Exact Energy of the Two and Three-Body Interactions in the Trigonometric Three-Body Problem via Jack Polynomials

The Hamiltonian of the three-body problem with two and three-body interactions of Calogero–Sutherland type written in an appropriate set of variables is shown to be a differential operator of Laplace–Beltrami HLB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{LB}$$\end{document} type. Its eigenfunction is expressed then in terms of Jack Polynomials. In the case of identical distinguishable fermions, the only possible partition and its corresponding occupation state in Fock space is exhibited. In the co-moving frame with the center of mass, the exact explicit excitation energy due to interactions is given in terms of the parameters of the problem. The excitation energy due to interactions is expressed in such a way that it is possible to distinguish the excitation energy due to the pairwise interactions neglecting the three-body interaction as well as the excitation energy due to the pure three-body interactions neglecting the pairwise interactions. Different possible levels of excitation energy due to different type of interactions are explicitly written.