We study fluctuations of polynomial linear statistics for discrete Schrödinger operators with a random decaying potential. We describe a decomposition of the space of polynomials into a direct sum of three subspaces determining the growth rate of the variance of the corresponding linear statistic. In particular, each one of these subspaces defines a unique critical value for the decay-rate exponent, above which the random variable has a limit that is sensitive to the underlying distribution and below which the random variable has asymptotically Gaussian fluctuations.

中文翻译：

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具有随机衰减势的薛定谔算子的光谱波动

我们研究具有随机衰减势的离散薛定谔算子的多项式线性统计的波动。我们将多项式空间分解为三个子空间的直接和，确定相应线性统计量的方差增长率。特别是，这些子空间中的每一个都定义了衰减率指数的唯一临界值，高于该值的随机变量具有对基础分布敏感的限制，低于该值的随机变量具有渐近高斯波动。