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

## TL;DR

This paper establishes a connection between the elliptic Painlevé Lax equation and van Diejen's elliptic Hamiltonian, extending the Painlevé-Calogero correspondence to a higher hierarchy level.

## Contribution

It demonstrates that a specialization of the elliptic Painlevé Lax equation yields the Schrödinger equation for van Diejen's elliptic Hamiltonian, generalizing previous results.

## Key findings

- Derived the Schrödinger equation from the Lax formulation.
- Extended the Painlevé-Calogero correspondence to higher hierarchy.
- Connected elliptic Painlevé equations with elliptic Calogero-Moser systems.

## 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.