From the von Neumann equation to the quantum Boltzmann equation II: Identifying the Born series

In a previous paper [Cal), the author studied a low density limit in the periodic von Neumann equation with potential, modified by a damping term. The model studied in [Cal], considered in dimensions d greater than or equal to 3, is deterministic. It describes the quantum dynamics of an electron in a periodic box (actually on a torus) containing one obstacle, when the electron additionally interacts with, say, an external bath of photons, The periodicity condition may be replaced by a Dirichlet boundary condition as well. In the appropriate low density asymptotics, followed by the limit where the damping vanishes, the author proved in [Cal] that the above system is described in the limit by a linear, space homogeneous, Boltzmann equation, with a cross-section given as an explicit power series expansion in the potential. The present paper continues the above study in that it identifies the cross-section previously obtained in [Cal] as the usual Born series of quantum scattering theory, which is the physically expected result. Hence we establish that a von Neumann equation converges, in the appropriate low density scaling, towards a linear Boltzmann equation with cross-section given by the full Born series expansion: we do not restrict ourselves to a weak coupling limit, where only the first term of the Born series would be obtained (Fermi's Golden Rule).