A Dirac field interacting with point nuclear dynamics

The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent one-body Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in H\u3c3 with \u3c3 08[1,32[. In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction of the propagator for the Dirac operator with several moving Coulomb singularities

On the probabilistic Cauchy theory for nonlinear dispersive PDEs

In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.Comment: 26 pages. To appear in Landscapes of Time-Frequency Analysis, Appl. Numer. Harmon. Ana

Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation

We consider the cubic fourth order nonlinear Schr\"odinger equation on the circle. In particular, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac34$, are quasi-invariant under the flow.Comment: 41 pages. To appear in Probab. Theory Related Field

A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations

42 pagesInternational audienceWe consider the defocusing nonlinear Schr\"odinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $\R^2$. Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure

Stability of equilibria for a Hartree equation for random fields

International audienceWe consider a Hartree equation for a random field, which describes the temporal evolution of infinitely many fermions. On the Euclidean space, this equation possesses equilibria which are not localized. We show their stability through a scattering result, with respect to localized perturbations in the not too focusing case in high dimensions d ≥ 4. This provides an analogue of the results of Lewin and Sabin [22], and of Chen, Hong and Pavlović [11] for the Hartree equation on operators. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger and Gross-Pitaevskii equations.On considère une équation de Hartree pour des champs aléatoires décrivant la dynamique d'un système infini de fermions. Sur l'espace euclidien, cette équation admet des équilibres non-localisés. On prouve la stabilité de ces derniers à travers un résultat de diffusion pour des perturbations localisées et en dimension supérieure à 4, lorsque la non-linéarité est faiblement focalisante. Cela fournit une contrepartie aléatoire aux résultats de Lewin et Sabin [22] et de Chen, Hong et Pavlović [11] qui traitent de l'équation de Hartree pour des opérateurs densités. La preuve s'appuie sur des techniques dispersives utilisées dans l'étude de la diffusion des équations de Schrödinger et de Gross-Pitaevskii

STRICHARTZ ESTIMATES FOR THE KLEIN-GORDON EQUATION IN A CONICAL SINGULAR SPACE

Consider a conical singular space X = C(Y) = (0, ∞)r ×Y with the metric g = dr 2 + r 2 h, where the cross section Y is a compact (n − 1)-dimensional closed Riemannian manifold (Y, h). We study the Klein-Gordon equations with inverse-square potentials in the space X, proving in particular global-in-time Strichartz estimates in this setting