Eigenvalue Fluctuations of 1-dimensional random Schr\"odinger operators

TL;DR

This paper extends previous work on eigenvalue fluctuations of 1D random Schrödinger operators with decaying potentials, identifying a critical decay rate that determines fluctuation behavior and establishing a CLT for certain regimes.

Contribution

It introduces a detailed analysis of eigenvalue fluctuations for Schrödinger operators with decaying potentials, including a phase transition at a critical decay exponent and CLT results.

Findings

Existence of a critical decay exponent  for fluctuation behavior.

Convergence in probability of trace fluctuations for  > .

Central Limit Theorem established for  .

Abstract

As an extension to the paper by Breuer, Grinshpon, and White \cite{B}, we study the linear statistics for the eigenvalues of the Schr\"odinger operator with random decaying potential with order ${\cal O}(x^{-\alpha})$ ($\alpha>0$) at infinity. We first prove similar statements as in \cite{B} for the trace of $f(H)$, where $f$ belongs to a class of analytic functions : there exists a critical exponent $\alpha_c$ such that the fluctuation of the trace of $f(H)$ converges in probability for $\alpha > \alpha_c$, and satisfies a CLT statement for $\alpha \le \alpha_c$, where $\alpha_c$ differs depending on $f$. Furthermore we study the asymptotic behavior of its expectation value.