We consider the problem of defining quantum Hamiltonians for a system of N-particles with zero range interactions, also known as point interactions. Such Hamiltonians ara characterized by the fact that the only physical parameter defining the interaction is the two body scattering length. In dimension three the mathematical construction of these Hamiltonians for N larger or equal than three is not trivial. A natural construction, based on the analogy with the two-particle case, leads to the so called Ter-Martirosyan Skornyakov (TMS) Hamiltonian and to the fall to the center phenomenon known as Thomas effect. Following a suggestion given by Minlos and Faddeev in a seminal paper published in 1962, reprised by Albeverio, Høegh-Krohn and Wu in 1981, we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non local interactions, also called separable potentials, with a suitably renormalized coupling constant.
The talk is based on a joint work with Giulia Basti, Domenico Finco, and Alessandro Teta.