Small-time controllability for the nonlinear Schrödinger equation on $\mathbb{R}^N$ via bilinear electromagnetic fields
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| Format: | Preprint |
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2023
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| author | Duca, Alessandro Pozzoli, Eugenio |
| author_facet | Duca, Alessandro Pozzoli, Eugenio |
| contents | We address the small-time controllability problem for a nonlinear Schrödinger equation (NLS) on $\mathbb{R}^N$ in the presence of magnetic and electric external fields. We choose a particular framework where the equation becomes $i\partial_t ψ= [-Δ+u_0(t)h_{\vec{0}}+\langle u(t), P\rangle +κ|ψ|^{2p}]ψ$. Here, the control operators are defined by the zeroth Hermite function $h_{\vec{0}}(x)$ and the momentum operator $P=i\nabla$. In detail, we study when it is possible to control the dynamics of (NLS) as fast as desired via sufficiently large control signals $u_0$ and $u$. We first show the existence of a family of quantum states for which this property is verified. Secondly, by considering some specific states belonging to this family, as a physical consequence we show the capability of controlling arbitrary changes of energy in bounded regions of the quantum system, in time zero. Our results are proved by exploiting the idea that the nonlinear term in (NLS) is only a perturbation of the linear problem when the time is as small as desired. The core of the proof, then, is the controllability of the bilinear equation which is tackled by using specific non-commutativity properties of infinite-dimensional propagators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_15819 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Small-time controllability for the nonlinear Schrödinger equation on $\mathbb{R}^N$ via bilinear electromagnetic fields Duca, Alessandro Pozzoli, Eugenio Optimization and Control Analysis of PDEs Quantum Physics 35Q55, 81Q93, 93B05 We address the small-time controllability problem for a nonlinear Schrödinger equation (NLS) on $\mathbb{R}^N$ in the presence of magnetic and electric external fields. We choose a particular framework where the equation becomes $i\partial_t ψ= [-Δ+u_0(t)h_{\vec{0}}+\langle u(t), P\rangle +κ|ψ|^{2p}]ψ$. Here, the control operators are defined by the zeroth Hermite function $h_{\vec{0}}(x)$ and the momentum operator $P=i\nabla$. In detail, we study when it is possible to control the dynamics of (NLS) as fast as desired via sufficiently large control signals $u_0$ and $u$. We first show the existence of a family of quantum states for which this property is verified. Secondly, by considering some specific states belonging to this family, as a physical consequence we show the capability of controlling arbitrary changes of energy in bounded regions of the quantum system, in time zero. Our results are proved by exploiting the idea that the nonlinear term in (NLS) is only a perturbation of the linear problem when the time is as small as desired. The core of the proof, then, is the controllability of the bilinear equation which is tackled by using specific non-commutativity properties of infinite-dimensional propagators. |
| title | Small-time controllability for the nonlinear Schrödinger equation on $\mathbb{R}^N$ via bilinear electromagnetic fields |
| topic | Optimization and Control Analysis of PDEs Quantum Physics 35Q55, 81Q93, 93B05 |
| url | https://arxiv.org/abs/2307.15819 |