  Eigenspaces of the quantum isotropic Harmonic Oscillator $\hat{H}_{\hbar} : =
- \frac{\hbar^2}{2} \Delta + \frac{||x||^2}{2}$ on $\mathbb{R}^d$ have
extremally high multiplicites and the eigenspace projections $\Pi_{\hbar,
E_N(\hbar)} $ have special asymptotic properties. This article gives a detailed
study of their Wigner distributions $W_{\hbar, E_N(\hbar)}(x, \xi)$.
Heuristically, if $E_N(\hbar) = E$, $W_{\hbar, E_N(\hbar)}(x, \xi)$ is the
`quantization' of the energy surface $\Sigma_E$, and should be like the
delta-function $\delta_{\Sigma_E}$ on $\Sigma_E$; rigorously, $W_{\hbar,
E_N(\hbar)}(x, \xi)$ tends in a weak* sense to $\delta_{\Sigma_E}$. But its
pointwise asymptotics and scaling asymptotics have more structure. The main
results give Bessel asymptotics of $W_{\hbar, E_N(\hbar)}(x, \xi)$ in the
interior $H(x, \xi) < E$ of $\Sigma_E$; interface Airy scaling asymptotics in
tubes of radius $\hbar^{2/3}$ around $\Sigma_E$, with $(x, \xi)$ either in the
interior or exterior of the energy ball; and exponential decay rates in the
exterior of the energy surface.
