  In the light of the Quantum Painleve-Calogero Correspondence established in
our previous papers [1,2], we investigate the inverse problem. We imply that
this type of the correspondence (Classical-Quantum Correspondence) holds true
and 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 Painleve list of equations. In
this sense the Classical-Quantum Correspondence can be regarded as an
alternative definition of the Painleve equations.
