On Double-Elliptic Integrable Systems. 1. A Duality Argument for the case of SU(2)

TL;DR

This paper introduces a new two-parameter family of elliptic 2-particle Hamiltonians for SU(2), exhibiting a duality that interchanges momentum and coordinate, extending known integrable models with elliptic dependence.

Contribution

It constructs a novel duality-invariant family of elliptic Hamiltonians with coordinate-dependent momentum modulus, unifying and extending existing integrable systems.

Findings

The Hamiltonians are closed under duality transformations.

The model generalizes Ruijsenaars-Calogero and Toda systems.

It demonstrates elliptic dependence in both coordinates and momenta.

Abstract

We construct a two parameter family of 2-particle Hamiltonians closed under the duality operation of interchanging the (relative) momentum and coordinate. Both coordinate and momentum dependence are elliptic, and the modulus of the momentum torus is a non-trivial function of the coordinate. This model contains as limiting cases the standard Ruijsenaars-Calogero and Toda family of Hamiltonians, which are at most elliptic in the coordinates, but not in the momenta.