Convergence of the hyperspherical-harmonics expansion with increasing number of particles for bosonic systems

A formal proof is given that for N -boson systems in which two-body interaction potentials are described by a single Gaussian the ratio of higher-order to the lowest-order hyper-radial potentials decreases at N → ∞ as N−1/2 or faster. As a result, for such potentials, the convergence of binding energies for ground and several lowest excited states, obtained in expansion of the N -body wave function over the hyperspherical-harmonics basis, improves with increasing number of bosons. For a phenomenological three-body repulsive potential, introduced to account for themissing hard core, the ratio of higher-order to lowest-order hyper-radial potential, corresponding to this three-body potential, also decreases as N−1/2 or faster when N → ∞. Although adding the three-body contributions leads to increased influence from the total nondiagonal couplings around the node of the lowest hyper-radial potential, the arguments are given that this should not dramatically deteriorate the convergence if the range of repulsion is properly chosen. This means that the hyperspherical-harmonics expansion with soft two-body and repulsive three-body effective forces may become an attractive tool for studying the spectra of many-body systems. It is suggested that fine tuning of the three-body repulsion to reproduce the binding energies over a large region of N is possible. In particular, it has been shown that an N -independent choice for the three-body repulsion exists for which the ground-state binding energies of N 112 atoms of helium, obtained in the lowest-order approximation of the hyperspherical-harmonics expansion, are close to the prediction of the Green’s function Monte Carlo method with a hard-core He-He potential.