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<p>We establish the Anderson localization and exponential dynamical localization for a class of quasi-periodic Schrödinger operators on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5288_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb Z^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">Z</mi> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> 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.</p>

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Localization for Lipschitz Monotone Quasi-periodic Schrödinger Operators on \(\mathbb Z^d\) via Rellich Functions Analysis

  • 期刊论文

  • 发表日期：

    2025年4月4日

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  • Hongyi Cao,
  • Yunfeng Shi,
  • Zhifei Zhang

摘要

We establish the Anderson localization and exponential dynamical localization for a class of quasi-periodic Schrödinger operators on \(\mathbb Z^d\) 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.