In this paper we consider the simplest model of quantum open systems, which is the case of one-body statistical physics with inflow boundary conditions at infinity. We develop a functional framework for some nonlinear dynamics, where the nonlinearities are (very) short-range perturbations of a given Hamiltonian. In this framework, one can prove the global in time existence and uniqueness of a solution for the time-dependent problem and the existence of steady states of which the form can be specified. This problem arises from the modelling of semiconductors and applications are reviewed at the end of the paper.