We prove decorrelation estimates for generalized lattice Anderson models on $\mathbb Z^d$ constructed with finite-rank perturbations in the spirit of Klopp [12]. These are applied to prove that the local eigenvalue statistics $\xi^\omega_{E}$ and $\xi^\omega_{E'}$, associated with two energies $E$ and $E'$ in the localization region and satisfying $|E - E'| > 4d$, are independent. That is, if $I,J$ are two bounded intervals, the random variables $\xi^\omega_{E}(I)$ and $\xi^\omega_{E'}(J)$, are independent and distributed according to a compound Poisson distribution whose Lévy measure has finite support. We also prove that the extended Minami estimate implies that the eigenvalues in the localization region have multiplicity at most the rank of the perturbation. The method of proof contains new ingredients that simplify the proof of the rank one case [12, 19, 21], extends to models for which the eigenvalues are degenerate, and applies to models for which the potential is not sign definite [20] in dimensions $d \geq 1$.

中文翻译：

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具有非第一扰动的随机Schrödinger算子的解相关估计

我们根据Klopp [12]的精神，用有限秩扰动构造的$ \ mathbb Z ^ d $证明了广义晶格Anderson模型的去相关估计。这些用于证明局部特征值统计$ \ xi ^ \ omega_ {E} $和$ \ xi ^ \ omega_ {E'} $，并与本地化区域中的两个能量$ E $和$ E'$相关，并且满足$ | E-E'| > 4d $，是独立的。也就是说，如果$ I，J $是两个有界区间，则随机变量$ \ xi ^ \ omega_ {E}（I）$和$ \ xi ^ \ omega_ {E'}（J）$都是独立且分布的根据一个复合的Poisson分布，其Lévy度量具有有限的支持。我们还证明了扩展的Minami估计意味着本地化区域中的特征值最多具有扰动等级的多重性。