Quantum limits of perturbed sub-Riemannian contact Laplacians in dimension 3

Víctor Arnaiz; Gabriel Rivière

doi:10.5802/jep.269

Quantum limits of perturbed sub-Riemannian contact Laplacians in dimension 3
[Limites quantiques pour les laplaciens sous-riemanniens de contact en dimension

3

]

Víctor Arnaiz ¹ ; Gabriel Rivière ²

¹ Laboratoire de Mathématiques Jean Leray, Nantes Université, UMR CNRS 6629, 2 rue de la Houssinière, 44322 Nantes Cedex 03, France
² Laboratoire de Mathématiques Jean Leray, Nantes Université, UMR CNRS 6629, 2 rue de la Houssinière, 44322 Nantes Cedex 03, France & Institut Universitaire de France, Paris, France

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 909-956.

Résumé
Abstract

Sur le fibré tangent unitaire d’une surface compacte riemannienne, nous considérons un sous-laplacien riemannien naturel associé à la structure de contact canonique. Dans la limite des grandes valeurs propres et pour des perturbations hypoelliptiques auto-adjointes de cet opérateur, nous étudions la façon dont la masse des fonctions propres s’échappe à l’infini dans l’espace co-tangent. En utilisant des méthodes semi-classiques, nous montrons que, dans ce régime sous-elliptique, les fonctions propres se concentrent sur certains ensembles de niveaux quantifiés le long de la direction du flot géodésique et qu’elles vérifient des propriétés d’invariance impliquant à la fois le champ de vecteurs géodésique et le terme de perturbation.

On the unit tangent bundle of a compact Riemannian surface, we consider a natural sub-Riemannian Laplacian associated with the canonical contact structure. In the large eigenvalue limit, we study the escape of mass at infinity in the cotangent space of eigenfunctions for hypoelliptic selfadjoint perturbations of this operator. Using semiclassical methods, we show that, in this subelliptic regime, eigenfunctions concentrate on certain quantized level sets along the geodesic flow direction and that they verify invariance properties involving both the geodesic vector field and the perturbation term.

Reçu le : 2023-06-30
Accepté le : 2024-07-30
Publié le : 2024-08-30

DOI : 10.5802/jep.269

Classification : 58J50, 35H10, 53C17
Keywords: Hypoelliptic operators, semiclassical analysis, contact flows
Mot clés : Opérateurs hypoelliptiques, analyse semi-classique, flots de contact

Affiliations des auteurs :

Víctor Arnaiz ¹ ; Gabriel Rivière ²

¹ Laboratoire de Mathématiques Jean Leray, Nantes Université, UMR CNRS 6629, 2 rue de la Houssinière, 44322 Nantes Cedex 03, France
² Laboratoire de Mathématiques Jean Leray, Nantes Université, UMR CNRS 6629, 2 rue de la Houssinière, 44322 Nantes Cedex 03, France & Institut Universitaire de France, Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Quantum limits of perturbed {sub-Riemannian} contact {Laplacians} in dimension~3},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Víctor Arnaiz; Gabriel Rivière. Quantum limits of perturbed sub-Riemannian contact Laplacians in dimension 3. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 909-956. doi : 10.5802/jep.269. https://jep.centre-mersenne.org/articles/10.5802/jep.269/

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