Sunrise Valley Quantum Research Seminars | "Differential and Difference Equations with Apparent Singularities and the Bethe Ansatz"

For a linear ODE in the complex domain, its singular point x=x0 is called an apparent singularity if all solutions to the ODE are meromorphic near this point. Equivalently, x=x0 is a regular singular point and the monodromy around x0 is trivial. What can one say about ODEs whose finite singular points are all apparent? I will give an answer for the case when x=∞ is an irregular singularity of the mildest kind (of Poincaré rank one). I will then discuss an analogous problem for difference equations. Finally, I'll indicate how these results can be applied to give a geometric interpretation of the algebraic Bethe ansatz for the quantum Gaudin and XXZ models. This is based on ongoing projects with N. Belousov, A. Liashyk, and Y. Xie.