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 Schrodinger-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. Keywords Asymptotic behaviour, Schrodinger-Poisson problem. 1 INTRODUCTION The Schrodinger-Poisson system (SPS) in (0, #) R 3 associated with a single particle in a vacuum can be written in terms of the wave function (x, t) and the potential V (x, t) as follows i~ # #t = - ~ 2 2m # + V , (x, 0) = #(x), lim |x|## # = 0, (1.1) where ~ stands for the Planck constant and m for the particle mass, and where the potential is related to the particle density |#(x, t)| 2 via the Poisson equation #V (x, t) = -#|#(x, t)| 2 , lim |x|## V =..