Interface asymptotics of eigenspace wigner distributions for the harmonic oscillator

Eigenspaces of the quantum isotropic Harmonic Oscillator pH :=-2 Δ H2/2 ||x||2 on ℝ^dhave extremally high multiplicites and the eigenspace projections Π;EN(h) have special asymptotic properties. This article gives a detailed study of their Wigner distributions W;EN(h)(x, ξ). Heuristically, if ENpq ΔE, W;ENpqpx; Σq is the 'quantization' of the energy surface ΣE, and should be like the delta-function ΔE on ΣE; rigorously, W;En(h)(x, ξ) tends in a weak∗ sense to δσE. But its pointwise asymptotics and scaling asymptotics have more structure. The main results give Bessel asymptotics of W;EN(h)(x, ξ) in the interior H(x, ξ) < E of ΣE; interface Airy scaling asymptotics in tubes of radius h2/3 around ΣE, with (x, ξ) either in the interior or exterior of the energy ball; and exponential decay rate sin the exterior of the energy surface.