  We introduce the theory of non-linear cosmological perturbations using the
correspondence limit of the Schr\"odinger equation. The resulting formalism is
equivalent to using the collisionless Boltzman (or Vlasov) equations which
remain valid during the whole evolution, even after shell crossing. Other
formulations of perturbation theory explicitly break down at shell crossing,
e.g. Eulerean perturbation theory, which describes gravitational collapse in
the fluid limit. This paper lays the groundwork by introducing the new
formalism, calculating the perturbation theory kernels which form the basis of
all subsequent calculations. We also establish the connection with conventional
perturbation theories, by showing that third order tree level results, such as
bispectrum, skewness, cumulant correlators, three-point function are exactly
reproduced in the appropriate expansion of our results. We explicitly show that
cumulants up to N=5 predicted by Eulerian perturbation theory for the dark
matter field $\delta$ are exactly recovered in the corresponding limit. A
logarithmic mapping of the field naturally arises in the Schr\"odinger context,
which means that tree level perturbation theory translates into (possibly
incomplete) loop corrections for the conventional perturbation theory. We show
that the first loop correction for the variance is $\sigma^2 = \sigma_L^2+
(-1.14+n)\sigma_L^4$ for a field with spectral index $n$. This yields 1.86 and
0.86 for $n=-3,-2$ respectively, and to be compared with the exact loop order
corrections 1.82, and 0.88. Thus our tree-level theory recovers the dominant
part of first order loop corrections of the conventional theory, while
including (partial) loop corrections to infinite order in terms of $\delta$.
