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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