Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

Abstract

We study local asymptotics for the spectral projector associated to a Schrödinger operator $-{\hslash }^2\Delta +V$ on ${\mathbb{R}}^n$ in the semiclassical limit as $\hslash \to 0$. We prove local uniform convergence of the rescaled integral kernel of this projector towards a universal model, inside the classically allowed region as well as on its boundary. This implies universality of microscopic fluctuations for the corresponding free fermions (determinantal) point processes, both in the bulk and around regular boundary points. Our results apply to a general class of smooth potentials in arbitrary dimension $n\ge 1$. These results are complemented by studying both macroscopic and mesoscopic fluctuations of the point process. We obtain tail bounds for macroscopic linear statistics and, provided $n\ge 2$, a central limit theorem for both macroscopic and mesoscopic linear statistics in the bulk.