Interface asymptotics of Wigner—Weyl distributions for the Harmonic Oscillator

We prove several types of scaling results for Wigner distributions of spectral projections of the isotropic Harmonic oscillator on ℝ^d. In prior work, we studied Wigner distributions Wh¯,EN(h¯)(x,ξ) of individual eigenspace projections. In this continuation, we study Weyl sums of such Wigner distributions as the eigenvalue E_N(h¯) ranges over spectral intervals [E− δ(h¯) , E+ δ(h¯)] of various widths δ(h¯) and as (x, ξ) ∈ T*ℝ^d ranges over tubes of various widths around the classical energy surface Σ._E ⊂ T*ℝ^d. The main results pertain to interface Airy scaling asymptotics around Σ_E, which divides phase space into an allowed and a forbidden region. The first result pertains to δ(h¯) = h¯ widths and generalizes our earlier results on Wigner distributions of individual eigenspace projections. Our second result pertains to δ(h¯) = h¯ ^{2 / 3} spectral widths and Airy asymptotics of the Wigner distributions in h¯ ^{2 / 3}-tubes around Σ_E. Our third result pertains to bulk spectral intervals of fixed width and the behavior of the Wigner distributions inside the energy surface, outside the energy surface and in a thin neighborhood of the energy surface.