Interface asymptotics of Wigner—Weyl distributions for the Harmonic Oscillator

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