Solving the Vlasov equation in two spatial dimensions with the
  Schr\"odinger method
  We demonstrate that the Vlasov equation describing collisionless
self-gravitating matter may be solved with the so-called Schr\"odinger method
(ScM). With the ScM, one solves the Schr\"odinger-Poisson system of equations
for a complex wave function in d dimensions, rather than the Vlasov equation
for a 2d-dimensional phase space density. The ScM also allows calculating the
d-dimensional cumulants directly through quasi-local manipulations of the wave
function, avoiding the complexity of 2d-dimensional phase space. We perform for
the first time a quantitive comparison of the ScM and a conventional Vlasov
solver in d=2 dimensions. Our numerical tests were carried out using two types
of cold cosmological initial conditions: the classic collapse of a sine wave
and those of a gaussian random field as commonly used in cosmological cold dark
matter N-body simulations. We compare the first three cumulants, that is, the
density, velocity and velocity dispersion, to those obtained by solving the
Vlasov equation using the publicly available code ColDICE. We find excellent
qualitative and quantitative agreement between these codes, demonstrating the
feasibility and advantages of the ScM as an alternative to N-body simulations.
We discuss, the emergence of effective vorticity in the ScM through the winding
number around the points where the wave function vanishes. As an application we
evaluate the background pressure induced by the non-linearity of large scale
structure formation, thereby estimating the magnitude of cosmological
backreaction. We find that it is negligibly small and has time dependence and
magnitude compatible with expectations from the effective field theory of large
scale structure.
