We develop a technique for proving number rigidity (in the sense of Ghosh and Peres in Duke Math J 166(10):1789–1858, 2017) of the spectrum of general random Schrödinger operators (RSOs). Our method makes use of Feynman–Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman–Kac formulas for RSOs with multiplicative noise (Gaudreau Lamarre in Semigroups for one-dimensional Schrödinger operators with multiplicative Gaussian noise, Preprint arXiv:1902.05047v3, 2019; Gaudreau Lamarre and Shkolnikov in Ann Inst Henri Poincaré Probab Stat 55(3):1402–1438, 2019; Gorin and Shkolnikov in Ann Probab 46(4):2287–2344, 2018) by Gorin, Shkolnikov, and the first-named author, we use this method to prove number rigidity for a class of one-dimensional continuous RSOs of the form \(-\frac{1}{2}\Delta +V+\xi \), where V is a deterministic potential and \(\xi \) is a stationary Gaussian noise. Our results require only very mild assumptions on the domain on which the operator is defined, the boundary conditions on that domain, the regularity of the potential V, and the singularity of the noise \(\xi \).

中文翻译：

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Feynman–Kac公式的随机薛定ding算子的光谱刚度

我们开发了一种证明一般随机Schrödinger算子（RSO）频谱的数字刚性的技术（就Dho Math J 166（10）：1789-1858，2017中的Ghosh和Peres而言）。我们的方法利用Feynman–Kac公式根据自相交本地时间来估计频谱的指数线性统计量的方差。受关于带乘性噪声的RSO的费曼-卡克公式的最新结果的启发（针对具有乘性高斯噪声的一维Schrödinger算子的半群中的Gaudreau Lamarre，预印本arXiv：1902.05047v3,2019年; Ann Inst HenriPoincaréProStat 55中的Gaudreau Lamarre和Shkolnikov （3）：1402-1438，2019； Gorin和Shkolnikov在Ann Probab 46（4）：2287-2344，2018）中，作者是Gorin，Shkolnikov和第一作者，\（-\ frac {1} {2} \ Delta + V + \ xi \），其中V是确定性电位，\（\ xi \）是平稳的高斯噪声。我们的结果只需要非常温和的假设就可以定义算子，该区域的边界条件，势V的规律性和噪声的奇异\（\ xi \）。