A geometrical interpretation of the Thomas theorem and the Efimov States

Abstract

Using a generalized Bohr model and the hyper-spherical formalism for a three-body system, we derive the Thomas theorem assuming a simple interaction depending on the range of the potential. We discuss the conditions for which an unbound two-body system produces a bound three-body system and derive universal energy functions. We apply our model to $^{4}$He and Triton atoms as well as to the triton nucleus. Using their scattering lengths and effective ranges, we are able to reproduce the two-body or the three-body binding energies with only one parameter fitted. Prediction for excited (Efimov) levels are also given and in particular we demonstrate that for some hyper-angles two equal minima appear which indicate a phase (shape) transition similar to the Landau's theory of phase transition. We suggest that the observed excited levels in two different experiments for the triton nucleus are indeed Efimov levels and there may be more surprises.