Inverse tunneling estimates and applications to the study of spectral statistics of random operators on the real line

In section 5, a proof of localization at the bottom of the spectrum for the displacement model in dimension 1 was added. This required a study of Lifshitz type asymptotics in the same energy region. The main theorem was improve and some assumptions weakened.We present a proof of Minami type estimates for one dimensional random Schrödinger operators valid at all energies in the localization regime provided a Wegner estimate is known to hold. The Minami type estimates are then applied to various models to obtain results on their spectral statistics. The heuristics underlying our proof of Minami type estimates is that close by eigenvalues of a one-dimensional Schrödinger operator correspond either to eigenfunctions that live far away from each other in space or they come from some tunneling phenomena. In the second case, one can undo the tunneling and thus construct quasi-modes that live far away from each other in space