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

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 $h^2\mathrm{\Delta }+V(x)$ that is perturbed by a smooth potential $h^{\alpha }{V}_{\omega }$ with $\alpha \in (0,1)$, and $V_{\omega }$ is a random potential that decorrelates on distances $h^{\beta }$, with $0<\beta <2\alpha$. 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.