On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta

TL;DR

This paper develops a large N limit construction for integrable many-body systems with elliptic momentum dependence, extending Cherednik and Nazarov-Sklyanin methods to the double elliptic case.

Contribution

It introduces a double-elliptization of Cherednik's construction and derives explicit expressions for operators used in the large N limit of elliptic integrable systems.

Findings

Explicit expression for double-elliptic Cherednik operators.

Construction of the large N limit using non-commuting operators.

Reduction to generating functions of Dell commuting Hamiltonians.

Abstract

The infinite number of particles limit in the dual to elliptic Ruijsenaars model (coordinate trigonometric degeneration of quantum double elliptic model) is proposed using the Nazarov-Sklyanin approach. For this purpose we describe double-elliptization of the Cherednik construction. Namely, we derive explicit expression in terms of the Cherednik operators, which reduces to the generating function of Dell commuting Hamiltonians on the space of symmetric functions. Although the double elliptic Cherednik operators do not commute, they can be used for construction of the $N\rightarrow \infty$ limit.