Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability
We prove quantitative spectral inequalities for the (anisotropic) Shubin operators on the whole Euclidean space, thus relating for functions from spectral subspaces associated to finite energy intervals their $L^2$-norm on the whole space to the $L^2$-norm on a suitable subset. A particular feature...
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.670/ |
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| author | Alphonse, Paul Seelmann, Albrecht |
| author_facet | Alphonse, Paul Seelmann, Albrecht |
| author_sort | Alphonse, Paul |
| collection | DOAJ |
| description | We prove quantitative spectral inequalities for the (anisotropic) Shubin operators on the whole Euclidean space, thus relating for functions from spectral subspaces associated to finite energy intervals their $L^2$-norm on the whole space to the $L^2$-norm on a suitable subset. A particular feature of our estimates is that the constant relating these $L^2$-norms is very explicit in geometric parameters of the corresponding subset of the whole space, which may become sparse at infinity and may even have finite measure. This extends results obtained recently by J. Martin and, in the particular case of the harmonic oscillator, by A. Dicke, I. Veselić, and the second author. We apply our results towards null-controllability of the associated parabolic equations, as well as to the ones associated to the (degenerate) Baouendi-Grushin operators acting on $\mathbb{R}^d \times \mathbb{T}^d$. |
| format | Article |
| id | doaj-art-c251c94cd7fc425ca114e5bd14606746 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-c251c94cd7fc425ca114e5bd146067462025-02-07T11:26:37ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G121635165910.5802/crmath.67010.5802/crmath.670Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllabilityAlphonse, Paul0Seelmann, Albrecht1Université de Lyon, ENSL, UMPA – UMR 5669, F-69364 Lyon, FranceTechnische Universität Dortmund, Fakultät für Mathematik, D-44221 Dortmund, GermanyWe prove quantitative spectral inequalities for the (anisotropic) Shubin operators on the whole Euclidean space, thus relating for functions from spectral subspaces associated to finite energy intervals their $L^2$-norm on the whole space to the $L^2$-norm on a suitable subset. A particular feature of our estimates is that the constant relating these $L^2$-norms is very explicit in geometric parameters of the corresponding subset of the whole space, which may become sparse at infinity and may even have finite measure. This extends results obtained recently by J. Martin and, in the particular case of the harmonic oscillator, by A. Dicke, I. Veselić, and the second author. We apply our results towards null-controllability of the associated parabolic equations, as well as to the ones associated to the (degenerate) Baouendi-Grushin operators acting on $\mathbb{R}^d \times \mathbb{T}^d$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.670/Spectral inequalitiesnull-controllabilityAgmon estimatesanisotropic Shubin operatorsBaouendi–Grushin operator |
| spellingShingle | Alphonse, Paul Seelmann, Albrecht Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability Comptes Rendus. Mathématique Spectral inequalities null-controllability Agmon estimates anisotropic Shubin operators Baouendi–Grushin operator |
| title | Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability |
| title_full | Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability |
| title_fullStr | Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability |
| title_full_unstemmed | Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability |
| title_short | Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability |
| title_sort | quantitative spectral inequalities for the anisotropic shubin operators and applications to null controllability |
| topic | Spectral inequalities null-controllability Agmon estimates anisotropic Shubin operators Baouendi–Grushin operator |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.670/ |
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