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
ΨN,0=φ0⊗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
φ0∈H(3/2)+a(R3) where
0≤a<1/2 and
O(N−1) for
a>1. Our proof is based on and extends the Fock space approach.
We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schrödinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in Pickl (Lett. Math. Phys. 97 (2) 151–164 2011 ) for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new mean-field limit for fermions with long-range interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(-Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(-Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the mean-field limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence.