This article is a review of recent results with Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty and Jan Philip Solovej. We consider a system of bosons with an interaction of intensity (mean-field regime). In the limit , we prove that the first order in the expansion of the eigenvalues of the many-particle Hamiltonian is given by the nonlinear Hartree theory, whereas the next order is predicted by the Bogoliubov Hamiltonian. We also discuss the occurrence of Bose-Einstein condensation in these...
Nous étudions le spectre du Hamiltonien d’un gaz de bosons, à la limite d’un grand nombre de particules et dans le régime de champ moyen (l’interaction est multipliée par ). Le premier terme du développement est donné par le modèle non linéaire de Hartree, alors que le second terme est donné par la théorie de Bogoliubov.
Dans cet exposé, je présente plusieurs modèles quantiques non linéaires permettant de décrire certaines étoiles. Je m’intéresse tout particulièrement à l’effondrement gravitationnel des étoiles trop lourdes, un phénomène modélisé par des solutions qui explosent en temps fini. Je montre l’existence de telles solutions et je décris plusieurs de leurs propriétés au temps d’explosion.
The method of choice for describing attractive quantum systems is Hartree−Fock−Bogoliubov (HFB) theory. This is a nonlinear model which allows for the description of , the main explanation for the superconductivity of certain materials at very low temperature. This paper is the first study of Hartree−Fock−Bogoliubov theory from the point of view of numerical analysis. We start by discussing its proper discretization and then analyze the convergence of the simple fixed point (Roothaan) algorithm....
We consider a multi-polaron model obtained by coupling the many-body Schrödinger equation for interacting electrons with the energy functional of a mean-field crystal with a localized defect, obtaining a highly non linear many-body problem. The physical picture is that the electrons constitute a charge defect in an otherwise perfect periodic crystal. A remarkable feature of such a system is the possibility to form a bound state of electrons via their interaction with the polarizable background....
We review recent results about the derivation and the analysis of two Hartree-Fock-type models for the polarization of vacuum. We pay particular attention to the variational construction of a self-consistent polarized vacuum, and to the physical agreement between our non-perturbative construction and the perturbative description provided by Quantum Electrodynamics.
We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation in a time-dependent potential and we show the existence of the wave operator in Schatten spaces.
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