Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 027, 5 pages      arXiv:1601.01181

A Simple Proof of Sklyanin's Formula for Canonical Spectral Coordinates of the Rational Calogero-Moser System

Tamás F. Görbe
Department of Theoretical Physics, University of Szeged, Tisza Lajos krt 84-86, H-6720 Szeged, Hungary

Received January 19, 2016, in final form March 08, 2016; Published online March 11, 2016

We use Hamiltonian reduction to simplify Falqui and Mencattini's recent proof of Sklyanin's expression providing spectral Darboux coordinates of the rational Calogero-Moser system. This viewpoint enables us to verify a conjecture of Falqui and Mencattini, and to obtain Sklyanin's formula as a corollary.

Key words: integrable systems; Calogero-Moser type systems; spectral coordinates; Hamiltonian reduction; action-angle duality.

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  1. Calogero F., Solution of the one-dimensional $N$-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436, Erratum, J. Math. Phys. 37 (1996), 3646.
  2. Calogero F., Classical many-body problems amenable to exact treatments, Lecture Notes in Physics. New Series m: Monographs, Vol. 66, Springer-Verlag, Berlin, 2001.
  3. Falqui G., Mencattini I., Bi-Hamiltonian geometry and canonical spectral coordinates for the rational Calogero-Moser system, arXiv:1511.06339.
  4. Kazhdan D., Kostant B., Sternberg S., Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), 481-507.
  5. Moser J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197-220.
  6. Perelomov A.M., Integrable systems of classical mechanics and Lie algebras. Vol. I, Birkhäuser Verlag, Basel, 1990.
  7. Ruijsenaars S.N.M., Action-angle maps and scattering theory for some finite-dimensional integrable systems. I. The pure soliton case, Comm. Math. Phys. 115 (1988), 127-165.
  8. Sklyanin E., Bispectrality and separation of variables in multiparticle hypergeometric systems, Talk given at the Workshop 'Quantum Integrable Discrete Systems', Cambridge, England, March 23-27, 2009.
  9. Sklyanin E., Bispectrality for the quantum open Toda chain, J. Phys. A: Math. Theor. 46 (2013), 382001, 8 pages, arXiv:1306.0454.
  10. Sutherland B., Beautiful models: 70 years of exactly solved quantum many-body problems, World Sci. Publ. Co., Inc., River Edge, NJ, 2004.

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