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

SIGMA 4 (2008), 051, 9 pages      arXiv:0806.1466      https://doi.org/10.3842/SIGMA.2008.051

Quantum Painlevé Equations: from Continuous to Discrete

Hajime Nagoya a, Basil Grammaticos b and Alfred Ramani c
a) Graduate School of Mathematical Sciences, The University of Tokyo, Japan
b) IMNC, Université Paris VII & XI, CNRS, UMR 8165, Bât. 104, 91406 Orsay, France
c) Centre de Physique Théorique, Ecole Polytechnique, CNRS, 91128 Palaiseau, France

Received March 05, 2008, in final form May 03, 2008; Published online June 09, 2008

Abstract
We examine quantum extensions of the continuous Painlevé equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlevé equations II, IV and V. From their auto-Bäcklund transformations we derive the contiguity relations which we interpret as the quantum analogues of the discrete Painlevé equations.

Key words: discrete systems; quantization; Painlevé equations.

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