Szeg\"o-type Theorems for One-Dimensional Schrodinger Operator with Random Potential (smooth case)

TL;DR

This paper extends Szeg"o-type theorems to one-dimensional Schr"odinger operators with random potentials, revealing a spectral Central Limit Theorem and asymptotic behavior of entanglement entropy in disordered fermions.

Contribution

It broadens the class of test functions for Szeg"o-type asymptotics and establishes a spectral CLT with a subleading term proportional to the square root of the interval length.

Findings

Subleading term follows a spectral Central Limit Theorem.

Asymptotic subleading term scales as L^{1/2}.

Results applied to entanglement entropy of disordered fermions.

Abstract

The paper is a continuation of work [15] in which the general setting for analogs of the Szeg\"o theorem for ergodic operators was given and several interesting cases were considered. Here we extend the results of [15] to a wider class of test functions and symbols which determine the Szeg\"o-type asymptotic formula for the one-dimensional Schrodinger operator with random potential. We show that in this case the subleading term of the formula is given by a Central Limit Theorem in the spectral context, hence the term is asymptotically proportional to $L^{1/2}$, where $L$ is the length of the interval on which the Schrodinger operator is initially defined. This has to be compared with the classical Szeg\"o formula, where the subleading term is bounded in $L$, $L \to \infty$. We prove an analog of standard Central Limit Theorem (the convergence of the probability of the corresponding event to the Gaussian Law) as well as an analog of the almost sure Central Limit Theorem (the convergence with probability 1 of the logarithmic means of the indicator of the corresponding event to the Gaussian Law). We illustrate our general results by establishing the asymptotic formula for the entanglement entropy of free disordered Fermions for non-zero temperature.