A Space-Time Wigner Function Approach to Long Time Schrödinger-Poisson Dynamics.

A long time description of electrostatic Schrodinger-Poisson states, satisfying i partial derivative t psi = -1/2 Delta(x)psi+(C/|x| *(x) |psi|(2)+ 1/2 |x|(2))., is provided in terms of a non-Markovian Wigner formalism through the choice of the simplest charge-preserving scale group, psi(e)(t, x) =psi(epsilon(-1)t, x), in which the position variable x is an element of R-3 remains unscaled while time t is an element of R+ is sent to infinity as epsilon -> 0. Typically, the introduction of this group of scale transformations leads to high frequency, time oscillatory states that may not converge in such a good topology as to deal with the nonlinear term. To overcome this drawback, the sequence of wavefunctions is Wignerized via the action of the so-called t-convoluted space-time Wigner transform. The main goal of this extended Wigner operator consists in producing an attenuating effect on the temporal oscillations as time grows, which in turn allows us to overcome the eventual lack of time compactness. In a certain sense, it transforms high frequency asymptotics into a low oscillating limit that allows us to find in a rigorous way the stationary Wigner-Poisson equation at the long time, as well as to recover some of the macroscopic features of the solutions to the original Schrodinger-Poisson problem.