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)) V(r(12),r(13),r(23)) r(ij)=|r(i)-r(j)| 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 Schrodinger 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 0 <= R <= 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.