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 $g=2\nu (\nu -1)$ with $1/2<\nu <3/2$ 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 $9\nu (\nu -1)/{\sin }^{2}3\varphi$. These are parametrized by a matrix $U\in U(2)$ satisfying ${\sigma }_{1}U{\sigma }_1=U$, 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 $\lambda$ enters the radial Hamiltonian through the potential $(\lambda -1/4)/r^2$ allowing a 1-parameter family of self-adjoint boundary conditions at $r=0$ if $\lambda <1$. For $0<\lambda <1$ our analysis of the radial Schrödinger equation is consistent with previous results on the possible energy spectra, while for $\lambda <0$ 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 $U=\pm 1_2,\pm {\sigma }_1$, which are explicitly solvable and are presented in detail. The choice $U=-1_2$ reproduces Calogero's quantization, while for the choice $U={\sigma }_1$ the system is smoothly connected to the harmonic oscillator in the limit $\nu \to 1$.

45 pages, 6 figures, LaTeX, v2: a reference and a note added, v3: merely a constant is corrected