Three particles in an external trap: Nature of the complete J=0 spectrum

Three bosonic, spin-polarized atoms in a spherical oscillator potential constitutes the simplest nontrivial Bose-Einstein condensate (BEC). The present paper develops the tools needed to understand the nature of the complete J=0 energy spectrum for this prototype system, assuming a sum of two-body potentials. The resulting spectrum is calculated as a function of the two-body scattering length a_sc, which documents the evolution of certain many-body levels that evolve from BEC-type to molecular-type as the scattering length is decreased. Implications for the behavior of the condensate excited-state spectrum and for condensate formation and decay are elucidated. The energy levels evolve smoothly, even through the regime where the number of two-body bound states N_b increases by 1, and a_{sc} switches from -infinity to infinity. We point out the possibility of suppressing three-body recombination by tuning the two-body scattering length to values that are larger than the size of the condensate ground state. Comparisons with mean-field treatments are presented.