Scaling asymptotics of spectral Wigner functions

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.