Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

We prove that certain natural random variables associated with the local eigenvalue statistics for generalized lattice Anderson models constructed with finite-rank perturbations are compound Poisson distributed. This distribution is characterized by the fact that the Lévy measure is supported on at most a finite set of positive integers determined by the rank. The proof relies on a Minami-type estimate for finite-rank perturbations. For Anderson-type continuum models on $${{\mathbb{R}^d}}$$^Rd, we prove a similar result for certain natural random variables associated with the local eigenvalue statistics. We prove that the compound Poisson distribution associated with these random variables has a Lévy measure whose support is at most the set of positive integers.