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Level spacing and Poisson statistics for continuum random Schrödinger operators
Journal of the European Mathematical Society ( IF 2.6 ) Pub Date : 2020-12-22 , DOI: 10.4171/jems/1033
Adrian Dietlein 1 , Alexander Elgart 2
Affiliation  

For continuum alloy-type random Schrödinger operators with signdefinite single-site bump functions and absolutely continuous single-site randomness we prove a probabilistic level-spacing estimate at the bottom of the spectrum. More precisely, given a finite-volume restriction of the random operator onto a box of linear size L, we prove that with high probability the eigenvalues below some threshold energy Esp keep a distance of at least e −(logL) for sufficiently large β > 1. This implies simplicity of the spectrum of the infinite-volume operator below Esp. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy E.

中文翻译:

连续谱随机薛定谔算子的水平间距和泊松统计

对于具有正定单点凹凸函数和绝对连续单点随机性的连续合金型随机薛定谔算子,我们证明了频谱底部的概率水平间距估计。更准确地说,给定随机算子在线性大小为 L 的盒子上的有限体积限制,我们证明,对于足够大的 β >,低于某个阈值能量 Esp 的特征值很可能保持至少 e -(logL) 的距离。 1. 这意味着 Esp 以下无限体积算子的频谱的简单性。在单点概率密度的 Lipschitz 连续性的附加假设下,我们还证明了由围绕参考能量 E 展开的特征值给出的点过程的 Minami 型估计和泊松统计。
更新日期:2020-12-22
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