Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2

Rañada, Manuel F.; Santander, Mariano

doi:10.1063/1.533014

Cited by 107 publications

(211 citation statements)

References 25 publications

(18 reference statements)

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“…In fact, the N = 2 restriction of this perturbed system does not appear in the classifications of MS systems on S 2 and H 2 given in [14,15,16]. Note also that the first perturbative term given by δ 1 = 0 can be considered as the constant curvature generalization of the (radial) Garnier system.…”

Section: Stereographic Projection: Poincaré Coordinatesmentioning

confidence: 99%

Superintegrable Anharmonic Oscillators on N-dimensional Curved Spaces

Ballesteros¹,

Enciso²,

Herranz³

et al. 2008

JNMP

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The maximal superintegrability of the intrinsic harmonic oscillator potential on N-dimensional spaces with constant curvature is revisited from the point of view of sl(2)-Poisson coalgebra symmetry. It is shown how this algebraic approach leads to a straightforward definition of a new large family of quasi-maximally superintegrable perturbations of the intrinsic oscillator on such spaces. Moreover, the generalization of this construction to those N-dimensional spaces with non-constant curvature that are endowed with sl(2)-coalgebra symmetry is presented. As the first examples of the latter class of systems, both the oscillator potential on an N-dimensional Darboux space as well as several families of its quasi-maximally superintegrable anharmonic perturbations are explicitly constructed.

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Section: Stereographic Projection: Poincaré Coordinatesmentioning

confidence: 99%

Superintegrable Anharmonic Oscillators on N-dimensional Curved Spaces

Ballesteros¹,

Enciso²,

Herranz³

et al. 2008

JNMP

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“…The idea is that if we call super-separable a system that admits Hamilton-Jacobi separation of variables (Schrödinger in the quantum case) in more than one coordinate system, then quadratic super-integrability (i.e., super-integrability with linear or quadratic constants of motion) can be considered as a property arising from super-separability. We note that these studies also include non-Euclidean Hamiltonian systems [10,11,12,13,14,15,16,17,18,19,20,21,22] and that in both cases, Euclidean and nonEuclidean, many of these systems are closely related with the harmonic oscillator.…”

Section: Super-integrable Systemsmentioning

confidence: 99%

“…On the other side each one of these three formalisms can be used for study of the κ-dependent version of the S-W system. In the language of L κ the potential, that was studied in [13,20], is given by…”

Section: Let Us Consider Thementioning

confidence: 99%

A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog

Cariñena

Rañada

Santander

2007

SIGMA

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Abstract. Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky-Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodinsky-Winternitz system in such bi-dimensional spaces of constant curvature. The quantization of the first system will be carried out and it is shown that it is super-solvable in the sense that the Schrödinger equation reduces, in three different coordinate systems, to two separate equations involving only one degree of freedom.Key words: deformed oscillator; integrability, super-integrability; Hamilton-Jacobi separability; Hamilton-Jacobi super-separability; quantum solvable systems 2000 Mathematics Subject Classification: 37J35; 34A34; 34C15; 70H06 Super-integrable systemsThere are few integrable systems in the Arnold-Liouville sense: Hamiltonian systems in a 2n-dimensional symplectic manifold for which there exist n functionally independent constants of motion f i in involution (including the Hamiltonian H itself), i.e.The system is said to be super-integrable when it is integrable and there exists a set of m > n functionally independent constants of motion, i.e. it possesses more independent first integrals than degrees of freedom. The existence of these additional first integrals gives rise to a higher degree of regularity in the phase space (e.g. there exist periodic orbits) since the trajectories are restricted to submanifolds of dimension lower than n. In particular, a system with n degrees of freedom possessing 2n − 1 independent first integrals is said to be maximally super-integrable. It is also well-known [1] that the only central potentials in which all bounded orbits are closed (periodic) are given by V = (1/2)ω 2 0 r 2 and V = −k/r. From a modern point of view the existence of closed trajectories is considered as a consequence of the existence of the maximal number of functionally independent integrals of motion; thus, the result obtained by Bertrand This paper is a contribution to the

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“…Likewise, their "deformed" counterpart (understood as spaces with non-constant curvature) have been deduced from the integrable metric (8). The explicit solution of the geodesic flows for all these spaces has been studied in [2], as well as a method to introduce (super)integrable potentials on them by adding a potential term of the form U (zJ (2) − ); in this way, some known potentials are recovered (appearing in the classifications [6,7]) and also new ones are obtained. We recall that another approach to superintegrability on 2D spaces of variable curvature can be found in [8,9].…”

Section: Introductionmentioning

confidence: 99%

Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations

2005

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The link between 3D spaces with (in general, non-constant) curvature and quantum deformations is presented. It is shown how the non-standard deformation of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians that represent geodesic motions on 3D manifolds with a non-constant curvature that turns out to be a function of the deformation parameter z. A different Hamiltonian defined on the same deformed coalgebra is also shown to generate a maximally superintegrable geodesic motion on 3D Riemannian and (2 + 1)D relativistic spaces whose sectional curvatures are all constant and equal to z. This approach can be generalized to arbitrary dimension. PACS: 02.30.lk, 02.20.Uw

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Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2

Abstract: On harmonic oscillators on the two-dimensional sphere S 2 and the hyperbolic plane H 2 . II.Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems

Cited by 107 publications

References 25 publications

Superintegrable Anharmonic Oscillators on N-dimensional Curved Spaces

Superintegrable Anharmonic Oscillators on N-dimensional Curved Spaces

A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog

Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations

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