Quantum ergodicity for sub-Riemannian Laplacians. I: the contact 3D case

Yves Colin de Verdière (1) , Luc Hillairet (2) , Emmanuel Trélat (3)

1 IF - Institut Fourier
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
3 LJLL - Laboratoire Jacques-Louis Lions

Let us consider a sub-Riemannian (sR) metric on a closed three-dimensional manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We establish a Quantum Ergodicity (QE) theorem for the eigenfunctions of any associated sR Laplacian under the assumption that the Reeb flow is ergodic. The limit measure is given by the normalized Popp measure. This is the first time that such a result is established for a hypoelliptic operator,whereas the usual Shnirelman theorem yields QE for the Laplace-Beltrami operator on a closed Riemannian manifold with ergodic geodesic flow. To prove our main theorem, we first establish a microlocal Weyl law, which allows us to identify the limit measure and to prove the microlocal concentration of the eigenfunctions on the characteristic manifold of the sR Laplacian. Then, we derive a Birkhoff normal form, thus showing that, in some sense, all contact 3D sR structures are microlocally equivalent. The quantum version of this normal form provides a useful microlocal factorization of the sR Laplacian. Using the normal form, the factorization and the ergodicity assumption, we finally establish a variance estimate, from which QE follows.

Mots clés: contact structure; Reeb vector field; Shnirelman theorem; quantum ergodicity; sub-Riemannian Laplacian

Domaines: Théorie spectrale [math.SP]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Yves Colin de Verdière, Luc Hillairet, Emmanuel Trélat. Quantum ergodicity for sub-Riemannian Laplacians. I: the contact 3D case. 2015. ⟨hal-01144257v1⟩