Lundi 14 mai 10:30-11:30 Alix Deleporte 
Concentration of eigenfunctions for semiclassical Toeplitz operators

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Lieu : Salle 3L15, IMO

Résumé : Toeplitz operators are a generalisation of the FBI point of view on pseudodifferential operators ; they also allow to quantize compact phase spaces, with applications to spin systems in the large spin limit. Motivated by the Heisenberg Antiferromagnet, we proved a series of results on the subprincipal effects on concentration for low-energy eigenfunctions in the general setting of Toeplitz operators. An example is the « miniwell » situation, studied by Helffer and Sjöstrand for Schrödinger operators but never in the general pseudodifferential case, where the principal symbol (classical energy) is minimal along a submanifold, but the ground state concentrates only at one point, a physical effect known as « quantum selection ».
In this talk, we will present Toeplitz operators, the physical problem of interest, and some of the tools used in our work.

Concentration of eigenfunctions for semiclassical Toeplitz operators  Version PDF


Mardi 17 avril 15:30-16:30 Valentina Franceschi  (LJLL, UMPC & Fondazione Ing. Aldo Gini (Padoue))
Essential self-adjointness of sub-elliptic laplacians

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Lieu : Salle 2L8, IMO

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.

Essential self-adjointness of sub-elliptic laplacians  Version PDF

Lundi 26 mars 14:00-15:00 Armen Shirikyan  (Université de Cergy-Pontoise)
Introduction élémentaire au théorème de fluctuation pour des systèmes dynamiques chaotiques

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Lieu : IMO - Salle 2L8

Résumé : Nous présentons un cadre général simple pour l’obtention de la relation de fluctuation dans des systèmes déterministes et stochastiques. Après avoir introduit quelques objets simples liés à la production d’entropie, nous montrons que la relation de fluctuation proposée par Evans–Searles, Gallavotti–Cohen et Lebowitz–Spohn est une conséquence du principe des grandes déviations (PGD). Nous passons ensuite à une classe de systèmes dynamiques chaotiques et étudions la validité du PGD. Sous des hypothèses assez générales (qui n’excluent pas des transitions de phase), on démontre que les mesures d’occupation satisfont le PGD avec une bonne fonction de taux pour laquelle une généralisation de la relation de fluctuation au niveau 3 est valable.
C’est un travail en collaboration avec N. Cuneo, V. Jaksic et C.-A. Pillet.

Introduction élémentaire au théorème de fluctuation pour des systèmes dynamiques chaotiques  Version PDF