Classical-Quantum Correspondence and Functional Relations for Painlevé Equations

In light of the quantum Painlevé–Calogero correspondence, we investigate the inverse problem. We imply that this type of the correspondence (classical-quantum correspondence) holds true, and we find out what kind of potentials arise from the compatibility conditions of the related linear problems. The latter conditions are written as functional equations for the potentials depending on a choice of a single function—the left-upper element of the Lax connection. The conditions of the correspondence impose restrictions on this function. In particular, it satisfies the heat equation. It is shown that all natural choices of this function (rational, hyperbolic, and elliptic) reproduce exactly the Painlevé list of equations. In this sense, the classical-quantum correspondence can be regarded as an alternative definition of the Painlevé equations.