Lax representation of the hyperbolic van Diejen dynamics with two coupling parameters

Abstract

In this paper, we construct a Lax pair for the classical hyperbolic van Diejen system with two independent coupling parameters. Built upon this construction, we show that the dynamics can be solved by a projection method, which in turn allows us to initiate the study of the scattering properties. As a consequence, we prove the equivalence between the first integrals provided by the eigenvalues of the Lax matrix and the family of van Diejen's commuting Hamiltonians. Also, at the end of the paper, we propose a candidate for the Lax matrix of the hyperbolic van Diejen system with three independent coupling constants.

38 pages

References in corpus (6)

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Cited by in corpus (9)

• Quantum Lax pairs via Dunkl and Cherednik operators
• On factorized Lax pairs for classical many-body integrable systems
• Trigonometric and elliptic Ruijsenaars-Schneider systems on the complex projective space
• Global description of action-angle duality for a Poisson-Lie deformation of the trigonometric ${\mathrm{BC}}_n$ Sutherland system
• Quantum-classical duality for Gaudin magnets with boundary
• Self-duality and scattering map for the hyperbolic van Diejen systems with two coupling parameters (with an appendix by S. Ruijsenaars)
• Quantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary
• Dunkl and Cherednik operators
• Lax matrices for a 1-parameter subfamily of van Diejen--Toda chains