Szegö-type theorems for one-dimensional Schrödinger operator with random potential (smooth case)

The paper is a continuation of work [15] in which the general setting for analogs of the Szegö 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 Szegö-type asymptotic formula for the one-dimensional Schrödinger operator with ergodic 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 to which the Schrödinger operator is initially restricted. This has to be compared with the classical Szegö 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.