Interface asymptotics of eigenspace Wigner distributions for the harmonic oscillator

Abstract

Eigenspaces of the quantum isotropic Harmonic Oscillator on have extremally high multiplicites and the eigenspace projections have special asymptotic properties. This article gives a detailed study of their Wigner distributions Heuristically, if is the “quantization” of the energy surface ΣE, and should be like the delta-function on ΣE; rigorously, tends in a weak* sense to But its pointwise asymptotics and scaling asymptotics have more structure. The main results give Bessel asymptotics of in the interior of ΣE; interface Airy scaling asymptotics in tubes of radius around ΣE, with either in the interior or exterior of the energy ball; and exponential decay rates in the exterior of the energy surface.