  We investigate the entire family of multi-center point interaction
Hamiltonians. We show that a large sub-family of these operators do not become
either singular or trivial when the positions of two or more scattering centers
tend to coincide. In this sense, they appear to be renormalised by default as
opposed to the "local" point interaction Hamiltonians usually considered in the
literature as the ones of physical interest. In the two-center case we study
the behaviour of the negative eigenvalues as a function of the center distance.
The result is used to analyze a formal Born-Oppenheimer approximation of a
three-particle system with two heavy and one light particle. We show that this
simplified model does not show any ultra-violet catastrophe and we prove that
the ratio of successive low energy eigenvalues follows the Efimov geometrical
law.
