Complex rational Ruijsenaars model. The two-particle case

We consider a complex rational degeneration of the hyperbolic Ruijsenaars model emerging in the limit ${\omega }_1+{\omega }_2\to 0$ (or $b\to \mathrm{ı}$ 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 ${\omega }_1-{\omega }_2\to 0$ (or $b\to 1$) limit. Additionally, two new degenerations to the complex Calogero-Sutherland type models are described.