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For a semi-linear Schr\"{o}dinger equation of Hartree type in three spatialdimensions, various approximations of singular, point-like perturbations areconsidered, in the form of potentials of very small range and very largemagnitude, obeying different scaling limits. The corresponding nets ofapproximate solutions represent actual generalised solutions for thesingular-perturbed Schr\"{o}dinger equation. The behaviour of such nets isinvestigated, comparing the distinct scaling regimes that yield, respectively,the Hartree equation with point interaction Hamiltonian vs the ordinary Hartreeequation with the free Laplacian. In the second case, the distinguished regimeadmitting a generalised solution in the Colombeau algebra is studied, and forsuch a solution compatibility with the classical Hartree equation isestablished, in the sense of the Colombeau generalised solution theory.

Generalised solutions to linear and non-linear Schr\"{o}dinger-type equations with point defect: Colombeau and non-Colombeau regimes