Inequivalent quantizations of the three-particle Calogero model constructed by separation of variables

Abstract

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 $9 ν(ν-1)/\sin^2 3ϕ$. These are parametrized by a matrix $U\in U(2)$ 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 Hamiltonian 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 $U=\pm {\bf 1}_2, \pm σ_1$, which are explicitly solvable and are presented in detail. The choice $U=-{\bf 1}_2$ reproduces Calogero's quantization, while for the choice $U=σ_1$ the system is smoothly connected to the harmonic oscillator in the limit $ν\to 1$.