Generalised Calogero-Moser Models and Universal Lax Pair Operators.

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  • Bordner A. J.
    Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502, Japan
  • Corrigan E.
    Department of Mathematical Sciences, University of Durham, South Road, Durham DH1–3LE, United Kingdom
  • Sasaki Ryu
    Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502, Japan

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Calogero-Moser models can be generalised for all of the finite reflection groups. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H3, H4, and the dihedral group I2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero-Moser models and for any choices of the potentials are constructed as linear combinations of the reflection operators. The consistency conditions are reduced to functional equations for the coefficient functions of the reflection operators in the Lax pair. There are only four types of such functional equations corresponding to the two-dimensional sub-root systems, (A2), (B2), (G2), and (I2(m)). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators, a powerful tool for solving quantum Calogero-Moser models.

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