In this paper we analyze the asymptotic dynamics of a system of N quantum
particles, in a weak coupling regime. Particles are assumed statistically independent at
the initial time.
Our approach follows the strategy introduced by the authors in a previous work
[BCEP1]: we compute the time evolution of the Wigner transform of the one-particle
reduced density matrix; it is represented by means of a perturbation series, whose expansion
is obtained upon iterating the Duhamel formula; this approach allows us to follow
the arguments developed by Lanford [L] for classical interacting particles evolving in a
low density regime.
We prove, under suitable assumptions on the interaction potential, that the complete
perturbation series converges term-by-term, for all times, towards the solution of
a Boltzmann equation.
The present paper completes the previous work [BCEP1]: it is proved there that a
subseries of the complete perturbation expansion converges uniformly, for short times,
towards the solution to the nonlinear quantum Boltzmann equation. This previous result
holds for (smooth) potentials having possibly non-zero mean value. The present text
establishes that the terms neglected at once in [BCEP1], on a purely heuristic basis,
indeed go term-by-term to zero along the weak coupling limit, at least for potentials
having zero mean.
Our analysis combines stationary phase arguments with considerations on the nature
of the various Feynman graphs entering the expansion.