On coupling kinetic and Schrödinger equations

We consider in this paper a system coupling a linear quantum Boltzmann equation and a defocussing cubic nonlinear Schrödinger equation. The Schrodinger equation reflects the dynamics of the wave function of the Bose-Einstein Condensate and the kinetic part of the system describes the evolution of the density function of the thermal cloud. An existence and uniqueness result for the system is supplied. We also prove the convergence to equilibrium of the density function of the thermal cloud and a scattering theory for the wave function of the condensate.