Enhanced Wegner and Minami Estimates and Eigenvalue Statistics of Random Anderson Models at Spectral Edges

摘要

We consider the discrete Anderson model and prove enhanced Wegner and Minami estimates where the interval length is replaced by the IDS computed on the interval. We use these estimates to improve on the description of finite volume eigenvalues and eigenfunctions obtained in Germinet and Klopp (J Eur Math Soc http://arxiv.org/abs/1011.1832, 2010). As a consequence of the improved description of eigenvalues and eigenfunctions, we revisit a number of results on the spectral statistics in the localized regime obtained in Germinet and Klopp (J Eur Math Soc http://arxiv.org/abs/1011.1832, 2010) and extend their domain of validity, namely: the local spectral statistics for the unfolded eigenvalues; the local asymptotic ergodicity of the unfolded eigenvalues. In dimension 1, for the standard Anderson model, the improvement enables us to obtain the local spectral statistics at band edge, that is in the Lifshitz tail regime. In higher dimensions, this works for modified Anderson models.