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

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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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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, 6720, Hungary

    Béla Gábor Pusztai

  2. MTA Lendület Holographic QFT Group, Wigner RCP, P.O.B. 49, Budapest 114, 1525, Hungary

    Béla Gábor Pusztai

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  1. Béla Gábor Pusztai
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Correspondence to Béla Gábor Pusztai.

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Communicated by P. Deift

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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