Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alix Deleporte, Gaultier Lambert

Abstract

We study local asymptotics for the spectral projector associated to a Schrödinger operator $-\hbar^2Δ+V$ 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.

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 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 for 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.

Paper Structure

This paper contains 45 sections, 54 theorems, 507 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Setting and assumptions.
  4. Pointwise Weyl law and macroscopic fluctuations
  5. Universal limit at microscopic scales
  6. Gaussian field at mesoscopic scales
  7. Discussion and related works
  8. Hermitian random matrices
  9. Free fermions on $\mathbb{R}^n$
  10. Other universality classes for fermionic systems
  11. Semiclassical projector asymptotics.
  12. A few perspectives
  13. Organisation and Notation.
  14. Acknowledgements
  15. Semiclassical techniques for Schrödinger operators
  16. ...and 30 more sections

Key Result

Theorem I.1

Let $(\mu,V)$ satisfy hyp:weak. There exists a constant $c>0$ so that for any $\epsilon\in (0,1]$ and for any $\hbar\in (0,1]$,

Theorems & Definitions (73)

  • Definition 1.1
  • Theorem I.1
  • Theorem I.2
  • Theorem I.3
  • Conjecture 1
  • Theorem II.1
  • Remark 1.2
  • Theorem II.2
  • Theorem II.3
  • Theorem III.1
  • ...and 63 more