Scaling Asymptotics of Wigner Distributions of Harmonic Oscillator Orbital Coherent States

The main result of this article gives scaling asymptotics of the Wigner distributions $W_{{\phi }_N^{\gamma },{\phi }_N^{\gamma }}$ of isotropic harmonic oscillator orbital coherent states ${\phi }_N^{\gamma }$ concentrating along Hamiltonian orbits $\gamma$ in shrinking tubes around $\gamma$ in phase space. In particular, these Wigner distributions exhibit a $\mathit{\text{hybrid}}$ semi-classical scaling. That is, simultaneously, we have an $\mathit{\text{Airy}}\mathit{\text{scaling}}$ when the tube has radius $N^{-2\mathrm{/}3}$ normal to the energy surface ${\mathrm{\Sigma }}_E$, and a $\mathit{\text{Gaussian}}\mathit{\text{scaling}}$ when the tube has radius $N^{-1\mathrm{/}2}$ tangent to ${\mathrm{\Sigma }}_E$.