Universal behavior in few-boson systems close to the unitary limit, where two bosons become unbound, has been intensively investigated in recent years both experimentally and theoretically. In this particular region, called the unitary window, details of the interparticle interactions are not important and observables, such as binding energies, can be characterized by a few parameters. With an increasing number of particles, the short-range repulsion, present in all atomic, molecular, and nuclear interactions, gradually induces deviations from the universal behavior. In the present paper, we discuss a simple way of incorporating nonuniversal behavior through one specific parameter which controls the smooth transition of the system from a universal to nonuniversal regime. Using a system of
N helium atoms as an example, we calculate their ground-state energies as trajectories within the unitary window and also show that the control parameters can be used to determine the energy per particle in homogeneous systems when $N\ensuremath{\rightarrow}\ensuremath{\infty}$.
Bosonic two-dimensional self-bound clusters consisting of
N atoms interacting through additive van der Waals potentials become unbound at a critical mass
m∗(N);
m∗(N) has been predicted to be independent of the size of the system. Furthermore, it has been predicted that the ground state energy
EN of the
N-atom system varies exponentially as the atomic mass approaches
m∗. This paper reports accurate numerical many-body calculations that allow these predictions to be tested. We confirm the existence of a universal critical mass
m∗, and show that the near-threshold behavior can only be described properly if a previously neglected term is included. We comment on the universality of the energy ratio
EN+1∕EN near threshold.