We prove global well-posedness for the Hartree equation with an interaction potential equal to the Dirac delta, that is
where is an integral operator, is the commutator and is the diagonal of the integral kernel of . 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 . 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].
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
où est un opérateur intégral, est le commutateur et est la diagonale du noyau intégral de . 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 . 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].
@article{SLSEDP_2015-2016____A12_0,
author = {Anne-Sophie de Suzzoni},
title = {Sur les syst\`emes de fermions \`a grand nombre de particules~: un~point de vue probabiliste},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:12},
pages = {1--12},
publisher = {Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique},
year = {2015-2016},
doi = {10.5802/slsedp.86},
language = {fr},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.86/}
}
TY - JOUR AU - Anne-Sophie de Suzzoni TI - Sur les systèmes de fermions à grand nombre de particules : un point de vue probabiliste JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:12 PY - 2015-2016 SP - 1 EP - 12 PB - Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.86/ DO - 10.5802/slsedp.86 LA - fr ID - SLSEDP_2015-2016____A12_0 ER -
%0 Journal Article %A Anne-Sophie de Suzzoni %T Sur les systèmes de fermions à grand nombre de particules : un point de vue probabiliste %J Séminaire Laurent Schwartz — EDP et applications %Z talk:12 %D 2015-2016 %P 1-12 %I Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.86/ %R 10.5802/slsedp.86 %G fr %F SLSEDP_2015-2016____A12_0
Anne-Sophie de Suzzoni. Sur les systèmes de fermions à grand nombre de particules : un point de vue probabiliste. Séminaire Laurent Schwartz — EDP et applications (2015-2016), Talk no. 12, 12 p. doi : 10.5802/slsedp.86. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.86/
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