Quantum Painlevé-Calogero correspondence

The Painlevé-Calogero correspondence is extended to auxiliary linear problems associated with Painlevé equations. The linear problems are represented in a new form which has a suggestive interpretation as a "quantized" version of the Painlevé-Calogero correspondence. Namely, the linear problem responsible for the time evolution is brought into the form of non-stationary Schrödinger equation in imaginary time, ∂tψ=((1/2) ∂x2+V(x,t))ψ, whose Hamiltonian is a natural quantization of the classical Calogero-like Hamiltonian H=(1/2) p2+V(x,t) for the corresponding Painlevé equation. In present paper, we present explicit constructions for the first five equations from the Painlevé list.