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

2019/03/31 by Masatoshi Noumi, S. N. M. Ruijsenaars, Simon Ruijsenaars +1 · 11 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Elliptic curve #Elliptic function #Hamiltonian (control theory) #Integrable system #Lax pair #Mathematical analysis #Mathematical physics #Mathematics #Matrix Theory and Algorithms #Nonlinear Waves and Solitons #Physics #Pure mathematics #math-ph #math.MP #math.QA #nlin.SI

paper · pdf · doi:10.3842/sigma.2020.063

published in Symmetry Integrability and Geometry Methods and Applications (National Academy of Sciences of Ukraine)

arxiv created 2020/07/08 · openalex publication_date 2020/07/08 · arxiv updated 2020/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

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 BC₁ 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.

Citations

• Heun equation and Painlevé equation
• First order analytic difference equations and integrable quantum systems
• Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations