Analytic representation of the Efimov effect in the helium trimer

Abstract

Exact solutions for the low-temperature helium dimer and trimer, $^{4}\mathrm{He}_{2}$ and $^{4}\mathrm{He}_{3}$, are derived, based on our $\ensuremath{\delta}$ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the $S$-wave alone giving an exact solution. The parameters ${\ensuremath{\lambda}}_{0}$ and ${r}_{0}$ are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.