A new look at the theory of point interactions

Abstract

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.