Scaling asymptotics of spectral Wigner functions

TL;DR

This paper establishes Airy scaling asymptotics for spectral Wigner functions at energy levels for a broad class of quantum Hamiltonians, extending previous results from specific cases to general confining potentials.

Contribution

The authors extend the proof of Airy scaling asymptotics of spectral Wigner functions from the harmonic oscillator to all confining quantum Hamiltonians with quadratic growth potentials.

Findings

Proved Airy scaling asymptotics for spectral Wigner functions across energy surfaces.

Extended previous results from harmonic oscillator to general confining potentials.

Provided a rigorous mathematical foundation for physicists' heuristic predictions.

Abstract

We prove that smooth Wigner-Weyl spectral sums at an energy level $E$ exhibit Airy scaling asymptotics across the classical energy surface $\Sigma_E$. This was proved earlier by the authors for the isotropic harmonic oscillator and the proof is extended in this article to all quantum Hamiltonians $-\hbar^2 \Delta + V$ 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 M.V. Berry, A. Ozorio de Almeida and other physicists.