Quantum Painlevé-Calogero correspondence

A. Zabrodin, A. Zotov

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30 Citations (Scopus)


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∂ 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 languageEnglish
Article number073507
JournalJournal of Mathematical Physics
Issue number7
Publication statusPublished - 12 Jul 2012
Externally publishedYes


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