Complex Harmonic-Oscillator Basis for the Relativistic Three-Body Problem

A complex harmonic-oscillator basis is employed for the three-body problem obeying S ₃-symmetry. Unlike a real basis it generates an additional quantum number (N _a ), in addition to the standard principal quantum number (N), and thus facilitates a more quantitative S ₃-classification of the various states than is usually possible. It is shown that certain bilinear forms with definite S ₃-symmetry properties, which can be constructed out of the linear harmonic-oscillator operators (a, a ^†) satisfy several uncoupled sets of SO(2, 1) algebras with spectra bounded from below. It is also briefly indicated how this S ₃-formalism can be adapted to the core structure of a more general relativistic three-particle system with unequal-mass kinematics through an appropriate choice of internal variables. Received May 11, 1994; revised November 3, 1994; accepted for publication November 23, 1994