Four-body problem in d-dimensional space: Ground state, (quasi)-exact-solvability. IV

Abstract

Due to its great importance for applications, we generalize and extend the approach of our previous papers to study aspects of the quantum and classical dynamics of a 4-body system with equal masses in d-dimensional space with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. The ground state (and some other states) in the quantum case and some trajectories in the classical case are of this type. We construct the quantum Hamiltonian for which these states are eigenstates. For d ≥ 3, this describes a 6-dimensional quantum particle moving in a curved space with special d-independent metric in a certain d-dependent singular potential, while for d = 1 it corresponds to a 3-dimensional particle and coincides with the A3 (4-body) rational Calogero model; the case d = 2 is exceptional and is discussed separately. The kinetic energy of the system has a hidden sl(7, R) Lie (Poisson) algebra structure, but for the special case d = 1, it becomes degenerate with hidden algebra sl(4, R). We find an exactly solvable 4-body S4-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasiexactly solvable 4-body sextic polynomial type potential with singular terms. The tetrahedron whose vertices correspond to the positions of the particles provides pure geometrical variables, volume variables, which lead to exactly solvable models. Their generalization to the n-body system as well as the case of nonequal masses is briefly discussed.