Higher order deformed elliptic Ruijsenaars operators

TL;DR

This paper introduces four new families of mutually commuting difference operators that include deformed elliptic Ruijsenaars operators, providing a rigorous proof of their quantum integrability and fundamental properties.

Contribution

It presents the first explicit construction and proof of commutativity for these deformed elliptic Ruijsenaars operators, extending previous trigonometric cases.

Findings

Operators form four infinite mutually commuting families

Proof of quantum integrability of the deformed Ruijsenaars model

Kernel function identities established

Abstract

We present four infinite families of mutually commuting difference operators which include the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators was previously introduced by Feigin and Silantyev. They provide a quantum mechanical description of two kinds of relativistic quantum mechanical particles which can be identified with particles and anti-particles in an underlying quantum field theory. We give direct proofs of the commutativity of our operators and of some other fundamental properties such as kernel function identities. In particular, we give a rigorous proof of the quantum integrability of the deformed Ruijsenaars model.