Asymptotic behaviour for the 3-D Schrödinger-Poisson System in the attractive case with positive energy

In this paper, we study the asymptotic behaviour of solutions to the three-dimensional Schrödinger-Poisson system in the attractive case with positive energy. In this case, it is proved that for a real initial condition, the solutions expand unboundedly as time goes to infinity. The proof of this result is based on the derivation of a dispersive equation relating density and linear momentum, as well as on optimal bounds for the kinetic energy.