Self-Duality and Scattering Map for the Hyperbolic van Diejen Systems with Two Coupling Parameters (with an Appendix by S. Ruijsenaars)

2017/01/30 by B. G. Pusztai, B.G. Pusztai
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Coupling (piping) #Duality (order theory) #Engineering #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Physics #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Scattering #hep-th #math-ph #math.MP #nlin.SI

paper · pdf · doi:10.1007/s00220-017-3035-2

48 pages

arxiv created 2017/01/30 · openalex publication_date 2017/12/16 · arxiv updated 2018/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• Solution of the One-Dimensional N-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials
• The master T-operator for the Gaudin model and the KP hierarchy
• Elliptic Ruijsenaars–Schneider model from the cotangent bundle over the two-dimensional current group
• The sine-Gordon solitons as an N-body problem
• Multiplicative quiver varieties and generalised Ruijsenaars–Schneider models