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.

Abstract

We prove that smooth Wigner–Weyl spectral sums at an energy level E exhibit Airy scaling asymptotics across the classical energy surface Σ E . 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.