Interface Asymptotics of Eigenspace Wigner distributions for the Harmonic Oscillator

Abstract

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.