Schr\"odinger method as N-body double and UV completion of dust
  We investigate large-scale structure formation of collisionless dark matter
in the phase space description based on the Vlasov 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 pressure- less 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 N-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 Schr\"odinger-Poisson equation (SPE) is the
sought-after function. We show that its evolution equation approximates the
Vlasov equation and therefore also the dust equations before shell-crossing,
but cures the shell-crossing singularities and is able to describe regions of
multi-streaming and virialization. This feature was already employed in
simulations by Widrow & Kaiser (1993). The coarse-grained Wigner ansatz allows
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 Schr\"odinger method (ScM) 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.
