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

TL;DR

This paper proves Minami type estimates for one-dimensional random Schrödinger operators across all energies in the localization regime, enabling analysis of spectral statistics through tunneling and eigenfunction localization.

Contribution

It introduces a new proof of Minami estimates applicable at all energies, leveraging tunneling phenomena and eigenfunction localization in one-dimensional models.

Findings

Minami estimates hold at all energies in the localization regime.

Spectral statistics can be derived using tunneling and eigenfunction separation.

Application to various models demonstrates the estimates' versatility.

Abstract

We present a proof of Minami type estimates for one dimensional random Schr\"odinger 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 Schr\"odinger 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.