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Prog. Theor. Phys. Vol. 102 No. 3 (1999) pp. 499-529

Generalised Calogero-Moser Models and Universal Lax Pair Operators

A. J. Bordner, E. Corrigan^*,*) and Ryu Sasaki

Yukawa Institute for Theoretical Physics, Kyoto University
Kyoto 606-8502, Japan
^*Department of Mathematical Sciences, University of Durham
South Road, Durham DH1-3LE, United Kingdom

(Received May 13, 1999)

Abstract:

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.

URL : http://ptp.ipap.jp/link?PTP/102/499/
DOI : 10.1143/PTP.102.499

^*)Address after October 1 1999: Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom.

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Citing Article(s) :

1. Journal of the Physical Society of Japan 70 (2001) pp. 3225-3237 :

   Spin Chains from Super-Models

   Tetsuo Deguchi and Pijush K. Ghosh

2. Progress of Theoretical Physics Vol. 102 No. 4 (1999) pp. 749-776 :

   Calogero-Moser Models. IV

   S. P. Khastgir, Ryu Sasaki and Kanehisa Takasaki

3. Progress of Theoretical Physics Vol. 103 No. 3 (2000) pp. 463-487 :

   Calogero-Moser Models. V

   Andrew J. Bordner, Nicholas S. Manton and Ryu Sasaki

4. Progress of Theoretical Physics Supplement No.135 (1999) pp. 75-93 :

   Seiberg-Witten Theory and Calogero-Moser Systems^*

   Eric D'Hoker and D. H. Phong

5. Progress of Theoretical Physics Supplement No.135 (1999) pp. 149-165 :

   Elliptic Ruijsenaars-Schneider and Calogero-Moser Models Represented by Sklyanin Algebra and sl(n) Gaudin Algebra

   Kai Chen, Heng Fan, Bo-yu Hou, Kang-jie Shi, Wen-li Yang and Rui-hong Yue

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