Three-body problem in 3D space: ground state, (quasi)-exact-solvability

TL;DR

This paper investigates the quantum and classical dynamics of a three-body system in 3D space, revealing exactly solvable models with special algebraic structures and potentials, focusing on states depending only on mutual distances.

Contribution

It introduces a new three-body quantum system with a hidden algebraic structure and constructs exactly solvable and quasi-exactly solvable potentials with additional integrals.

Findings

Ground state and planar trajectories are functions of mutual distances.

Hamiltonian corresponds to a particle in a curved space with special metric.

Identifies exactly solvable and quasi-exactly-solvable potentials with extra integrals.

Abstract

We study aspects of the quantum and classical dynamics of a $3$-body system in 3D space with interaction depending only on mutual distances. The study is restricted to solutions in the space of relative motion which are functions of mutual distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories in the classical case are of this type. The quantum (and classical) system for which these states are eigenstates is found and its Hamiltonian is constructed. It corresponds to a three-dimensional quantum particle moving in a curved space with special metric. The kinetic energy of the system has a hidden $sl(4,R)$ Lie (Poisson) algebra structure, alternatively, the hidden algebra $h^{(3)}$ typical for the $H_3$ Calogero model. We find an exactly solvable three-body generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential; both models have an extra integral.