Complex harmonic-oscillator basis for the relativistic three-body problem

Abstract

A complex harmonic-oscillator basis is employed for the three-body problem obeying S₃-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 S₃-classification of the various states than is usually possible. It is shown that certain bilinear forms with definite S₃-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 S₃-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.