Self-duality and scattering map for the hyperbolic van Diejen systems with two coupling parameters (with an Appendix by S. Ruijsenaars). (English) Zbl 1422.70008
Summary: 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.
MSC:
| 70G65 | Symmetries, Lie group and Lie algebra methods for problems in mechanics |
| 70E55 | Dynamics of multibody systems |
| 70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
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