Clock statistics for 1d Schr\"odinger operators

TL;DR

This paper investigates the local eigenvalue statistics of one-dimensional Schrödinger operators with random potentials, establishing clock convergence results for both i.i.d. and correlated cases, advancing understanding of spectral properties in disordered systems.

Contribution

It proves clock convergence for eigenvalues of 1d Schrödinger operators with alloy-type random potentials, including correlated cases, extending previous results to more general randomness.

Findings

Proves clock convergence for eigenvalues with alloy-type potentials.

Extends results to correlated random potentials with exponential decay.

Enhances understanding of spectral statistics in disordered quantum systems.

Abstract

We study the 1d Schr\"odinger operators with alloy type random supercritical decaying potential and prove the clock convergence for the local statistics of eigenvalues. We also consider, besides the standard i.i.d. case, more general ones with exponentially decaying correlations.