and recover the final phase space distribution from the wavefunction.

Note also that another possible route, where the Hartree equation is used instead of the Schrödinger equation, has been explored by [@Aschbacher2001] and [@Frohlich2010].

The time evolution of the Schrödinger-Poisson system is done using an explicit finite-differences scheme for the wave function and a FFT algorithm to solve the Poisson equation. We try to improve upon their algorithm for the time evolution as will be described below. [@Widrow1993] have made several simulations using this Schrödinger method obtaining results in agreement with usual [$N$-body ]{}simulations. They also claim that their method is computationally comparable to [$N$-body ]{}simulations making it a promising tool for cosmological purposes.

These authors choose to use one single wave function to represent the distribution function. This has important consequences on the validity of equation (\[eq:1\]). By using one single wave function, the phase space distribution built from it can not be everywhere positive and the authors have thus to add an additional Gaussian smoothing. We alleviate this shortcoming by using more than one wave function and a different transformation from wave functions to phase space distribution.

Their choice of Gaussian smoothed density also led them to a 