Four-body (an)harmonic oscillator in d-dimensional space: S-states, (quasi)-exact-solvability, hidden algebra sl (7)

Abstract

As a generalization and extension of our previous paper [Turbiner et al., J. Phys. A: Math. Theor. 53, 055302 (2020)], in this work, we study a quantum four-body system in ${\mathbb{R}}^d$ (d ≥ 3) with quadratic and sextic pairwise potentials in the relative distances, r_ij ≡ |r_i − r_j|, between particles. Our study is restricted to solutions in the space of relative motion with zero total angular momentum (S-states). In variables ${\rho }_{ij}\equiv r_{ij}^2$ , the corresponding reduced Hamiltonian of the system possesses a hidden sl(7; R) Lie algebra structure. In the ρ-representation, it is shown that the four-body harmonic oscillator with arbitrary masses and unequal spring constants is exactly solvable. We pay special attention to the case of four equal masses and to atomic-like (where one mass is infinite and three others are equal), molecular two-center (two masses are infinite and two others are equal), and molecular three-center (three infinite masses) cases. In particular, exact results in the molecular case are compared with those obtained within the Born–Oppenheimer approximation. The first and second order symmetries of non-interacting system are searched. In addition, the reduction to the lower dimensional cases d = 1, 2 is discussed. It is shown that for the four-body harmonic oscillator case, there exists an infinite family of eigenfunctions that depend on the single variable, which is the moment of inertia of the system.