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We investigate large-scale structure formation of collisionless dark matter in the phase space description based on the Vlasov (or collisionless Boltzmann) equation whose nonlinearity is induced solely by gravitational interaction according to the Poisson equation. Determining the time evolution of density and peculiar velocity demands solving the full Vlasov hierarchy for the moments of the phase space distribution function. In the presence of long-range interaction no consistent truncation of the hierarchy is known apart from the pressureless fluid (dust) model, which is incapable of describing virialization due to the occurrence of shell-crossing singularities and the inability to generate vorticity and higher cumulants like velocity dispersion. Our goal is to find a simple ansatz for the phase space distribution function that approximates the full Vlasov distribution function without pathologies in a controlled way and therefore can serve as theoretical TV-body double and as a replacement for the dust model. We argue that the coarse-grained Wigner probability distribution obtained from a wave function fulfilling the Schrödinger-Poisson equation (SPE) is the sought-after function. We show that its evolution equation approximates the Vlasov equation and therefore also the dust fluid equations before shell crossing, but cures the shell-crossing singularities and is able to describe regions of multistreaming and virialization. This feature was already employed in cosmological simulations of large-scale structure formation by Widrow and Kaiser (1993). The coarse-grained Wigner ansatz allows us to calculate all higher moments from density and velocity analytically, thereby incorporating nonzero higher cumulants in a self-consistent manner. On this basis we are able to show that the Schrödinger method automatically closes the corresponding hierarchy such that it suffices to solve the SPE in order to directly determine density and velocity and all higher cumulants. [ABSTRACT FROM AUTHOR]