Calogero-Sutherland hyperbolic system and Heckman-Opdam $\mathfrak{gl}_n$ hypergeometric function

TL;DR

This paper establishes the equivalence of two integral representations for wave functions in the hyperbolic Calogero-Sutherland system, linking them to Heckman-Opdam hypergeometric functions through analysis of Baxter operators.

Contribution

It introduces a novel connection between integral representations of Calogero-Sutherland wave functions and Heckman-Opdam hypergeometric functions, using Baxter operators and asymptotic analysis.

Findings

Proved equivalence of two integral representations.

Linked wave functions to Heckman-Opdam hypergeometric functions.

Analyzed asymptotics and special values of integral representations.

Abstract

We prove equivalence of two integral representations for the wave functions of hyperbolic Calogero-Sutherland system. For this we study two families of Baxter operators related to hyperbolic Calogero-Sutherland and rational Ruijsenaars models; the first one as a limit from hyperbolic Ruijsenaars system, while the second one independently. Besides, computing asymptotics of integral representations and also the value at zero point, we identify them with renormalized Heckman-Opdam $\mathfrak{gl}_n$ hypergeometric function.