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

SIGMA 16 (2020), 063, 16 pages      arXiv:1903.09738      https://doi.org/10.3842/SIGMA.2020.063
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

The Elliptic Painlevé Lax Equation vs. van Diejen's 8-Coupling Elliptic Hamiltonian

Masatoshi Noumi a, Simon Ruijsenaars b and Yasuhiko Yamada a
a)  Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
b)  School of Mathematics, University of Leeds, Leeds LS2 9JT, UK

Received April 20, 2020, in final form June 26, 2020; Published online July 08, 2020

Abstract
The 8-parameter elliptic Sakai difference Painlevé equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schrödinger equation for the $BC_1$ 8-parameter 'relativistic' Calogero-Moser Hamiltonian due to van Diejen. This amounts to a generalization of previous results concerning the Painlevé-Calogero correspondence to the highest level in the two hierarchies.

Key words: Painlevé-Calogero correspondence; elliptic difference Painlevé equation; Ruijsenaars-van Diejen Hamiltonian.

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