Decorrelation estimates for random Schr\"odinger operators with non rank one perturbations

TL;DR

This paper establishes decorrelation estimates for generalized lattice Anderson models with finite-rank perturbations, demonstrating independence of local eigenvalue statistics at different energies and bounding eigenvalue multiplicity, extending previous results to more complex models.

Contribution

It introduces new methods that simplify the rank one case analysis, extend to degenerate eigenvalues, and apply to non-sign definite potentials in higher dimensions.

Findings

Eigenvalue statistics at different energies are independent when separated by more than 4d.

Eigenvalues in the localization region have multiplicity at most the perturbation rank.

Eigenvalues follow a compound Poisson distribution with finite support Lévý measure.

Abstract

We prove decorrelation estimates for generalized lattice Anderson models on $Z^d$ constructed with finite-rank perturbations in the spirit of Klopp \cite{klopp}. These are applied to prove that the local eigenvalue statistics $\xi^\omega_{E}$ and $\xi^\omega_{E^\prime}$, 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 L\'evy 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 \cite{klopp,shirley,trinh}, extends to models for which the eigenvalues are degenerate, and applies to models for which the potential is not sign definite \cite{tautenhahn-veselic1} in dimensions $d \geq 1$.