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

TL;DR

The paper establishes that two integral representations for the wave functions of the hyperbolic Calogero--Sutherland system are equivalent and identifies them with the renormalized Heckman--Opdam $\mathfrak{gl}_n$ hypergeometric function. It achieves this by developing Baxter-operator formalisms for both the hyperbolic Ruijsenaars system (via a nonrelativistic limit) and its rational degeneration, proving commutativity and duality through kernel identities and Mellin–Barnes representations, and deriving robust asymptotics and bounds to justify limiting procedures. A central outcome is the explicit linkage between the wave functions and the HO function, including dual integral representations and the Harish-Chandra series, normalized at the origin. These results unify nonrelativistic Calogero–Sutherland theory with the Heckman–Opdam framework, providing concrete kernels, dualities, and asymptotics that enhance both spectral theory and special functions.

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.

Theorems & Definitions (35)

• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Theorem 3.1
• Proposition 3.4
• Remark 4.1
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 4.4