Global multiplicity bounds and spectral statistics for random operators

Abstract

In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on R. We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory.In particular for higher rank Anderson models with pure-point spectrum, with the randomness having support equal to R, there is a uniform lower bound on spectral multiplicity and in case this is larger than one the local statistics is not Poisson.

Decorrelation estimates for random Schrödinger operators with non rank one perturbations

“…Further investigations on the simplicity of eigenvalues for Anderson-type models may be found in the article by Naboko, Nichols, and Stolz [18], Mallick [13], Mallick and Krishna [14], and Mallick and Narayanan [15]. Mallick [13] proves that the singular spectrum is simple for a class of Anderson models with higher rank perturbation extending the results of Naboko, Nichols, and Stolz [18].…”

“…Mallick [13] proves that the singular spectrum is simple for a class of Anderson models with higher rank perturbation extending the results of Naboko, Nichols, and Stolz [18]. Mallick and Krishna [14] prove that, for higher rank Anderson models with the single site potential having support in the whole real line, the Minami estimate implies simplicity of the pure point spectrum away from the continuous spectrum. They a also show that in the case of higher multiplicity spectrum the spectral statistics cannot be Poisson but must be compound Poisson.…”