Eigenvalue Fluctuations of 1-dimensional random Schrödinger operators

As an extension to the paper by Breuer, Grinshpon, and White \cite{B}, we study the linear statistics for the eigenvalues of the Schrödinger operator with random decaying potential with order $\mathcal{O}(x^{-\alpha })$ ($\alpha >0$) 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.