From quantum to classical theories: the origin of the ODE/IM correspondence

We exemplify our procedure on how to derive an Ordinary Differential Equation from a quantum system for the case of the quantum sine-Gordon model. In specific, we move from the functional relations and properties satisfied by 𝑄-functions and derive a Lax pair, thus reversing the usual construction in the Ordinary Differential Equation/Integrable Model correspondence. The link between quantum and classical theory is provided by a Marchenko-like equation, which descends from the quantum 𝑇𝑄-system and is the basis for deriving the Lax pair. We discuss a special case of our construction, in which the classical problem is related to a Painlevé equation, and the massless (conformal) limit. Ostensibly, the whole construction can be modified by introducing moduli and further generalisations.