Superintegrability of the Calogero-Moser system

Wojciechowski, S.

doi:10.1016/0375-9601(83)90018-x

Cited by 196 publications

(215 citation statements)

References 5 publications

Supporting

Mentioning

208

Contrasting

Unclassified

Order By: Relevance

“…The √ r factor in (2.21) gets cancelled in the similarity transformation (2.12), giving 8) and the angular wave function reads…”

Section: Jhep10(2015)191mentioning

confidence: 99%

“…Originally defined for the root system of A 1 ⊕ A n−1 , the Calogero model was quickly generalized for any finite Coxeter group of rank n [4]. In particular the rational version is remarkable for its conformal properties and its maximal superintegrability [8,9] (see also [10]). Moreover, when the coupling constant g is integral, the quantum model enjoys additional and algebraically independent conserved quantities, which make it 'analytically integrable' [11,12].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The tetrahexahedric angular Calogero model

Correa

Lechtenfeld

2015

J. High Energ. Phys.

View full text Add to dashboard Cite

Abstract:The spherical reduction of the rational Calogero model (of type A n−1 and after removing the center of mass) is considered as a maximally superintegrable quantum system, which describes a particle on the (n−2)-sphere subject to a very particular potential. We present a detailed analysis of the simplest non-separable case, n=4, whose potential is singular at the edges of a spherical tetrahexahedron. A complete set of independent conserved charges and of Hamiltonian intertwiners is constructed, and their algebra is elucidated. They arise from the ring of polynomials in Dunkl-deformed angular momenta, by classifying the subspaces invariant and antiinvariant under all Weyl reflections, respectively.

show abstract

“…The √ r factor in (2.21) gets cancelled in the similarity transformation (2.12), giving 8) and the angular wave function reads…”

Section: Jhep10(2015)191mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The tetrahexahedric angular Calogero model

Correa

Lechtenfeld

2015

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…A classical superintegrable system H = ͚ ij g ij p i p j + V͑x͒ on an n-dimensional local Riemannian manifold is one that admits 2n − 1 functionally independent symmetries ͑i.e., constants of the motion͒ S k , k =1, ... ,2n − 1 with S 1 = H. That is, ͕H , S k ͖ = 0 where ͕f,g͖ = ͚ j=1 n ‫ץ͑͑‬ x j f‫ץ‬ p j g − ‫ץ‬ p j f‫ץ‬ x j g͒͒ is the Poisson bracket for functions f͑x , p͒ , g͑x , p͒ on phase space. [1][2][3][4][5][6][7][8] Note that 2n − 1 is the maximum possible number of functionally independent symmetries and, locally, such symmetries always exist. The main interest is in symmetries that are polynomials in the p k and are globally defined, except for lower dimensional singularities such as poles and branch points.…”

Section: Introduction and Examplesmentioning

confidence: 99%

Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

Kalnins

Kress

Miller

2005

Journal of Mathematical Physics

112

193

View full text Add to dashboard Cite

This paper is the first in a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. Many examples of such systems are known, and lists of possible systems have been determined for constant curvature spaces in two and three dimensions, as well as few other spaces. Observed features of these systems are multiseparability, closure of the quadratic algebra of second-order symmetries at order 6, use of representation theory of the quadratic algebra to derive spectral properties of the quantum Schrödinger operator, and a close relationship with exactly solvable and quasi-exactly solvable systems. Our approach is, rather than focus on particular spaces and systems, to use a general theoretical method based on integrability conditions to derive structure common to all systems. In this first paper we consider classical superintegrable systems on a general twodimensional Riemannian manifold and uncover their common structure. We show that for superintegrable systems with nondegenerate potentials there exists a standard structure based on the algebra of 2 ϫ 2 symmetric matrices, that such systems are necessarily multiseparable and that the quadratic algebra closes at level 6. Superintegrable systems with degenerate potentials are also analyzed. This is all done without making use of lists of systems, so that generalization to higher dimensions, where relatively few examples are known, is much easier.

show abstract

“…if the distribution D is integrable, we have a degenerate integrable system and a commutative diagram (3). An example of such system is the (spinless) Calogero-Moser system [30]. For other examples see [2][5].…”

Section: 2mentioning

confidence: 99%

“…One should emphasize that degenerate integrability is a special structure which is stronger than Liouville integrability: invariant tori now have dimension k < n. In the extreme case of k = 1 all trajectories are periodic. A degenerately integrable system may also be Liouville integrable, as in the case of spinless Calogero-Moser system [30].…”

Section: Introductionmentioning

confidence: 99%

Degenerately Integrable Systems

Reshetikhin

2016

J Math Sci

View full text Add to dashboard Cite

Abstract. The subject of this paper is degenerate integrability in Hamiltonian mechanics. It starts with a short survey of degenerate integrability. The first section contains basic notions. It is followed by a number of examples which include the Kepler system, Casimir models, spin Calogero models, spin Ruijsenaars models, and integrable models on symplectic leaves of Poisson Lie groups. The new results are degenerate integrability of relativistic spin Ruijsenaars and Calogero-Moser systems and the duality between them.

show abstract

Superintegrability of the Calogero-Moser system

Cited by 196 publications

References 5 publications

The tetrahexahedric angular Calogero model

The tetrahexahedric angular Calogero model

Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

Degenerately Integrable Systems

Contact Info

Product

Resources

About