Three-body Generalizations of the Sutherland Problem

TL;DR

This paper introduces exactly solvable three-body generalizations of the Sutherland problem, revealing hidden symmetries and expressing eigenfunctions via Jack polynomials, advancing the understanding of multi-particle quantum systems.

Contribution

It presents new exactly solvable three-body Hamiltonians with explicit eigenfunctions and uncovers hidden $sl(3,R)$ symmetry, extending the Sutherland model to include three-body interactions.

Findings

Eigenfunctions expressed in terms of Jack polynomials

Hidden $sl(3,R)$ symmetry explains solvability

Inclusion of both two- and three-body potentials

Abstract

The three-particle Hamiltonian obtained by replacing the two-body trigonometric potential of the Sutherland problem by a three-body one of a similar form is shown to be exactly solvable. When written in appropriate variables, its eigenfunctions can be expressed in terms of Jack symmetric polynomials. The exact solvability of the problem is explained by a hidden $sl(3,R)$ symmetry. A generalized Sutherland three-particle problem including both two- and three-body trigonometric potentials and internal degrees of freedom is then considered. It is analyzed in terms of three first-order noncommuting differential-difference operators, which are constructed by combining SUSYQM supercharges with the elements of the dihedral group~$D_6$. Three alternative commuting operators are also introduced.