Abstract
In this paper, we construct global action-angle variables for a certain two-parameter family of hyperbolic van Diejen systems. Following Ruijsenaars’ ideas on the translation invariant models, the proposed action-angle variables come from a thorough analysis of the commutation relation obeyed by the Lax matrix, whereas the proof of their canonicity is based on the study of the scattering theory. As a consequence, we show that the van Diejen system of our interest is self-dual with a factorized scattering map. Also, in an appendix by S. Ruijsenaars, a novel proof of the spectral asymptotics of certain exponential type matrix flows is presented. This result is of crucial importance in our scattering-theoretical analysis.
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S. Ruijsenaars—School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK. E-mail: siru@maths.leeds.ac.uk.
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Pusztai, B.G. Self-Duality and Scattering Map for the Hyperbolic van Diejen Systems with Two Coupling Parameters (with an Appendix by S. Ruijsenaars). Commun. Math. Phys. 359, 1–60 (2018). https://doi.org/10.1007/s00220-017-3035-2
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DOI: https://doi.org/10.1007/s00220-017-3035-2