Enhanced Wegner and Minami estimates and eigenvalue statistics of random Anderson models at spectral edges

François Germinet (1) , Frédéric Klopp (2)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 IMJ - Institut de Mathématiques de Jussieu

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 a previous paper. 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 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.

Mots clés: random Schrödinger operators; Wegner estimate; MInami estimate; eigenvalue statistics

Domaines: Physique mathématique [math-ph]

François Germinet, Frédéric Klopp. Enhanced Wegner and Minami estimates and eigenvalue statistics of random Anderson models at spectral edges. 2011. ⟨hal-00638598⟩