Interface Asymptotics of Eigenspace Wigner distributions for the Harmonic Oscillator

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 $W_{\mathrm{\hslash },E_N(\mathrm{\hslash })}(x,\xi )$. Heuristically, if $E_N(\mathrm{\hslash })=E$, $W_{\mathrm{\hslash },E_N(\mathrm{\hslash })}(x,\xi )$ is the `quantization’ of the energy surface ${\mathrm{\Sigma }}_E$, and should be like the delta-function ${\delta }_{{\mathrm{\Sigma }}_E}$ on ${\mathrm{\Sigma }}_E$; rigorously, $W_{\hbar, E_N(\hbar)}(x, \xi)$tendsinaweak\ast senseto$\delta_{\Sigma_E}$. But its pointwise asymptotics and scaling asymptotics have more structure. The main results give Bessel asymptotics of $W_{\mathrm{\hslash },E_N(\mathrm{\hslash })}(x,\xi )$ in the interior $H(x,\xi )<E$ of ${\mathrm{\Sigma }}_E$; interface Airy scaling asymptotics in tubes of radius ${\mathrm{\hslash }}^{2\mathrm{/}3}$ around ${\mathrm{\Sigma }}_E$, with $(x,\xi )$ either in the interior or exterior of the energy ball; and exponential decay rates in the exterior of the energy surface.