### Description

Hamiltonians with contact (or zero-range) interactions are useful models to analyze the behaviour of quantum systems at low energy in different contexts. In this talk we discuss the mathematical aspects of the construction of such Hamiltonians in dimension three as self-adjoint and lower bounded operators in the appropriate Hilbert space. We first consider the case of a system made of three identical bosons. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The construction is then extended to the case of an arbitrary number of interacting bosons. The talk is based on a series of works in collaboration with G. Basti, C. Cacciapuoti, D. Ferretti, R. Figari, D. Finco and H. Saberbaghi.