In this work we investigate small clusters of bosons using the hyperspherical harmonic basis. We consider systems with $A=2,3,4,5,6$ particles interacting through a soft interparticle potential. In order to make contact with a real system, we use an attractive Gaussian potential that reproduces the values of the dimer binding energy and the atom-atom scattering length obtained with one of the most widely used ${}^4$He-${}^4$He interactions, the LM2M2 potential of Aziz and Slaman. The intensity of the potential is varied in order to explore the clusters’ spectra in different regions with large positive and large negative values of the two-body scattering length. In addition, we include a repulsive three-body force to reproduce the trimer binding energy. With this model, consisting in the sum of a two- and three-body potential, we have calculated the spectrum of the four-, five-, and six-particle systems. In all the regions explored, we have found that these systems present two states, one deep and one shallow close to the $A-1$ threshold. Some universal relations between the energy levels are extracted; in particular, we have estimated the universal ratios between thresholds of the three-, four-, and five-particle continua using the two-body Gaussian potential. They agree with recent measurements and theoretical predictions.

• Received 5 June 2012

DOI:https://doi.org/10.1103/PhysRevA.86.042513