L(2) solutions to the Schrodinger-Poisson system

We solve a system of infinitely many coupled Schrodinger equations with self-consistent Coulomb potential for an initial data which is only L(2). For this purpose, we establish Strichartz' type inequalities in the framework of vector-valued wave functions (density matrices). Also, several smoothing effects are shown, which we compare with the Vlasov-Poisson System, semi-classical limit of the Schrodinger-Poisson System.