Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 081, 25 pages      arXiv:2103.09681      https://doi.org/10.3842/SIGMA.2021.081

Quantization of Calogero-Painlevé System and Multi-Particle Quantum Painlevé Equations II-VI

Fatane Mobasheramini a and Marco Bertola ab
a) Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montreal, QC H3G 1M8, Canada
b) SISSA, Area of Mathematics, via Bonomea 265, Trieste, Italy

Received March 19, 2021, in final form August 31, 2021; Published online September 07, 2021

Abstract
We consider the isomonodromic formulation of the Calogero-Painlevé multi-particle systems and proceed to their canonical quantization. We then proceed to the quantum Hamiltonian reduction on a special representation to radial variables, in analogy with the classical case and also with the theory of quantum Calogero equations. This quantized version is compared to the generalization of a result of Nagoya on integral representations of certain solutions of the quantum Painlevé equations. We also provide multi-particle generalizations of these integral representations.

Key words: quantization of Painlevé; Calogero-Painlevé; Harish-Chandra isomorphism.

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