Quantum Painlev\'e Equations: from Continuous to Discrete

TL;DR

This paper explores quantum versions of Painlevé equations, deriving their discrete analogues through auto-Bäcklund transformations, advancing the understanding of quantum integrable systems.

Contribution

It introduces quantum extensions of Painlevé II, IV, and V equations and derives their discrete analogues via auto-Bäcklund transformations.

Findings

Quantum Painlevé equations formulated for non-commuting variables

Discrete Painlevé equations derived as quantum analogues

Framework connects continuous and discrete quantum integrable systems

Abstract

We examine quantum extensions of the continuous Painlev\'e equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlev\'e equations II, IV and V. From their auto-B\"acklund transformations we derive the contiguity relations which we interpret as the quantum analogues of the discrete Painlev\'e equations.