Bose Einstein condensate in the Lowest Landau Level : Hamiltonian dynamics

Ce texte est une prépublication de l'IRMAR.Dans un article antérieur avec A. Aftalion et X. Blanc, les propriétés d'hypercontractivité du semigroupe des dilatations dans des espaces de fonctions entières est apparu comme un outil essentiel de l'étude du modèle de plus bas niveau de Landau pour les condensats de Bose Einstein. Ce travail portait sur le problème stationnaire. Nous abordons ici la question de la dynamique Hamiltonienne et les problèmes de stabilité spectrale. A MODIFIED VERSION OF THIS PREPRINT HAS BEEN PUBLISHED IN REVIEWS IN MATHEMATICAL PHYSICS

Adiabatic evolution of 1D shape resonances: an artificial interface conditions approach

Artificial interface conditions parametrized by a complex number $\theta_{0}$ are introduced for 1D-Schr\"odinger operators. When this complex parameter equals the parameter $\theta\in i\R$ of the complex deformation which unveils the shape resonances, the Hamiltonian becomes dissipative. This makes possible an adiabatic theory for the time evolution of resonant states for arbitrarily large time scales. The effect of the artificial interface conditions on the important stationary quantities involved in quantum transport models is also checked to be as small as wanted, in the polynomial scale $(h^N)_{N\in \N}$ as $h\to 0$, according to $\theta_{0}$.Comment: 60 pages, 13 figure

Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries

This article is concerned with maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic Laplacian.Comment: 186 page

Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion

We consider non-reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized. We solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them. We discuss in particular the role of the prefactor in the exponential convergence estimate. Our rigorous results are illustrated by numerical experiments

Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary.

Prépublication 04-40, IRMAR, UMR-CNRS 6625, Université de Rennes 1 (Août 2004)This article is a continuation of previous works by Bovier-Eckhoff-Gayrard-Klein, Bovier-Gayrard-Klein and Helffer-Klein-Nier. It is concerned with the analysis of the exponentially small eigenvalues of a semiclassical Witten Laplacian. We consider here the case of riemanian manifolds with boundary with a Dirichlet realization of the Witten Laplacian. A modified version of this preprint has been published in Mémoires de la SMF vol. 105, (2006

Mean field limit for bosons and infinite dimensional phase-space analysis

International audienceThis article proposes the construction of Wigner measures in the infinite dimensional bosonic quantum field theory, with applications to the derivation of the mean field dynamics. Once these asymptotic objects are well defined, it is shown how they can be used to make connections between different kinds of results or to prove new ones

Mean field propagation of infinite dimensional Wigner measures with a singular two-body interaction potential

49 pagesInternational audienceWe consider the quantum dynamics of many bosons systems in the mean field limit with a singular pair-interaction potential, including the attractive or repulsive Coulombic case in three dimensions. By using a measure transportation technique, we show that Wigner measures propagate along the nonlinear Hartree flow. Such property was previously proved only for bounded potentials in our previous works with a slightly different strategy

Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach

Nous calculons de façon précise les premières valeurs propres, exponentiellement petites, du Laplacien de Witten sur les 0-formes. Ces quantités sont liées au taux de retour à l\'équilibre pour des processus stochastiques réversibles.Lápproche proposée ici repose sur la structure du complexe de Witten et donne des résultats plus précis et généraux que les méthodes probabilistes