Good Lie Brackets for classical and quantum harmonic oscillators

We study the small-time controllability problem on the Lie groups $SL_2(\mathbb{R})$ and $SL_2(\mathbb{R})\ltimes H_{d}(\mathbb{R})$ with Lie bracket methods (here $H_{d}(\mathbb{R})$ denotes the $(2d+1)$-dimensional real Heisenberg group). Then, using unitary representations of $SL_2(\mathbb{R})\ltimes H_{d}(\mathbb{R})$ on $L^2(\mathbb{R}^d,\mathbb{C})$ and $L^p(T^*\mathbb{R}^d,\mathbb{R}), p\in[1,\infty)$, we recover small-time approximate reachability properties of the Schr\"odinger PDE for the quantum harmonic oscillator, and find new small-time approximate reachability properties of the Liouville PDE for the classical harmonic oscillator.