Universality for free fermions and the local Weyl law for semiclassical Schr\"odinger operators

TL;DR

This paper establishes universal local spectral asymptotics for semiclassical Schrödinger operators, demonstrating that the rescaled spectral projector converges to a universal model, leading to universal fluctuation behavior in free fermion point processes.

Contribution

It proves the local uniform convergence of the spectral projector's kernel to a universal limit for general smooth potentials in any dimension, extending universality results to boundary regions.

Findings

Universal local asymptotics for spectral projectors in the semiclassical limit.

Universality of microscopic fluctuations for free fermions in bulk and boundary.

Central limit theorem for linear statistics in the bulk for dimensions n≥2.

Abstract

We study local asymptotics for the spectral projector associated to a Schr\"odinger operator $-\hbar^2\Delta+V$ on $\mathbb{R}^n$ in the semiclassical limit as $\hbar\to0$. 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 for 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 $n\geq 2$, a central limit theorem for both macroscopic and mesoscopic linear statistics in the bulk.