Interface Asymptotics of Eigenspace Wigner distributions for the Harmonic Oscillator

Abstract

Eigenspaces of the quantum isotropic Harmonic Oscillator ${\hat{H}}_{\hslash }:=-\frac{{\hslash }^2}{2}\hat{I}”+\frac{||x|{|}^2}{2}$ on ${\mathbb{R}}^d$ have extremally high multiplicites and the eigenspace projections ${\hat{I}}_{\hslash ,E_N(\hslash )}$ have special asymptotic properties. This article gives a detailed study of their Wigner distributions $W_{\hslash ,E_N(\hslash )}(x,\hat{I}¾)$. Heuristically, if $E_N(\hslash )=E$, $W_{\hslash ,E_N(\hslash )}(x,\hat{I}¾)$ is the `quantization' of the energy surface $\hat{I}{\pounds }_E$, and should be like the delta-function $\hat{I}{´}_{\hat{I}{\pounds }_E}$ on $\hat{I}{\pounds }_E$; rigorously, $W_{\hslash ,E_N(\hslash )}(x,\hat{I}¾)$ tends in a weak* sense to $\hat{I}{´}_{\hat{I}{\pounds }_E}$. But its pointwise asymptotics and scaling asymptotics have more structure. The main results give Bessel asymptotics of $W_{\hslash ,E_N(\hslash )}(x,\hat{I}¾)$ in the interior $H(x,\hat{I}¾)<E$ of $\hat{I}{\pounds }_E$; interface Airy scaling asymptotics in tubes of radius ${\hslash }^{2/3}$ around $\hat{I}{\pounds }_E$, with $(x,\hat{I}¾)$ either in the interior or exterior of the energy ball; and exponential decay rates in the exterior of the energy surface.

28 pages, 3 figures, v2. Statement of Theorem 1.5 slightly modified. Several references added