Level statistics of one-dimensional Schr\"odinger operators with random decaying potential

TL;DR

This paper investigates the spectral statistics of one-dimensional Schrödinger operators with decaying random potentials, revealing different limiting behaviors depending on the decay rate, including clock and circular beta-ensemble limits.

Contribution

It characterizes the level statistics for Schrödinger operators with decaying potentials, identifying phase transitions in eigenvalue distributions based on decay rate.

Findings

For decay rate $eta > 1/2$, eigenvalues exhibit clock process statistics.

At critical decay rate $eta=1/2$, eigenvalues follow the circular $eta$-ensemble.

Eigenvalue fluctuations tend to Gaussian in the supercritical case.

Abstract

We study the level statistics of one-dimensional Schr\"odinger operator with random potential decaying like $x^{-\alpha}$ at infinity. We consider the point process $\xi_L$ consisting of the rescaled eigenvalues and show that : (i)(ac spectrum case) for $\alpha > \frac 12$, $\xi_L$ converges to a clock process, and the fluctuation of the eigenvalue spacing converges to Gaussian. (ii)(critical case) for $\alpha = \frac 12$, $\xi_L$ converges to the limit of the circular $\beta$-ensemble.