We use the method of eigenvalue level spacing developed by Dietlein and Elgart [Level spacing and Poisson statistics for continuum random Schrödinger operators, J. Eur. Math. Soc. (JEMS)23(4) (2021) 1257–1293] to prove that the local eigenvalue statistics (LES) for the Anderson model on ${\mathbb{Z}}^d$, with uniform higher-rank $m\ge 2$, single-site perturbations, is given by a Poisson point process with intensity measure $n(E_0)ds$, where $n(E_0)$ is the density of states at energy $E_0$ in the region of localization near the spectral band edges. This improves the result of Hislop and Krishna [Eigenvalue statistics for random Schrödinger operators with non-rank one perturbations, Comm. Math. Phys.340(1) (2015) 125–143], who proved that the LES is a compound Poisson process with Lévy measure supported on the set $\{1,2,\dots ,m\}$. Our proofs are an application of the ideas of Dietlein and Elgart to these higher-rank lattice models with two spectral band edges, and illustrate, in a simpler setting, the key steps of the proof of Dietlein and Elgart.

我们使用 Dietlein 和 Elgart 开发的特征值水平间距方法 [连续随机薛定谔算子的水平间距和泊松统计，J. Eur. 数学。苏克。(JEMS) 23 (4) (2021) 1257–1293]证明安德森模型的局部特征值统计量（LES）${\mathbb{Z}}^d$，具有统一的更高等级$米\ge 2$，单点扰动，由具有强度测量的泊松点过程给出$n（{乙}_0）ds$， 在哪里$n（{乙}_0）$是能量状态密度${乙}_0$在靠近光谱带边缘的定位区域中。这改进了 Hislop 和 Krishna [具有非阶一扰动的随机薛定谔算子的特征值统计，Comm.] 的结果。数学。物理。340 (1) (2015) 125–143]，证明了 LES 是一个在集合上支持 Lévy 测度的复合泊松过程$\{1,2,\dots \dots ,米\}$。我们的证明是将 Dietlein 和 Elgart 的思想应用于这些具有两个谱带边缘的高阶晶格模型，并以更简单的设置说明 Dietlein 和 Elgart 证明的关键步骤。