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

Alphonse, Paul; Seelmann, Albrecht

Comptes Rendus. Mathématique, Tome 362 (2024) no. G12, pp. 1635-1659

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 ${ℝ}^d\times {𝕋}^d$.

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 ${ℝ}^d\times {𝕋}^d$.

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

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