The Elliptic Painlev\'e Lax Equation vs. van Diejen's 8-Coupling Elliptic Hamiltonian

Understand

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.