Three-body harmonic molecule

In this study, the quantum three-body harmonic system with finite rest length R and zero total angular momentum L = 0 is explored. It governs the near-equilibrium S-states eigenfunctions $\psi(r_{12},r_{13},r_{23})$ of three identical point particles interacting by means of any pairwise confining potential $V(r_{12},r_{13},r_{23})$ that entirely depends on the relative distances $r_{ij} = |\mathbf{r}_i-\mathbf{r}_j|$ between particles. At R = 0, the system admits a complete separation of variables in Jacobi-coordinates, it is (maximally) superintegrable and exactly-solvable. The whole spectra of excited states is degenerate, and to analyze it a detailed comparison between two relevant Lie-algebraic representations of the corresponding reduced Hamiltonian is carried out. At R > 0, the problem is not even integrable nor exactly-solvable and the degeneration is partially removed. In this case, no exact solutions of the Schrödinger equation have been found so far whilst its classical counterpart turns out to be a chaotic system. For R > 0, accurate values for the total energy E of the lowest quantum states are obtained using the Lagrange-mesh method. Concrete explicit results with not less than eleven significant digits for the states $N = 0,1,2,3$ are presented in the range $0\leqslant R \leqslant 4.0$ a.u. In particular, it is shown that (I) the energy curve $E = E(R)$ develops a global minimum as a function of the rest length R, and it tends asymptotically to a finite value at large R, and (II) the degenerate states split into sub-levels. For the ground state, perturbative (small-R) and two-parametric variational results (arbitrary R) are displayed as well. An extension of the model with applications in molecular physics is briefly discussed.