Unique Continuation Principle for Spectral Projections of Schrödinger Operators and Optimal Wegner Estimates for Non-ergodic Random Schrödinger Operators
Unique Continuation Principle for Spectral Projections of Schrödinger Operators and Optimal Wegner Estimates for Non-ergodic Random Schrödinger Operators
复制标题
薛定谔算子谱投影的独特连续原理和非遍历随机薛定谔算子的最优韦格纳估计
DOI：
10.1007/s00220-013-1795-x
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发表时间：
2012
影响因子：
2.4
通讯作者：
A. Klein
中科院分区：
文献类型：
--
作者：
A. Klein
我们证明了薛定谔算子谱投影的独特连续原理。我们考虑 L2(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} 上的薛定谔算子 H = −Δ + V \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm L}^2(\mathbb{R}^d)}$$\end{document} ，并令 HΛ 表示其对有限框的限制 Λ 具有狄利克雷或周期性边界条件。我们证明了类型 χI (HΛ) WχI (HΛ) ≥ κ χI (HΛ) 的唯一连续估计，其中 κ > 0 对于适当的势 W ≥ 0 和区间 I。作为应用，我们获得了一类具有合金型随机势的非遍历随机薛定谔算子（“弯曲的”安德森哈密顿算子）在所有能量上的最佳韦格纳估计。我们还证明了最优韦格纳估计位于谱的底部，并且预期对无序的依赖性（韦格纳估计随着无序的增加而改善），即使对于通常的（遍历）安德森哈密顿量来说也是一个新结果。这些估计用于证明安德森哈密顿量在光谱底部的固定间隔内处于高度无序状态。
We prove a unique continuation principle for spectral projections of Schrödinger operators. We consider a Schrödinger operator H = −Δ + V on L2(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm L}^2(\mathbb{R}^d)}$$\end{document} , and let HΛ denote its restriction to a finite box Λ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the type χI (HΛ) WχI (HΛ) ≥ κ χI (HΛ) with κ > 0 for appropriate potentials W ≥ 0 and intervals I. As an application, we obtain optimal Wegner estimates at all energies for a class of non-ergodic random Schrödinger operators with alloy-type random potentials (‘crooked’ Anderson Hamiltonians). We also prove optimal Wegner estimates at the bottom of the spectrum with the expected dependence on the disorder (the Wegner estimate improves as the disorder increases), a new result even for the usual (ergodic) Anderson Hamiltonian. These estimates are applied to prove localization at high disorder for Anderson Hamiltonians in a fixed interval at the bottom of the spectrum.