Calogero model

*Calogero* models comprise several integrable systems in classical mechanics and their relatives; they are describing interacting systems of $n$ particles on a circle or on a line. Basic variants include Calogero-Moser system, Calogero-Sutherland system and “relativistic” Ruijsenaars model?s. Calogero-Moser system was a historical source of the study of Dunkl operators and Cherednik’s double Hecke algebras. The Calogero-Moser system can also be related to the rational $n$-soliton solutions of rational KP equation; the elucidation of the relation between the soliton solutions and solutions of the Calogero-Moser system is rather deep and is called **Calogero-Moser correspondence**. Its modern formulation involves noncommutative algebraic geometry.

Some special functions come out of analysis of Calogero models, like Jack polynomials.

- Pavel Etingof,
*Lectures on Calogero-Moser systems*, pdf - D. Kazhdan, B. Kostant, S. Sternberg,
*Hamiltonian group actions and dynamical systems of Calogero type*, Comm. Pure Appl. Math.**31**(1978), 481-508, MR478225, doi - H. Airault, H. P. McKean, J. Moser,
*Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem*, Comm. Pure Appl. Math.**30**(1977), no. 1, 95–148; MR0649926, doi - E. Mukhin, V. Tarasov, A. Varchenko,
*KZ characteristic variety as the zero set of classical Calogero-Moser Hamiltonians*, arxiv/1201.3990 - Sergio Andraus, Makoto Katori, Seiji Miyashita,
*Calogero-Moser systems as a diffusion-scaling transform of Dunkl processes on the line*, arxiv/1211.6561

On Calogero-Moser correspondence:

- I. M. Kričever,
*Rational solutions of the Kadomcev-Petviašvili equation and the integrable systems of $N$ particles on a line*, Funkcional. Anal. i Priložen.**12**(1978), no. 1, 76–78, MR488139 - Takahiro Shiota,
*Calogero-Moser hierarchy and KP hierarchy*, J. Math. Phys.**35**(1994), no. 11, 5844–5849, MR95i:58095 - George Wilson,
*Collisions of Calogero-Moser particles and an adelic Grassmannian*, With an appendix by I. G. Macdonald. Invent. Math.**133**(1998), no. 1, 1–41, MR99f:58107, doi - Yu. Berest, G. Wilson,
*Ideal classes of the Weyl algebra and noncommutative projective geometry*(with an Appendix by M. Van den Bergh), Internat. Math. Res. Notices**26**(2002), 1347–1396. - Yu. Berest, G. Wilson,
*Mad subalgebras of rings of differential operators on curves*, Advances in Math.**212**no. 1 (2007), 163–190. - V. Ginzburg,
*Non-commutative symplectic geometry, quiver varieties, and operads*, Math. Res. Lett. 8 (2001), 377-400. - Yuri Berest, Oleg Chalykh,
*$A_\infty$-modules and Calogero-Moser spaces*, J. Reine Angew. Math.**607**(2007), 69–112, MR2009f:16019, doi

Revised on November 29, 2012 20:24:39
by Zoran Škoda
(193.51.104.65)