We prove a scattering result near certain steady states for a Hartree equation for a random field. This equation describes the evolution of a system of infinitely many particles. It is an analogous formulation of the usual Hartree equation for density matrices. We treat dimensions and , extending our previous result. We reach a large class of interaction potentials, which includes the nonlinear Schrödinger equation. This result has an incidence in the density matrices framework. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger equation, and on the use of explicit low frequency cancellations as done by Lewin and Sabin. To relate to density matrices, we use Strichartz estimates for orthonormal systems from Frank and Sabin, and Leibniz rules for integral operators.
Nous montrons un résultat de diffusion au voisinage de certains états d’équilibre pour une équation d’Hartree pour un champ aléatoire. Cette équation décrit l’évolution d’un système composé d’une infinité de particules. Elle est analogue à la formulation usuelle de l’équation d’Hartree pour des matrices de densité. Nous traitons le cas des dimensions deux et trois, étendant ainsi notre résultat précédent. Nous considérons une large classe de potentiels d’interaction, qui inclue l’équation de Schrödinger non linéaire. Notre résultat en implique un analogue dans le cadre des matrices de densité. La preuve repose sur des techniques dispersives utilisées pour la diffusion pour les équations de Schrödinger non linéaires, et sur des annulations dans les basses fréquences similaires à celles de Lewin et Sabin. Pour relier notre résultat au matrices de densité, nous utilisons des estimations de Strichartz pour des systèmes orthonormaux de Frank et Sabin, et des règles de Leibniz pour des opérateurs intégraux.
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Accepted:
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Mots-clés : Hartree equation, nonlinear Schrödinger equation, density matrices, random fields, stability, scattering
@article{AHL_2022__5__429_0,
author = {Collot, Charles and de Suzzoni, Anne-Sophie},
title = {Stability of steady states for {Hartree} and {Schr\"odinger} equations for infinitely many particles},
journal = {Annales Henri Lebesgue},
pages = {429--490},
publisher = {\'ENS Rennes},
volume = {5},
year = {2022},
doi = {10.5802/ahl.127},
language = {en},
url = {http://www.numdam.org/articles/10.5802/ahl.127/}
}
TY - JOUR AU - Collot, Charles AU - de Suzzoni, Anne-Sophie TI - Stability of steady states for Hartree and Schrödinger equations for infinitely many particles JO - Annales Henri Lebesgue PY - 2022 SP - 429 EP - 490 VL - 5 PB - ÉNS Rennes UR - http://www.numdam.org/articles/10.5802/ahl.127/ DO - 10.5802/ahl.127 LA - en ID - AHL_2022__5__429_0 ER -
%0 Journal Article %A Collot, Charles %A de Suzzoni, Anne-Sophie %T Stability of steady states for Hartree and Schrödinger equations for infinitely many particles %J Annales Henri Lebesgue %D 2022 %P 429-490 %V 5 %I ÉNS Rennes %U http://www.numdam.org/articles/10.5802/ahl.127/ %R 10.5802/ahl.127 %G en %F AHL_2022__5__429_0
Collot, Charles; de Suzzoni, Anne-Sophie. Stability of steady states for Hartree and Schrödinger equations for infinitely many particles. Annales Henri Lebesgue, Volume 5 (2022), pp. 429-490. doi : 10.5802/ahl.127. http://www.numdam.org/articles/10.5802/ahl.127/
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