Sur les systèmes de fermions à grand nombre de particules : un point de vue probabiliste

de Suzzoni, Anne-Sophie

Séminaire Laurent Schwartz — EDP et applications (2015-2016), Exposé no. 12, 12 p.

On cherche à démontrer le caractère globalement bien posé pour l’équation de Hartree, avec un potentiel d’interaction égal au delta de Dirac, c’est-à-dire

$$i{\partial }_t\gamma =[-▵+{\rho }_{\gamma },\gamma ]$$

où $\gamma$ est un opérateur intégral, $[·,·]$ est le commutateur et ${\rho }_{\gamma }$ est la diagonale du noyau intégral de $\gamma$. On étudie cette équation autour de certains de ses états stationnaires. La difficulté principale vient du fait que les états stationnaires ne sont pas de classe trace alors que l’espace naturel pour la résolution de l’équation est l’espace des opérateurs positifs tels que $\mathrm{Tr}\left(\right(1-▵\left)\gamma \right)<\infty$. Pour s’affranchir de cette difficulté, on prend un point de vue probabiliste : on introduit une équation sur des processus aléatoires puis on traduit les résultats qu’on obtient pour cette équation en résultats pour l’équation de Hartree. Ce manuscrit résume partiellement [12].

We prove global well-posedness for the Hartree equation with an interaction potential equal to the Dirac delta, that is

$$i{\partial }_t\gamma =[-▵+{\rho }_{\gamma },\gamma ]$$

where $\gamma$ is an integral operator, $[·,·]$ is the commutator and ${\rho }_{\gamma }$ is the diagonal of the integral kernel of $\gamma$. We study this equation around some of its stationary states. The main difficulty comes from the fact that the stationary states are not trace class whereas the natural space to solve the equation is the space of positive operators such that $\mathrm{Tr}\left(\right(1-▵\left)\gamma \right)<\infty$. To solve this problem, we take a probabilistic point of view: we introduce an equation on random processes before translating results on this equation into results on the Hartree equation. This paper is a partial summary of [12].

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