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

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.