# Self-duality and scattering map for the hyperbolic van Diejen systems with two coupling parameters (with an appendix by S. Ruijsenaars)

## TL;DR

This paper constructs global action-angle variables for a two-parameter hyperbolic van Diejen system, demonstrating its self-duality and a factorized scattering map, with a key spectral asymptotics result included.

## Contribution

It introduces a new method to analyze the van Diejen system using scattering theory and action-angle variables, establishing self-duality and detailed scattering properties.

## Key findings

- The van Diejen system is self-dual.
- The scattering map is factorized.
- Spectral asymptotics of exponential matrix flows are characterized.

## 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.