Splitting the degeneracy of harmonically-bound identical spinless bosons: General derivation

The Gartenhaus-Schwartz model of N particles interacting via pairwise harmonic potentials $\text{V}{}_{\mathrm{ij}}\text{=}\text{1}\text{2}\text{kR}{}_{\mathrm{ij}}{}^2$ is extended by addition of an arbitrary pairwise interaction small enough to be treated by degenerate perturbation theory. The theory is worked out for a rarefied fluid of N spinless bosons, to give the splittings of the degeneracies of the U(3N?3) symmetry inherent in the Gartenhaus-Schwartz model. Following Moshinsky and Kramer the external field is formally introduced in order to reduce the zeroth-order problem to the 3N-dimensional harmonic oscillator. For systems with more particles than quanta, a recipe is given to construct states in which all the excitation is in internal degrees of freedom and none is in translation. Analytical solutions are obtained for states with no more than two quanta in excitation of relative motion of any one pair of particles.