Smooth spectral statistics of random perturbations of Schrödinger operators near Anosov energy levels

Abstract: We investigate the spectral statistics of random perturbations of semiclassical Schrödinger operators on compact manifolds, near energy levels corresponding to chaotic classical dynamics. The prototypical example is that of an operator $h<sup>2Δ+V(x)$ that is perturbed by a smooth potential $h<sup>αV_ω$ with $α\in (0,1)$, and $V_ω$ is a random potential that decorrelates on distances $h<sup>β$, with $0<β<2α$. We show that for a generic perturbation, the spectral fluctuations of the smoothed counting function of eigenvalues obey a universal behavior at a certain mesoscopic scale, which is coherent with the predictions of random matrix theory.