Three-body problem in $d$-dimensional space: ground state, (quasi)-exact-solvability

Alexander V Turbiner, Willard Miller, Jr., M. A. Escobar-Ruiz

Published 2017-07-05Version 1

As a straightforward generalization and extension of our previous paper, J. Phys. A50 (2017) 215201 [TMA:2016] we study aspects of the quantum and classical dynamics of a $3$-body system with equal masses, each body with $d$ degrees of freedom, 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. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories (in interaction plane) in the classical case are of this type. The quantum (and classical) system for which these states are eigenstates is obtained and its Hamiltonian is derived. It corresponds to a three-dimensional quantum particle moving in a curved space with special $d$-dimension-independent metric in a certain $d$-dependent singular potential, while at $d=1$ it elegantly degenerates to two-dimensional particle moving in flat space. The kinetic energy of the system is $d$-independent, it has a hidden $sl(4,R)$ Lie (Poisson) algebra structure, alternatively, the hidden algebra $h^{(3)}$ typical for the $H_3$ Calogero model like in $d=3$ case. We find an exactly-solvable three-body $S^3$-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential with singular terms. For both models an extra first order integral exists. For $d=1$ the whole family of 3-body (two-dimensional) Calogero-Moser-Sutherland systems as well as TTW model are reproduced. It is shown that a straightforward generalization of 3-body (rational) Calogero model to $d>1$ leads to two primitive quasi-exactly-solvable problems. The extension to the case of non-equal masses is straightforward and is briefly discussed.