Energetics and structural properties of three-dimensional bosonic clusters near threshold

We treat three-dimensional bosonic clusters wih up to N=40 atoms, interacting additively through two-body Van der Waals potentials, in the near-threshold regime. Our study inludes super-borromean systems with N atoms for which all subsystems are unbound. We determine the energetics and structural properties as the expectation value of the interparticle distance as a function of the coupling strength. It has been shown that the coupling strength g*(N), for which the N-body systems becomes unbound, is bounded by the coupling constant g*(N-1) for which the next smaller system with N-1 atoms becomes unbound, i.e., g*(N) >= (N-1)/Ng*(N-1). By fitting our numerically determined ground state energies to a simple functional form with three fitting parameters, we determine the relationship between g*(N) and g*(N-1). Our trimer and tetramer energies fall on the so-called Tjon line, which has been studied in nuclear physics. We confirm the existence of generalized Tjon lines for larger clusters. Signatures of the universal behavior of weakly-bound three-dimensional clusters can possibly be observed in ultracold Bose gases.