Essential self-adjointness of sub-elliptic laplacians

Mardi 17 avril 15:30-16:30 - Valentina Franceschi - LJLL, UMPC & Fondazione Ing. Aldo Gini (Padoue)

Résumé : The aim of this seminar is to present recent results obtained in [2] in collaboration with D. Prandi (CNRS, CentraleSupélec, Gif-sur-Yvette, France) and L. Rizzi (CNRS & Institut Fourier, Grenoble, France) about essential self-adjointness of sub-elliptic laplacians.
These are hypoelliptic operators defined on a manifold M, that are naturally associated to a geometric structure on it. In the case when such a structure is Riemannian and complete, the associated Laplace-Beltrami operator is indeed essentially self-adjoint [3]. This amounts to say that the solutions to the Schrödinger equation on M are well defined without imposing any boundary conditions.
Our purpose is to address the case when the structure is sub-Riemannian : this can be thought of as a generalization of the Riemannian case, under anisotropic constraints on the directions of motion on M. In particular, singularities may appear, encoded in the blow up of an intrinsic measure, whose definition depends only on the geometry.
In this case the problem is still open and a standing conjecture, formulated by Boscain and Laurent in [1], asserts that the sub-elliptic Laplacian is essentially self-adjoint.
We will explain our results supporting the conjecture and underline the cases that are not included in our analysis. The results in [2] are a generalization of the ones in [4].
[1] U. Boscain and C. Laurent, The Laplace-Beltrami operator in almost-Riemannian geometry, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 5, 1739–1770.
[2] V. Franceschi, D. Prandi, and L. Rizzi, On the essential self-adjointness of sub-Laplacians, ArXiv e-prints (2017), available at
[3] Matthew P. Gaffney, Hilbert space methods in the theory of harmonic integrals, Trans. Amer. Math. Soc. 78 (1955), 426–444.
[4] D. Prandi, L. Rizzi, and M. Seri, Quantum confinement on non-complete Riemannian manifolds, Journal of Spectral Theory - to appear.

Lieu : Salle 2L8, IMO

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