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

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

Three particles in an external trap: Nature of the complete<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>J</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mn /></mml:math>spectrum

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