Complex rational Ruijsenaars model. The two-particle case

TL;DR

This work analyzes a complex rational degeneration of the two-particle hyperbolic Ruijsenaars model in the limit $\omega_1+\omega_2\to 0$ (equivalently $b\to i$), deriving complex hypergeometric wave functions via Mellin-Barnes integrals and establishing their dual integral representations and reflection symmetry. It develops a parallel complex degeneration of Baxter $Q$-operators and identifies a separate $\omega_1-\omega_2\to 0$ limit leading to a related complex Calogero-Sutherland type system, along with two additional complex CS degenerations. The authors construct explicit eigenfunctions $\Phi^g_{\lambda_1,\lambda_2}(x_1,x_2)$ and their center-of-mass reductions $F^{r,h}_{\alpha,\beta}$, demonstrate bispectral relations and dual representations, and derive complex-limit versions of reflection and $Q$-operator commutativity. The results extend the known bispectral structure of Ruijsenaars systems to complex degenerate regimes, revealing rich connections to complex gamma functions, two-dimensional cylinder geometries, and new integrable quantum-mechanical models with potential links to conformal field theory and representation theory.

Abstract

We consider a complex rational degeneration of the hyperbolic Ruijsenaars model emerging in the limit $ω_1+ω_2\to 0$ (or $b\to \imath$ in $2d$ CFT) and investigate in detail the two-particle case. Corresponding wave functions are described by complex hypergeometric functions in the Mellin-Barnes representation. Their dual integral representation and reflection symmetry in the coupling constant are established. Besides, a complex limit of the hyperbolic Baxter $Q$-operators is considered. Another complex degeneration of the hyperbolic Ruijsenaars model is obtained by taking a special $ω_1-ω_2\to 0$ (or $b\to 1$) limit. Additionally, two new degenerations to the complex Calogero-Sutherland type models are described.