The Linear Boltzmann Equation as the Low Density Limit of a Random SCHRÖDINGER Equation

We study the long time evolution of a quantum particle interacting with a random potential in the Boltzmann-Grad low density limit. We prove that the phase space density of the quantum evolution defined through the Husimi function converges weakly to a linear Boltzmann equation. The Boltzmann collision kernel is given by the full quantum scattering cross-section of the obstacle potential.