Quantization of Calogero-Painlev\'e System and Multi-Particle Quantum Painlev\'e Equations II-VI

TL;DR

This paper develops a quantum version of multi-particle Calogero-Painlevé systems through isomonodromic formulation and Hamiltonian reduction, extending classical results and integral representations to the quantum multi-particle context.

Contribution

It introduces a novel quantization approach for Calogero-Painlevé systems and generalizes integral representations of quantum Painlevé solutions to multi-particle cases.

Findings

Quantized multi-particle Calogero-Painlevé systems via isomonodromic formulation.

Derived quantum Hamiltonian reduction to radial variables.

Extended integral representations to multi-particle quantum Painlevé equations.

Abstract

We consider the isomonodromic formulation of the Calogero-Painlev\'e 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 Painlev\'e equations. We also provide multi-particle generalizations of these integral representations.