Contact interactions for many-particle quantum systems in dimension three

We discuss a class of regularized zero-range Hamiltonians for three different problems satisfying a bosonic symmetry in dimension three. Following the standard approach in defining such Hamiltonians in three dimensions, one comes up with the so-called Ter-Martirosyan Skornyakov Hamiltonian that turns out to be unbounded from below (Thomas collapse occurs in case of usual two-body point interactions since zero-range interactions become too singular when three or more particles get close). In order to avoid this energetical instability, we consider a many-body repulsion meant to weaken the strength of the interaction when more than two particles coincide. More precisely, developing a suggestion made in the early '60s by Minlos and Faddeev, we introduce an effective scattering length depending on the positions of the particles. In case of a three-boson problem (or a Bose gas of non-interacting particles interacting only with an impurity) such a function vanishes as a third particle gets closer to the couple of interacting particles. Similarly, dealing with an interacting Bose gas, we also take into account a four-body repulsion in order to handle the ultraviolet singularity associated with the collapse of two distinct couples of interacting particles. We show that the Hamiltonians corresponding to these regularizations are self-adjoint and bounded from below, provided that the strength of the many-body force is large enough. Moreover, we compare our results with the ones obtained in the early '80s by Albeverio et al, which exploits an alternative method based on Dirichlet forms, providing the construction of a one-parameter family of many-body regularized zero-range Hamiltonians. In particular, we prove that such a class of regularized Hamiltonians is a special case of what can be obtained with our approach.