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

  • Alix Deleporte

    Université Paris-Saclay, CNRS, Orsay, France
  • Gaultier Lambert

    Universität Zürich, Zürich, Switzerland; KTH Royal Institute of Technology, Stockholm, Sweden
Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators cover
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Abstract

We study local asymptotics for the spectral projector associated to a Schrödinger operator on in the semiclassical limit as . 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 . 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 , a central limit theorem for both macroscopic and mesoscopic linear statistics in the bulk.

Cite this article

Alix Deleporte, Gaultier Lambert, Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1447