Revisiting Quantum Mechanical Zero-Range Potentials

Inthiscontributionwemakeabriefoverviewofhistoryandrecentresults in the theory of many quantum particles interacting via zero-range forces. We recall the regularisation mechanism suggested by several authors in the past in order to avoid the “fall to the center” problem in three-body systems. Following those suggestions a family of three-body point interaction Hamiltonians bounded from below were made available recently. We conclude showing that a similar kind of ultraviolet problem is already present in the theory of point interaction Hamiltonians in one-body Quantum Mechanics. A careful look to the entire family of many center point interaction Hamiltonians shows that the great majority of them do not become either singular or trivial when the positions of two or more scattering centers tend to coincide. In this sense, those Hamiltonians appear to be renormalised by default as opposed to the “local” point interaction Hamiltonians usually considered in the literature since the early days of Quantum Mechanics. The renormalization mechanism turns out to be very similar to the one used in the three-body problem.