Uncertainty Principle and Geometric Condition for the Observability of Schrödinger Equations
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912283418951680 |
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| author | Wei, Longben Duan, Zhiwen Xu, Hui |
| author_facet | Wei, Longben Duan, Zhiwen Xu, Hui |
| contents | We provide necessary and sufficient geometric conditions for the exact observability of the Schrödinger equation with inverse-square potentials on the half-line. These conditions are derived from a Logvinenko-Sereda type theorem for generalized Fourier transform. Specifically, the generalized Fourier transform associated with the Schrödinger operator with inverse-square potentials on the half-line is the well-known Hankel transform. We present a necessary and sufficient condition for a subset $Ω$, such that a function whose Hankel transform is supported in a given interval can be bounded, in the $L^2$-norm, from above by its restriction to $Ω$, with a constant independent of the position of the interval. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_09592 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uncertainty Principle and Geometric Condition for the Observability of Schrödinger Equations Wei, Longben Duan, Zhiwen Xu, Hui Analysis of PDEs Mathematical Physics 35J10, 93B07, 42A65, 42C20 We provide necessary and sufficient geometric conditions for the exact observability of the Schrödinger equation with inverse-square potentials on the half-line. These conditions are derived from a Logvinenko-Sereda type theorem for generalized Fourier transform. Specifically, the generalized Fourier transform associated with the Schrödinger operator with inverse-square potentials on the half-line is the well-known Hankel transform. We present a necessary and sufficient condition for a subset $Ω$, such that a function whose Hankel transform is supported in a given interval can be bounded, in the $L^2$-norm, from above by its restriction to $Ω$, with a constant independent of the position of the interval. |
| title | Uncertainty Principle and Geometric Condition for the Observability of Schrödinger Equations |
| topic | Analysis of PDEs Mathematical Physics 35J10, 93B07, 42A65, 42C20 |
| url | https://arxiv.org/abs/2307.09592 |