Quantum vs Classical Integrability in Ruijsenaars-Schneider Systems

TL;DR

This paper explores the similarities between quantum and classical integrability in Ruijsenaars-Schneider systems, showing that properties like eigenvalues and equilibrium features mirror those in Calogero-Moser systems.

Contribution

It demonstrates that key integrability features of Calogero-Moser systems extend to Ruijsenaars-Schneider systems based on classical root systems.

Findings

Classical equilibrium properties are similar to those in Calogero-Moser systems.

Eigenvalues of Lax matrices at equilibrium are integer-valued.

Features like minimum energies and oscillation frequencies are preserved.

Abstract

The relationship (resemblance and/or contrast) between quantum and classical integrability in Ruijsenaars-Schneider systems, which are one parameter deformation of Calogero-Moser systems, is addressed. Many remarkable properties of classical Calogero and Sutherland systems (based on any root system) at equilibrium are reported in a previous paper (Corrigan-Sasaki). For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all "integer valued". In this paper we report that similar features and results hold for the Ruijsenaars-Schneider type of integrable systems based on the classical root systems.