Physics – Mathematical Physics
Scientific paper
2010-11-27
Archive for Rational Mechanics and Analysis, Vol. 203 (2012), 455-497
Physics
Mathematical Physics
v3. Revised according to the referee report. Accepted to appear in Archive for Rational Mechanics and Analysis. Comments are w
Scientific paper
10.1007/s00205-011-0453-8
In this paper, we consider the Hamiltonian evolution of N weakly interacting Bosons. Assuming triple collisions, its mean field approximation is given by a quintic Hartree equation. We construct a second order correction to the mean field approximation using a kernel k(t,x,y) and derive an evolution equation for k. We show the global existence for the resulting evolution equation for the correction and establish an apriori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work in [22,12,13] where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order O(1/$\sqrt{N}$).
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