Integrability of the $C_{n}$ and $BC_{n}$ Ruijsenaars-Schneider models

TL;DR

This paper constructs Lax pairs and involutive Hamiltonians for the $C_{n}$ and $BC_{n}$ Ruijsenaars-Schneider models with trigonometric and rational potentials, and derives their Calogero-Moser limits.

Contribution

It provides explicit Lax pairs and Hamiltonians for these models, extending the understanding of their integrability and connection to Calogero-Moser systems.

Findings

Lax pairs for $C_{n}$ and $BC_{n}$ RS models are constructed.

Involutive Hamiltonians are explicitly given.

Nonrelativistic limits yield Lax pairs for Calogero-Moser systems.

Abstract

We study the $C_{n}$ and $BC_{n}$ Ruijsenaars-Schneider(RS) models with interaction potential of trigonometric and rational types. The Lax pairs for these models are constructed and the involutive Hamiltonians are also given. Taking nonrelativistic limit, we also obtain the Lax pairs for the corresponding Calogero-Moser systems.