continuation and Wegner lemma results, using Bernoulli decompositions and modifications of the arguments therein. This leads to a localization result at the bottom of the spectrum.