Scaling Asymptotics of Wigner Distributions of Harmonic Oscillator Orbital Coherent States

TL;DR

This paper analyzes the detailed asymptotic behavior of Wigner distributions for harmonic oscillator orbital coherent states, revealing a hybrid semi-classical scaling involving Airy and Gaussian regimes near Hamiltonian orbits.

Contribution

It provides the first detailed asymptotic description of Wigner distributions for orbital coherent states, highlighting a novel hybrid scaling behavior in phase space.

Findings

Wigner distributions exhibit Airy scaling normal to energy surface

Gaussian scaling tangent to energy surface

Asymptotic behavior depends on tube radius around Hamiltonian orbits

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 $\Sigma_E$.