Quantum Painleve-Calogero Correspondence

Abstract

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, $\p_t ψ=(1/2\, \p_x^2 +V(x,t))ψ$, whose Hamiltonian is a natural quantization of the classical Calogero-like Hamiltonian $H=1/2p^2+V(x,t)$ for the corresponding Painleve equation.

55 pages, references added

References in corpus (4)

• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• On the Linearization of the Painleve' III-VI Equations and Reductions of the Three-Wave Resonant System
• Random Matrix Theory and the Sixth Painlevé Equation
• Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painleve I transcendent

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