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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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| _version_ | 1866929759855837184 |
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| author | Agrachev, Andrei Kazandjian, Bettina Pozzoli, Eugenio |
| author_facet | Agrachev, Andrei Kazandjian, Bettina Pozzoli, Eugenio |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11307 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Good Lie Brackets for classical and quantum harmonic oscillators Agrachev, Andrei Kazandjian, Bettina Pozzoli, Eugenio Optimization and Control 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. |
| title | Good Lie Brackets for classical and quantum harmonic oscillators |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2503.11307 |