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The Elliptic Painlevé Lax Equation vs. van Diejen's 8-Coupling Elliptic Hamiltonian (1903.09738v2)
Published 23 Mar 2019 in math-ph, math.MP, math.QA, and nlin.SI
Abstract: The 8-parameter elliptic Sakai difference Painlev\'e equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schr\"odinger 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 Painlev\'e-Calogero correspondence to the highest level in the two hierarchies.