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 $W_{{\phi }_N^{\gamma },{\phi }_N^{\gamma }}$ 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 $\text{hybrid}$ semi-classical scaling. That is, simultaneously, we have an $\text{Airy scaling}$ when the tube has radius $N^{-2/3}$ normal to the energy surface ${\Sigma }_E$, and a $\text{Gaussian scaling}$ when the tube has radius $N^{-1/2}$ tangent to ${\Sigma }_E$.

22 pages, 7 figures

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