Observability estimates for the Schr{\"o}dinger equation in the plane with periodic bounded potentials from measurable sets
The goal of this article is to obtain observability estimates for Schr{\"o}dinger equations in the plane R 2. More precisely, considering a 2$\pi$Z 2-periodic potential V $\in$ L $\infty$ (R 2), we prove that the evolution equation i$\partial$tu = --$\Delta$u + V (x)u, is observable from any 2$...
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| Main Authors | , |
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| Format | Journal Article |
| Language | English |
| Published |
17.04.2023
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| Abstract | The goal of this article is to obtain observability estimates for
Schr{\"o}dinger equations in the plane R 2. More precisely, considering a
2$\pi$Z 2-periodic potential V $\in$ L $\infty$ (R 2), we prove that the
evolution equation i$\partial$tu = --$\Delta$u + V (x)u, is observable from any
2$\pi$Z 2-periodic measurable set, in any small time T > 0. We then extend
Ta{\"u}ffer's recent result [T{\"a}u22] in the two-dimensional case to less
regular observable sets and general bounded periodic potentials. The
methodology of the proof is based on the use of the Floquet-Bloch transform,
Strichartz estimates and semiclassical defect measures for the obtention of
observability inequalities for a family of Schr{\"o}dinger equations posed on
the torus R 2 /2$\pi$Z 2 . |
|---|---|
| AbstractList | The goal of this article is to obtain observability estimates for
Schr{\"o}dinger equations in the plane R 2. More precisely, considering a
2$\pi$Z 2-periodic potential V $\in$ L $\infty$ (R 2), we prove that the
evolution equation i$\partial$tu = --$\Delta$u + V (x)u, is observable from any
2$\pi$Z 2-periodic measurable set, in any small time T > 0. We then extend
Ta{\"u}ffer's recent result [T{\"a}u22] in the two-dimensional case to less
regular observable sets and general bounded periodic potentials. The
methodology of the proof is based on the use of the Floquet-Bloch transform,
Strichartz estimates and semiclassical defect measures for the obtention of
observability inequalities for a family of Schr{\"o}dinger equations posed on
the torus R 2 /2$\pi$Z 2 . |
| Author | Martin, Jérémy Balc'H, Kévin Le |
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| BackLink | https://doi.org/10.48550/arXiv.2304.08050$$DView paper in arXiv |
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| Snippet | The goal of this article is to obtain observability estimates for
Schr{\"o}dinger equations in the plane R 2. More precisely, considering a
2$\pi$Z 2-periodic... |
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| SubjectTerms | Mathematics - Analysis of PDEs Mathematics - Optimization and Control |
| Title | Observability estimates for the Schr{\"o}dinger equation in the plane with periodic bounded potentials from measurable sets |
| URI | https://arxiv.org/abs/2304.08050 |
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