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Integrable time-discretisation of the Ruijsenaars-Schneider model

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Abstract

An exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the Calogero-Moser system, that was introduced for the first time by Ruijsenaars and Schneider. For the discrete-time model the equations of motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2 XYZ Heisenberg magnet. We present a Lax pair, the sympletic structure and prove the involutivity of the invariants. Exact solutions are investigated in the rational and hyperbolic (trigonometric) limits of the system that is given in terms of elliptic functions. These solutions are connected with discrete soliton equations. The results obtained allow us to consider the Bethe Ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding Bethe wavefunction.

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Author notes

  1. V. B. Kuznetsov

    Present address: Department of Mathematical and Computational Physics, Institute of Physics, St. Petersburg University, 198904, St. Petersburg, Russia

Authors and Affiliations

  1. Department of Applied Mathematical Studies, The University of Leeds, LS2 9JT, Leeds, UK

    F. W. Nijhoff

  2. Dipartimento di Fisica E. Amaldi, Università di Roma III, and Istituto di Fisica Nucleare, Sezione di Roma, P. le Aldo Moro 2, I-00185, Roma, Italy

    O. Ragnisco

  3. Faculteit voor Wiskunde en Informatica, Universiteit van Amsterdam, Platage Muidergracht 24, NL-1018 TV, Amsterdam, The Netherlands

    V. B. Kuznetsov

Authors
  1. F. W. Nijhoff
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  2. O. Ragnisco
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  3. V. B. Kuznetsov
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Additional information

Communicated by M. Jimbo

Supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)

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Nijhoff, F.W., Ragnisco, O. & Kuznetsov, V.B. Integrable time-discretisation of the Ruijsenaars-Schneider model. Commun.Math. Phys. 176, 681–700 (1996). https://doi.org/10.1007/BF02099255

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  • DOI: https://doi.org/10.1007/BF02099255

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