  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.
