Localization and unique continuation for non-stationary Schrödinger operators on the 2D lattice

2024/05/17 by Hurtado, Omar · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2405.10636

Abstract

We extend methods of Ding and Smart from their breakthrough paper in 2020 which showed Anderson localization for certain random Schrödinger operators on ℓ²(ℤ²) 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.

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• Localization of the 1D Non-Stationary Anderson Model

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• Localization for random operators with non-monotone potentials with exponentially decaying correlations