Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

Ragnisco, Orlando; Ballesteros, Ángel; Herranz, Francisco J.; Musso, Fabio

doi:10.3842/sigma.2007.026

Cited by 4 publications

(5 citation statements)

References 42 publications

(99 reference statements)

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“…• And, finally, the last possible generalization is to consider Poisson-Hopf algebra deformations of sl(2, R) [28,62,63] which convey an additional quantum deformation parameter q = e z giving rise to a deformed classical Hamiltonian H λ,N . In this case, the deformation parameter z would determine superintegrable perturbations of the initial (underformed) Hamiltonian (6.3).…”

Section: Discussionmentioning

confidence: 99%

Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

2023

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We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by where q i and p i are generic canonical variables, γ n are arbitrary coefficients, and N ∈ N . For N = 2, being both γ 1 , γ 2 different from zero, this reduces to the classical Zernike system. We prove that always provides a superintegrable system (for any value of γ n and N) by obtaining the corresponding constants of the motion explicitly, which turn out to be of higher-order in the momenta. Such generic results are not only applied to the Euclidean plane, but also to the sphere and the hyperbolic plane. In the latter curved spaces, is expressed in geodesic polar coordinates showing that such a new superintegrable Hamiltonian can be regarded as a superposition of the isotropic 1 : 1 curved (Higgs) oscillator with even-order anharmonic curved oscillators plus another superposition of higher-order momentum-dependent potentials. Furthermore, the symmetry algebra determined by the constants of the motion is also studied, giving rise to a ( 2 N − 1 ) th-order polynomial algebra. As a byproduct, the Hamiltonian is interpreted as a family of superintegrable perturbations of the classical Zernike system. Finally, it is shown that (and so the Zernike system as well) is endowed with a Poisson s l ( 2 , R ) -coalgebra symmetry which would allow for further possible generalizations that are also discussed.

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Section: Discussionmentioning

confidence: 99%

Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

2023

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“…and the common integrals (2.2) are just the mth (left and right) coproducts of the Casimir of sl(2, R). This set of (2N − 3) integrals is "universal" for any Hamiltonian function defined by H = H(q 2 , p 2 , q · p) so that this always provides, at least, a quasi-MS system [32,33]. Therefore the Hamiltonians shown in Table 1 are distinguished systems since they have "additional" symmetries.…”

Section: 1mentioning

confidence: 99%

“…Since both of them are central potentials, the angular momentum integrals (2.2) are valid for both cases, that is, S (m) U ≡ S (m) and S U,(m) ≡ S (m) in (1.2). We recall that, in fact, the spherical symmetry of a central potential on E N directly provides such (2N − 3) independent angular momentum integrals, so they characterize a quasi-MS system [32,33]. However what makes rather special the harmonic oscillator and KC systems is the existence of one more independent integral, which is extracted from a new set of integrals that ensure their MS property and is related to the fact that these two systems are the only ones fulfilling the classical Bertand's theorem [34].…”

Section: Harmonic Oscillator and Kepler Potentials On Euclidean Spacementioning

confidence: 99%

Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform

Ballesteros

Enciso

Herranz³

et al. 2011

SIGMA

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Abstract. The Stäckel transform is applied to the geodesic motion on Euclidean space, through the harmonic oscillator and Kepler-Coloumb potentials, in order to obtain maximally superintegrable classical systems on N -dimensional Riemannian spaces of nonconstant curvature. By one hand, the harmonic oscillator potential leads to two families of superintegrable systems which are interpreted as an intrinsic Kepler-Coloumb system on a hyperbolic curved space and as the so-called Darboux III oscillator. On the other, the Kepler-Coloumb potential gives rise to an oscillator system on a spherical curved space as well as to the Taub-NUT oscillator. Their integrals of motion are explicitly given. The role of the (flat/curved) Fradkin tensor and Laplace-Runge-Lenz N -vector for all of these Hamiltonians is highlighted throughout the paper. The corresponding quantum maximally superintegrable systems are also presented.

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“…, N) define two sets of N integrals in involution. Proofs, technical details and further generalizations can be found in [1,2] but, at this point, some remarks concerning the symmetry and superintegrability properties of H (N) are in order.…”

Section: Introductionmentioning

confidence: 99%

“…The Poisson-coalgebraic "dynamical" symmetry underlying all the superintegrable Hamiltonian systems that we shall present in the sequel can be summarized as the following quite general result [1,2]: Let (q, p) = (q 1 , . .…”

Section: Introductionmentioning

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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Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

Cited by 4 publications

References 42 publications

Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform

Superintegrable Anharmonic Oscillators on N-dimensional Curved Spaces

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