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

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 $B{C}_1$ 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.

References in corpus (5)

• Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications
• A Lax Formalism for the Elliptic Difference Painlevé Equation
• Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case
• An Elliptic Garnier System from Interpolation
• Heun equation and Painlevé equation

Cited by in corpus (7)

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• Quantum Representation of Affine Weyl Groups and Associated Quantum Curves
• The rational Heun operator and Wilson biorthogonal functions
• Elliptic Racah polynomials
• Weyl invariant $E_8$ Jacobi forms and $E$-strings
• q-Heun equation and initial-value space of q-Painlevé equation
• Quantum Mirror Map for Del Pezzo Geometries