Geodesic
A Simple Proof of Sklyanin's Formula for Canonical Spectral Coordinates of the Rational Calogero–Moser System
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: integrable systems; Calogero–Moser type systems; spectral coordinates; Hamiltonian reduction; action-angle duality. 
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Tamás F. Görbe. A Simple Proof of Sklyanin's Formula for Canonical Spectral Coordinates of the Rational Calogero–Moser System. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a26/

[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 | DOI | MR | DOI | MR | Zbl

[2] Calogero F., Classical many-body problems amenable to exact treatments, Lecture Notes in Physics, 66, Springer-Verlag, Berlin, 2001 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[5] Moser J., “Three integrable Hamiltonian systems connected with isospectral deformations”, Adv. Math., 16 (1975), 197–220 | DOI | MR | Zbl

[6] Perelomov A. M., Integrable systems of classical mechanics and Lie algebras, v. I, Birkhäuser Verlag, Basel, 1990 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[8] Sklyanin E., “Bispectrality and separation of variables in multiparticle hypergeometric systems”, Quantum Integrable Discrete Systems (Cambridge, England, March 23–27, 2009), Talk given at the Workshop

[9] Sklyanin E., “Bispectrality for the quantum open Toda chain”, J. Phys. A: Math. Theor., 46 (2013), 382001, 8 pp., arXiv: 1306.0454 | DOI | MR | Zbl

[10] Sutherland B., Beautiful models: 70 years of exactly solved quantum many-body problems, World Sci. Publ. Co., Inc., River Edge, NJ, 2004 | DOI | MR | Zbl