Quantum Painleve-Calogero Correspondence

The Painleve-Calogero correspondence is extended to auxiliary linear problems associated with Painleve equations. The linear problems are represented in a new form which has a suggestive interpretation as a "quantized" version of the Painleve-Calogero correspondence. Namely, the linear problem responsible for the time evolution is brought into the form of non-stationary Schrodinger equation in imaginary time, $\p_t \psi =(1/2\, \p_x^2 +V(x,t))\psi$, whose Hamiltonian is a natural quantization of the classical Calogero-like Hamiltonian $H=1/2\, p^2 +V(x,t)$ for the corresponding Painleve equation.