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 $\mathbb R^d$. In prior work, we studied Wigner distributions $W_{\hbar, E_N(\hbar)}(x, \xi)$ of individual eigenspace projections. In this continuation, we study Weyl sums of such Wigner distributions as the eigenvalue $E_N(\hbar)$ ranges over spectral intervals $[E - \delta(\hbar), E + \delta(\hbar)]$ of various widths $\delta(\hbar)$ and as $(x, \xi) \in T^*\mathbb R^d$ ranges over tubes of various widths around the classical energy surface $\Sigma_E \subset T^*\mathbb R^d$. The main results pertain to interface Airy scaling asymptotics around $\Sigma_E$, which divides phase space into an allowed and a forbidden region. The first result pertains to $\delta(\hbar) = \hbar$ widths and generalizes our earlier results on Wigner distributions of individual eigenspace projections. Our second result pertains to $\delta(\hbar) = \hbar^{2/3}$ spectral widths and Airy asymptotics of the Wigner distributions in $\hbar^{2/3}$-tubes around $\Sigma_E$. 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.