Les séminaires auront lieu à l'
Institut Henri Poincaré, habituellement en salle 314.
Le séminaire est financé par le GDR "Dynamique quantique" du CNRS.
Pour tout renseignement complémentaire, veuillez contacter les organisateurs, Hakim Boumaza, Mathieu Lewin ou Stéphane Nonnenmacher.
|11h15 - 12h15||Gueorgui Popov (Nantes)
||K.A.M. tori isospectral deformations and spectral rigidity
We are interested in the spectral rigidity of the Laplace-Beltrami operator in the case when the corresponding classical Hamiltonian system is either completely integrable, or close to a nondegenerate completely integrable system. By the Kolmogorov-Arnold-Moser (K.A.M.) theorem, there exists a large family of invariant tori of the classical system with Diophantine rotation vectors. We show that the average action on the K.A.M. tori is an isospectral invariant. As an application, we obtain spectral rigidity for two-dimensional billiard tables in the presence of elliptic "bouncing ball" geodesics. We obtain as well infinitesimal rigidity of Liouville billiard tables in dimension two and three. The proof is based on a construction of smooth families of quasi-modes associated with the K.A.M. tori.
|14h - 15h||Mostafa Sabri (Strasbourg)
||Quantum ergodicity for the Anderson model on regular graphs
I will discuss a result of delocalization for the Anderson model on the regular tree (Bethe lattice). The Anderson model is a random Schrödinger operator, where we add a random i.i.d. perturbation to the adjacency matrix. Localization at high disorder is well understood today for a wide variety of models, both in the sense of a.s. pure point spectrum with exponentially decaying eigenfunctions, and in a dynamical sense. Delocalization remains a great challenge; only for tree models, it was proved by Klein, Aizenman, Warzel and generalized by others, that for weak disorder, large parts of the spectrum are purely absolutely continuous, and the dynamical transport is ballistic. A form of delocalization was also obtained by Geisinger. In this work, we try to complete the picture by proving that in such a regime, the eigenfunctions are also delocalized in space, in
the sense that if we consider a sequence of regular graphs converging to the regular tree, then the eigenfunctions become asymptotically uniformly distributed (as opposed to the exponential decay in the localization regime). The precise result is a quantum ergodicity theorem.
This is a joint work with Nalini Anantharaman.
|15h15 - 16h15|| Andrea Mantile (Reims)
|| On the simultaneous identification of scattering parameters for classical waves with potential and transmission conditions.
We consider the stationary waves scattering for 3D Schrödinger-type operators with singular perturbations supported on surfaces. This is a multiple scattering problem from obstacles and potentials, whose solutions depend on locations and shapes of the obstacles, the related transmission impedances and the background potentials. The corresponding inverse problem consists in determining these scattering parameters from a complete set of far-field data at a fixed energy. In this talk we provide, under suitable a priori bounds assumptions, a uniqueness result for this problem.
This is a joint work with A. Posilicano and M. Sini.