Exactly solvable dynamical systems in the neighborhood of the Calogero model

TL;DR

This paper explores the algebraic structure of Calogero models, identifying exactly solvable Sutherland models for small particle numbers and extending these results to general N.

Contribution

It explicitly constructs quadratic Lie algebra forms for N=3 and N=4 that correspond to solvable Sutherland models, generalizing to all N.

Findings

Quadratic Lie algebra forms are explicitly constructed for N=3 and N=4.

These forms correspond to exactly solvable Sutherland models.

Results extend to all particle numbers N.

Abstract

The Hamiltonian of the $N$-particle Calogero model can be expressed in terms of generators of a Lie algebra for a definite class of representations. Maintaining this Lie algebra, its representations, and the flatness of the Riemannian metric belonging to the second order differential operator, the set of all possible quadratic Lie algebra forms is investigated. For N=3 and N=4 such forms are constructed explicitly and shown to correspond to exactly solvable Sutherland models. The results can be carried over easily to all $N$.