Scaling asymptotics of spectral Wigner functions* * SZ was partially supported by NSF Grant DMS-1810747. BH was funded by NSF CAREER Grant DMS-2143754 as well as NSF Grants DMS-1855684, DMS-2133806 and an ONR MURI on Foundations of Deep Learning.

Boris Hanin, Steve Zelditch

Research output: Contribution to journalArticlepeer-review

Abstract

We prove that smooth Wigner-Weyl spectral sums at an energy level E exhibit Airy scaling asymptotics across the classical energy surface ΣE . This was proved earlier by the authors for the isotropic harmonic oscillator and the proof is extended in this article to all quantum Hamiltonians −ℏ 2Δ + V where V is a confining potential with at most quadratic growth at infinity. The main tools are the Herman-Kluk initial value parametrix for the propagator and the Chester-Friedman-Ursell normal form for complex phases with a one-dimensional cubic degeneracy. This gives a rigorous account of Airy scaling asymptotics of spectral Wigner distributions of Berry, Ozorio de Almeida and other physicists.

Original languageEnglish (US)
Article number414003
JournalJournal of Physics A: Mathematical and Theoretical
Volume55
Issue number41
DOIs
StatePublished - Oct 14 2022

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modeling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)

Keywords

  • Airy
  • asymptotics
  • spectral
  • Wigner-Weyl

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