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<p>We extend methods of Ding and Smart (Invent Math 219(2):467–506, 2020) which showed Anderson localization for certain random Schrödinger operators on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\ell ^2(\mathbb {Z}^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

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Localization and Unique Continuation for Non-stationary Schrödinger Operators on the 2D Lattice

  • 期刊论文

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  • 发表日期：

    2026年3月9日

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  • Omar Hurtado

摘要

We extend methods of Ding and Smart (Invent Math 219(2):467–506, 2020) which showed Anderson localization for certain random Schrödinger operators on \(\ell ^2(\mathbb {Z}^2)\) ℓ 2 ( 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.