In this paper, we investigate the dynamics of a system of $N$ weakly interacting bosons with singular three-body interactions in three dimensions. By assuming factorized initial data $Ψ_{N,0}=φ_{0}^{\otimes N}$ and triple collisions, we prove that in the many-particle limit, its mean-field approximation converges to quintic Hartree dynamics. Moreover, we prove that the rate of convergence towards the mean-field quintic Hartree evolution is of $O(N^{-(1+4a)/(3+2a)})$ for $φ_{0}\in H^{(3/2)+a}(\mathbb{R}^{3})$ where $0\leq a<1/2$ and $O(N^{-1})$ for $a>1$ . Our proof is based on and extends the Fock space approach.In this paper, we investigate the dynamics of a system of $N$ weakly interacting bosons with singular three-body interactions in three dimensions. By assuming factorized initial data $Ψ_{N,0}=φ_{0}^{\otimes N}$ and triple collisions, we prove that in the many-particle limit, its mean-field approximation converges to quintic Hartree dynamics. Moreover, we prove that the rate of convergence towards the mean-field quintic Hartree evolution is of $O(N^{-(1+4a)/(3+2a)})$ for $φ_{0}\in H^{(3/2)+a}(\mathbb{R}^{3})$ where $0\leq a<1/2$ and $O(N^{-1})$ for $a>1$. Our proof is based on and extends the Fock space approach.

arXiv:2006.13040Jun 22, 20202 citations

RATE OF CONVERGENCE TOWARDS HARTREE DYNAMICS

Aug 18, 20160 citations

Fluctuations Around Hartree Dynamics

Niels Benedikter, Marcello Porta, Benjamin Schlein

SpringerBriefs in mathematical physicsOct 6, 20150 citations

In this paper, we investigate the dynamics of a system of

N

weakly interacting bosons with singular three-body interactions in three dimensions. By assuming factorized initial data

Ψ_{N,0}=φ_{0}^{\otimes N}

and triple collisions, we prove that in the many-particle limit, its mean-field approximation converges to quintic Hartree dynamics. Moreover, we prove that the rate of convergence towards the mean-field quintic Hartree evolution is of

O(N^{-(1+4a)/(3+2a)})

for

φ_{0}\in H^{(3/2)+a}(\mathbb{R}^{3})

where

0\leq a<1/2

and

O(N^{-1})

for

a>1

. Our proof is based on and extends the Fock space approach.