  For a semi-linear Schr\"{o}dinger equation of Hartree type in three spatial
dimensions, various approximations of singular, point-like perturbations are
considered, in the form of potentials of very small range and very large
magnitude, obeying different scaling limits. The corresponding nets of
approximate solutions represent actual generalised solutions for the
singular-perturbed Schr\"{o}dinger equation. The behaviour of such nets is
investigated, comparing the distinct scaling regimes that yield, respectively,
the Hartree equation with point interaction Hamiltonian vs the ordinary Hartree
equation with the free Laplacian. In the second case, the distinguished regime
admitting a generalised solution in the Colombeau algebra is studied, and for
such a solution compatibility with the classical Hartree equation is
established, in the sense of the Colombeau generalised solution theory.
