Generalised Calogero-Moser models and universal Lax pair operators

TL;DR

This paper develops universal Lax pair operators for generalized Calogero-Moser models based on all finite reflection groups, including non-crystallographic types, by reducing consistency conditions to specific functional equations.

Contribution

It constructs universal Lax pair operators for all generalized Calogero-Moser models across all finite reflection groups, including non-crystallographic systems, using reflection operators and functional equations.

Findings

Lax pairs are expressed as linear combinations of reflection operators.

Functional equations are classified into four types based on root subsystems.

Spectral parameter dependence resembles Dunkl operators.

Abstract

Calogero-Moser models can be generalised for all of the finite reflection groups. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H_3, H_4, and the dihedral group I_2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero-Moser models and for any choices of the potentials are constructed as linear combinations of the reflection operators. The consistency conditions are reduced to functional equations for the coefficient functions of the reflection operators in the Lax pair. There are only four types of such functional equations corresponding to the two-dimensional sub-root systems, A_2, B_2, G_2, and I_2(m). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators, a powerful tool for solving quantum Calogero-Moser models.