Quantum Lax pairs via Dunkl and Cherednik operators

TL;DR

This paper constructs quantum Lax pairs for Calogero--Moser and Ruijsenaars systems using Dunkl and Cherednik operators, unifying classical and quantum integrable models across various root systems and solving a long-standing problem in elliptic BC_n systems.

Contribution

It introduces a novel method linking Dunkl and Cherednik operators to quantum Lax pairs for broad classes of integrable systems, including elliptic cases.

Findings

Constructed quantum Lax pairs for Calogero--Moser systems with arbitrary Weyl groups.

Produced quantum Lax pairs for Ruijsenaars systems using Cherednik operators.

Calculated a Lax matrix for the elliptic BC_n (van Diejen) system with nine coupling constants.

Abstract

We establish a direct link between Dunkl operators and quantum Lax matrices $\mathcal L$ for the Calogero--Moser systems associated to an arbitrary Weyl group $W$ (or an arbitrary finite reflection group in the rational case). This interpretation also provides a companion matrix $\mathcal A$ so that $\mathcal L, \mathcal A$ form a quantum Lax pair. Moreover, such an $\mathcal A$ can be associated to any of the higher commuting quantum Hamiltonians of the system, so we obtain a family of quantum Lax pairs. These Lax pairs can be of various sizes, matching the sizes of orbits in the reflection representation of $W$, and in the elliptic case they contain a spectral parameter. This way we reproduce universal classical Lax pairs by D'Hoker-Phong and Bordner-Corrigan-Sasaki, and complement them with quantum Lax pairs in all cases (including the elliptic case, where they were not previously known). The same method, with the Dunkl operators replaced by the Cherednik operators, produces quantum Lax pairs for the generalised Ruijsenaars systems for arbitrary root systems. As one of the main applications, we calculate a Lax matrix for the elliptic $BC_n$ case with nine coupling constants (van Diejen system), thus providing an answer to a long-standing open problem.