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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2504.16641 |
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| _version_ | 1866912343219240960 |
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| author | Boussaïd, Nabile Duca, Alessandro |
| author_facet | Boussaïd, Nabile Duca, Alessandro |
| contents | The local exact controllability of the one-dimensional bilinear Schr{ö}dinger equation with Dirichlet boundary conditions has been extensively studied in subspaces of H 3 since the seminal work of K. Beauchard. Our first objective is to revisit this result and establish the controllability in H 1 0 for suitable discontinuous control potentials. In the second part, we consider the equation in the presence of periodic boundary conditions and a constant magnetic field. We prove the local exact controllability of periodic H 1 -states, thanks to a Zeeman-type effect induced by the magnetic field which decouples the resonant spectrum. Finally, we discuss open problems and partial results for the Neumann case and the harmonic oscillator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16641 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $H^1$ local exact controllability of some one-dimensional bilinear Schr{ö}dinger equations Boussaïd, Nabile Duca, Alessandro Analysis of PDEs The local exact controllability of the one-dimensional bilinear Schr{ö}dinger equation with Dirichlet boundary conditions has been extensively studied in subspaces of H 3 since the seminal work of K. Beauchard. Our first objective is to revisit this result and establish the controllability in H 1 0 for suitable discontinuous control potentials. In the second part, we consider the equation in the presence of periodic boundary conditions and a constant magnetic field. We prove the local exact controllability of periodic H 1 -states, thanks to a Zeeman-type effect induced by the magnetic field which decouples the resonant spectrum. Finally, we discuss open problems and partial results for the Neumann case and the harmonic oscillator. |
| title | $H^1$ local exact controllability of some one-dimensional bilinear Schr{ö}dinger equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.16641 |