Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability

Paul Alphonse; Albrecht Seelmann

doi:10.5802/crmath.670

Article de recherche - Équations aux dérivées partielles, Théorie du contrôle

Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability
[Inégalités spectrales quantitatives pour les opérateurs de Shubin anisotropes et applications en contrôlabilité à zéro]

Paul Alphonse ¹ ; Albrecht Seelmann ²

¹ Université de Lyon, ENSL, UMPA – UMR 5669, F-69364 Lyon, France
² Technische Universität Dortmund, Fakultät für Mathematik, D-44221 Dortmund, Germany

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1635-1659.

Résumé
Abstract

On démontre des inégalités spectrales quantitatives pour les opérateurs de Shubin (anisotropes) sur tout l’espace euclidien, reliant ainsi pour les fonctions des sous-espaces spectraux associés à des intervalles d’énergie finie leur norme $L^{2}$ sur l’espace entier à la norme $L^{2}$ sur un sous-ensemble approprié. Une caractéristique particulière de nos estimations est que la constante reliant ces normes $L^{2}$ est très explicite en les paramètres géométriques du sous-ensemble de l’espace entier correspondant, qui peut devenir clairsemé à l’infini et même avoir une mesure finie. On étend ainsi des résultats obtenus récemment par J. Martin et, dans le cas particulier de l’oscillateur harmonique, par A. Dicke, I. Veselić et le deuxième auteur. Nous appliquons nos résultats à la contrôlabilité à zéro des équations paraboliques associées, ainsi qu’à celles associées aux opérateurs (dégénérés) de Baouendi-Grushin agissant sur $ℝ^{d} \times 𝕋^{d}$ .

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 $ℝ^{d} \times 𝕋^{d}$ .

Reçu le : 2023-12-18
Révisé le : 2024-07-18
Accepté le : 2024-07-19
Publié le : 2024-11-25

DOI : 10.5802/crmath.670

Classification : 35P05, 93B05, 35P10
Keywords: Spectral inequalities, null-controllability, Agmon estimates, anisotropic Shubin operators, Baouendi–Grushin operator
Mots-clés : Inégalités spectrales, contrôlabilité à zéro, estimées d’Agmon, opérateurs de Shubin anisotropes, opérateur de Baouendi–Grushin

Affiliations des auteurs :

Paul Alphonse ¹ ; Albrecht Seelmann ²

¹ Université de Lyon, ENSL, UMPA – UMR 5669, F-69364 Lyon, France
² Technische Universität Dortmund, Fakultät für Mathematik, D-44221 Dortmund, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Quantitative spectral inequalities for the anisotropic {Shubin} operators and applications to null-controllability},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1635--1659},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2024},
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     language = {en},
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Paul Alphonse; Albrecht Seelmann. Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1635-1659. doi : 10.5802/crmath.670. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.670/

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