Stable pairs on local curves and Bethe roots

We give an explicit formula for the descendent stable pair invariants of all (absolute) local curves in terms of certain power series called Bethe roots, which also appear in the physics/representation theory literature. We derive new explicit descriptions for the Bethe roots which are of independent interest. From this we derive rationality, functional equation and a characterization of poles for the full descendent stable pair theory of local curves as conjectured by Pandharipande and Pixton. We also sketch how our methods give a new approach to the spectrum of quantum multiplication on $\mathsf{Hilb}^n(\mathbf{C}^2)$.