  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.
