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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.11307 |
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Table of Contents:
- 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 Schrödinger PDE for the quantum harmonic oscillator, and find new small-time approximate reachability properties of the Liouville PDE for the classical harmonic oscillator.