A variational approach to the Schrodinger-Poisson system: Asymptotic behaviour, breathers, and stability

In this paper a variational formulation of the three-dimensional Schrodinger Poisson system is proposed with the aim of solving the open problem of the asymptotic behaviour in time of the solutions in the case of attractive Coulomb forces, A dispersive equation relating density and linear moment dispersions is found. Optimal bounds for the kinetic energy are obtained which leads to study the asymptotic behaviour in time for the solutions in the attractive case with positive energy. The description of the asymptotic behaviour properties of the solutions such as a the existence of breathing mode solution, i.e. a changing size oscillatory wave function, in the case of attractive potential with negative kinetic energy are also given. A study of the stability of stationary solutions is proposed using a Liapunov functional and also starting from a perturbation of an associated time-independent solution of the Schrodinger-Poisson equation (linear stability).