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 12<ν<32 and then the angular Hamiltonian is shown to admit a 2-parameter family of inequivalent quantizations compatible with the dihedral D6 symmetry of its potential term 9ν(ν−1)/sin23ϕ. These are parametrized by a matrix U∈U(2) satisfying σ1Uσ1=U, and in all cases we describe the qualitative features of the angular eigenvalues and classify the eigenstates under the D6 symmetry and its S3 subgroup generated by the particle exchanges. The angular eigenvalue λ enters the radial Hamiltonian through the potential (λ−14)/r2 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=±12,±σ1, which are explicitly solvable and are presented in detail. The choice U=−12 reproduces Calogero's quantization, while for the choice U=σ1 the system is smoothly connected to the harmonic oscillator in the limit ν→1.