X iv : m at h - ph / 0 41 20 95 v 2 2 8 Fe b 20 05 Inequivalent quantizations of the three - particle Calogero model constructed by separation of variables

We quantize the 1-dimensional 3-body problem with harmonic and inverse square pair potential by separating the Schrödinger equation following the classic work of Calogero, but allowing all possible self-adjoint boundary conditions for the angular and radial Hamiltonians. The inverse square coupling constant is taken to be g = 2ν(ν − 1) with 1 2 < ν < 3 2 and then the angular Hamiltonian is shown to admit a 2-parameter family of inequivalent quantizations compatible with the dihedral D 6 symmetry of its potential term 9ν(ν − 1)/ sin 2 3φ. These are parametrized by a matrix U ∈ U (2) satisfying σ 1 U σ 1 = U , and in all cases we describe the qualitative features of the angular eigenvalues and classify the eigenstates under the D 6 symmetry and its S 3 subgroup generated by the particle exchanges. The angular eigenvalue λ enters the radial Hamil-tonian through the potential (λ − 1 4)/r 2 allowing a 1-parameter family of self-adjoint boundary conditions at r = 0 if λ < 1. For 0 < λ < 1 our analysis of the radial Schrödinger equation is consistent with previous results on the possible energy spectra, while for λ < 0 it shows that the energy is not bounded from below rejecting those U 's admitting such eigenvalues as physically impermissible. The permissible self-adjoint angular Hamiltonians include, for example, the cases U = ±1 2 , ±σ 1 , which are explicitly solvable and are presented in detail. The choice U = −1 2 reproduces Calogero's quan-tization, while for the choice U = σ 1 the system is smoothly connected to the harmonic oscillator in the limit ν → 1.