From quantum to classical theories: the origin of the ODE/IM correspondence
November 21, 2022
June 15, 2023
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.
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