Scaling Asymptotics of Wigner Distributions of Harmonic Oscillator Orbital Coherent States

Abstract

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