Good Lie Brackets for classical and quantum harmonic oscillators

We study the small-time controllability problem on the Lie groups $S{L}_2(\mathbb{R})$ and $S{L}_2(\mathbb{R})⋉H_d(\mathbb{R})$ with Lie bracket methods (here $H_d(\mathbb{R})$ denotes the $(2d+1)$-dimensional real Heisenberg group). Then, using unitary representations of $S{L}_2(\mathbb{R})⋉H_d(\mathbb{R})$ on $L^2({\mathbb{R}}^d,\mathbb{C})$ and $L^p(T^{\ast }{\mathbb{R}}^d,\mathbb{R}),p\in [1,\mathrm{\infty })$, we recover small-time approximate reachability properties of the Schrödinger PDE for the quantum harmonic oscillator, and find new small-time approximate reachability properties of the Liouville PDE for the classical harmonic oscillator.