ODE/IM correspondence
In mathematical physics, the ODE/IM correspondence is a link between ordinary differential equations and integrable models. It was first found in 1998 by Patrick Dorey and Roberto Tateo. In this original setting it relates the spectrum of a certain integrable model of magnetism known as the XXZ-model to solutions of the one-dimensional Schrödinger equation with a specific choice of potential, where the position coordinate is considered as a complex coordinate.
Since then, such a correspondence has been found for many more ODE/IM pairs.