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 J. Phys. A: Math. Theor. 53 055302 [1], in this work we study a quantum 4-body system in Rd (d ≥ 3) with quadratic and sextic pairwise potentials in the relative distances, rij ≡ |ri − rj |, between particles. Our study is restricted to solutions in the space of relative motion with zero total angular momentum (S-states). In variables ρij ≡ r2 ij , 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 4-body harmonic oscillator with arbitrary masses and unequal spring constants is exactly-solvable (ES). We pay special attention to the case of four equal masses and to atomic-like (where one mass is infinite, three others are equal), molecular two-center (two masses are infinite, two others are equal) and molecular threecenter (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. Also, the reduction to the lower dimensional cases d = 1, 2 is discussed. It is shown that for four body harmonic oscillator case there exists an infinite family of eigenfunctions which depend on the single variable which is the moment-of-inertia of the system.