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 W (h) over bar, E-N ((h) over bar) (x, xi) of individual eigenspace projections. In this continuation, we study Weyl sums of such Wigner distributions as the eigenvalue E-N ((h) over bar) ranges over spectral intervals [E - delta((h) over bar), E + delta((h) over bar)] of various widths delta((h) over bar) and as (x, xi) is an element of T*R-d ranges over tubes of various widths around the classical energy surface Sigma(E) subset of T * 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((h) over bar) = (h) over bar widths and generalizes our earlier results on Wigner distributions of individual eigenspace projections. Our second result pertains to delta((h) over bar) = (h) over bar (2/3) spectral widths and Airy asymptotics of the Wigner distributions in (h) over bar (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.