  Known quantum and classical perturbative long-distance corrections to the
Newton potential are extended into the short-distance regime using evolution
equations for a `running' gravitational coupling, which is used to construct
examples non-perturbative potentials for the gravitational binding of two
particles. Model-I is based on the complete set of the relevant Feynman
diagrams. Its potential has a singularity at a distance below which it becomes
complex and the system gets black hole-like features. Model-II is based on a
reduced set of diagrams and its coupling approaches a non-Gaussian fixed point
as the distance is reduced. Energies and eigenfunctions are obtained and used
in a study of time-dependent collapse (model-I) and bouncing (both models) of a
spherical wave packet. The motivation for such non-perturbative `toy' models
stems from a desire to elucidate the mass dependence of binding energies found
25 years ago in an explorative numerical simulation within the dynamical
triangulation approach to quantum gravity. Models I \& II suggest indeed an
explanation of this mass dependence, in which the Schwarzschild scale plays a
role. An estimate of the renormalized Newton coupling is made by matching with
the small-mass region. Comparison of the dynamical triangulation results for
mass renormalization with `renormalized perturbation theory' in the continuum
leads to an independent estimate of this coupling, which is used in an improved
analysis of the binding energy data.
