Spectral statistics for random Schr\"odinger operators in the localized regime

TL;DR

This paper investigates the eigenvalue and eigenfunction statistics of random Schrödinger operators in the localized regime, demonstrating Poisson behavior and independence of eigenvalues in large volumes, with detailed distributional results.

Contribution

It proves that eigenvalues behave as independent identically distributed variables in the localized phase, establishing Poisson statistics for eigenvalues and localization centers.

Findings

Eigenvalues are approximately i.i.d. in large volumes.

Eigenvalues exhibit Poisson statistics locally and globally.

Distribution of level spacings and localization centers follows Poisson law.

Abstract

We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$. Restrict it to some large cube $\Lambda$. Consider now $I_\Lambda$, a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $\Lambda$ grows to infinity. We prove that, with probability one in the large volume limit, the eigenvalues of the random Hamiltonian restricted to the cube inside the interval are given by independent identically distributed random variables, up to an error of size an arbitrary power of the volume of the cube. As a consequence, we derive * uniform Poisson behavior of the locally unfolded eigenvalues, * a.s. Poisson behavior of the joint distibutions of the unfolded energies and unfolded localization centers in a large range of scales. * the distribution of the unfolded level spacings, locally and globally, * the distribution of the unfolded localization centers, locally and globally.