We discuss a class of regularized zero-range Hamiltonians for different problems satisfying a bosonic symmetry in dimension three, corresponding to the investigation of suitable self-adjoint extensions of the free Laplacian restricted on the space of regular functions vanishing along the coincidence hyperplane (i.e. the hyperplanes where the contact interaction occurs).
Following the standard procedure in defining such operators, one comes up with the so-called Ter-Martirosyan Skornyakov Hamiltonian which turns out to be unbounded from below in three dimensions (Thomas collapse takes place in case of usual two-body point interactions since zero-range forces become too singular when three or more particles get close to each other).
In order to avoid this energetical instability, we consider a class of self-adjoint extensions taking account of a many-body repulsion meant to weaken the strength of the interaction when more than two particles coincide. More precisely, following a suggestion coming from 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 lower-bounded, provided that the strength of the many-body force is large enough.