## Abstract

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, ∂ _{t}Ψ=(1/2∂ ^{2} _{x} +V (X,t))Ψwhose Hamiltonian is a natural quantization of the classical Calogero-like Hamiltonian H = 1/2p ^{2}+V(x,t)for the corresponding Painlevé equation. In present paper, we present explicit constructions for the first five equations from the Painlevé list.

Original language | English |
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Article number | 073507 |

Journal | Journal of Mathematical Physics |

Volume | 53 |

Issue number | 7 |

DOIs | |

Publication status | Published - 12 Jul 2012 |

Externally published | Yes |