Eigenvalue fluctuations of 1-dimensional random Schrödinger operators

As an extension to the paper by Breuer et al., Ann. Henri Poincare 22, 3763 (2021), we study the linear statistics for the eigenvalues of the Schrödinger operator with random decaying potential with order O(x−α) (α > 0) at infinity. We first prove similar statements as in Breuer et al., Ann. Henri Poincare 22, 3763 (2021) for the trace of f(H), where f belongs to a class of analytic functions: there exists a critical exponent αc such that the fluctuation of the trace of f(H) converges in probability for α > αc, and satisfies a central limit theorem statement for α ≤ αc, where αc differs depending on f. Furthermore we study the asymptotic behavior of its expectation value.