Decorrelation Estimates for Random Discrete Schrödinger Operators in Dimension One and Applications to Spectral Statistics

Abstract

The purpose of the present work is to establish decorrelation estimates at distinct energies for some random Schrödinger operator in dimension one. In particular, we establish the result for some random operators on the continuum with alloy-type potential. These results are used to give a description of the spectral statistics.

Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

“…We note that in one-dimension there are stronger results and the condition |E − E ′ | > 4d is not needed. Our results are inspired by the work of Klopp [9] for the Anderson models on Z d and of Shirley [18] for related models on Z d . The condition |E − E ′ | > 4d requires that the two energies be fairly far apart.…”

“…The proof of this fact follows the argument of Klein and Molchanov [8]. For d = 1, Shirley [18] proved that the usual Minami estimate holds for the dimer model (m k = 2) so the eigenvalues are almost surely simple.…”

“…He applied them to show that the local eigenvalue point processes at distinct energies converge to independent Poisson processes (in dimensions d 1 the energies need to be far apart as is the case for the models studied here). Shirley [18] extended the family of one-dimensional lattice models for which the decorrelation estimate may be proved to include alloy-type models with correlated random variables, hopping models, and certain one-dimensional quantum graphs.…”

“…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 [9,18,20], extends to models for which the eigenvalues are degenerate, and applies to models for which the potential is not sign definite [19] in dimensions d 1. Ann.…”

Decorrelation estimates for some continuous and discrete random Schrödinger operators in dimension one and applications to spectral statistics

“…Theorem 1.4 (with S = ∅) was first proved in [10] for the discrete Anderson model in dimension one. Then, it was proved for other discrete models in dimension one in [18,16,15] and for the first time for a continuous model in [15], but with the covering condition. In the present article, we prove for the first time decorrelation estimates and therefore Theorom 1.1, without the covering condition.…”