Group Analysis of Differential Equations and Integrable Systems − 2018

Robert Conte
Centre de mathématiques et de leurs applications, École normale supérieure de Cachan, CNRS, Université Paris-Saclay, Cachan, France
Department of mathematics, the University of Hong Kong, Pokfulam road, Hong Kong

Quantum correspondence for Painlevé VI

From any Lax pair of ${\rm P_{VI}}$, one can build a generalized heat equation with coefficients independent of the nonlinear field [3,4]. Its identification to the time-dependent Schrödinger equation of quantum mechanics ("quantum correspondence"') is easy in the elliptic representation of ${\rm P_{VI}}$ [5], but ad hoc in its usual rational representation [4]. We identify [1,2] the reason for this difficulty: the classical Hamiltonian (not unique) must be associated to one of the four tau-functions of Chazy.

[1] R. Conte, Surfaces de Bonnet et équations de Painlevé, C.R. Math. Acad. Sci. Paris 342 (2017) 40-44.
[2] R. Conte, Generalized Bonnet surfaces and Lax pairs of ${\rm P_{VI}}$, J. Math. Phys. 58 (2017) 103508.
[3] D.P. Novikov, The 2x2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system, Teoreticheskaya i Matematicheskaya Fizika 161 (2009) 191-203; Theor. Math. Phys. 161 (2009) 1485-1496.
[4] B.I. Suleimanov, Hamiltonian property of the Painlevé equations and the method of isomonodromic deformations, Diff. equ. 30 (1994) 726-732.
[5] A. Zabrodin and A. Zotov, Quantum Painlevé-Calogero correspondence for Painlevé VI, J. Math. Phys. 53 (2012) 073508.