Reduction of superintegrable systems: The anisotropic harmonic oscillator

Rodrı́guez, Miguel Á.; Tempesta, Piergiulio; Winternitz, P.

doi:10.1103/physreve.78.046608

Cited by 76 publications

(130 citation statements)

References 35 publications

(50 reference statements)

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“…23) which matches what was obtained for the Cartesian solutions in (5.13) upon letting2(N ′ + ℓ) = m 1 + m 2 . (5.24) Equation (5.16) can alternatively be presented in the formH|ℓ, N ′ , s 1 , s 2 , k 1 , k 2 = E|ℓ, N ′ , s 1 , s 2 , k 1 , k 2 (5.25)where the |ℓ, N ′ , s 1 , s 2 , k 1 , k 2 are 4-component spinors, whose entries depends on the value of s 1 , s 2 :…”

supporting

confidence: 86%

“…This is in effect a system of two singular oscillators, and such oscillators are associated to the algebra su(1, 1) instead of osp(1|2). These have been examined in detail and shown to be superintegrable in [23]. The fact that many four-dimensional oscillators were a priori considered is immaterial under this reduction process.…”

Section: Dimensional Reductionmentioning

confidence: 99%

“…A further observation is that this new model can be obtained as well through dimensional reduction. It was shown in [23] that harmonic oscillators in 2d dimensions can be reduced to maximally superintegrable systems of singular oscillators in d dimensions with integrals of motion inherited from those in the higher dimensions. This will be seen to prevail for the model with dual −1 Hahn symmetry upon projecting uncoupled harmonic oscillators with internal degrees of freedom from four to two dimensions.…”

Section: Introductionmentioning

confidence: 99%

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Superintegrability and the dual −1 Hahn algebra in superconformal quantum mechanics

2020

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A two-dimensional superintegrable system of singular oscillators with internal degrees of freedom is identified and exactly solved. Its symmetry algebra is seen to be the dual −1 Hahn algebra which describes the bispectral properties of the polynomials with the same name that are essentially the Clebsch-Gordan coefficients of the superconformal algebra osp(1|2). It is also shown how this superintegrable model is obtained under dimensional reduction from a set of uncoupled harmonic oscillators in four dimensions. * julien.gaboriaud@umontreal.ca

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supporting

confidence: 86%

Section: Dimensional Reductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Superintegrability and the dual −1 Hahn algebra in superconformal quantum mechanics

2020

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“…In this way, we have recovered all the well-known results on the (super)integrability of anisotropic oscillators [1][2][3] which can be summarized as follows. …”

Section: The Classical Factorization Methodssupporting

confidence: 64%

The anisotropic oscillator on curved spaces: A new exactly solvable model

et al. 2016

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a b s t r a c tWe present a new exactly solvable (classical and quantum) model that can be interpreted as the generalization to the twodimensional sphere and to the hyperbolic space of the twodimensional anisotropic oscillator with any pair of frequencies ω x and ω y . The new curved Hamiltonian H κ depends on the curvature κ of the underlying space as a deformation/contraction parameter, and the Liouville integrability of H κ relies on its separability in terms of geodesic parallel coordinates, which generalize the Cartesian coordinates of the plane. Moreover, the system is shown to be superintegrable for commensurate frequencies ω x : ω y , thus mimicking the behaviour of the flat Euclidean case, which is always recovered in the κ → 0 limit. The additional constant of motion in the commensurate case is, as expected, of higher-order in the momenta and can be explicitly deduced by performing the classical factorization of the Hamiltonian. The known 1 : 1 and 2 : 1 anisotropic curved oscillators are recovered as particular cases of H κ , meanwhile all the remaining ω x : ω y curved oscillators define new superintegrable systems. Furthermore, the quantum HamiltonianĤ κ is fully constructed and studied by following a quantum factorization approach. In the case of commensurate frequencies, the HamiltonianĤ κ turns out to be quantum superintegrable and leads to a new exactly solvable quantum model. Its corresponding spectrum, that exhibits a maximal degeneracy, is explicitly given as * Corresponding author. Á. Ballesteros et al. / Annals of Physics 373 (2016) an analytical deformation of the Euclidean eigenvalues in terms of both the curvature κ and the Planck constanth. In fact, such spectrum is obtained as a composition of two one-dimensional (either trigonometric or hyperbolic) Pösch-Teller set of eigenvalues.

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“…Moreover, the 'additional' integral that ensures the superintegrability of the system is, in fact, a second order function arising from a true sixth order integral. More explicitly, if we set λ 1 = 0 then it is known [6] Therefore these latter expressions seem to point out that the construction of the S 2 and H 2 analogues of some other Euclidean commensurate oscillators could be feasible.…”

Section: Discussionmentioning

confidence: 99%

The anisotropic oscillator on the 2D sphere and the hyperbolic plane

2013

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An integrable generalization on the two-dimensional sphere S 2 and the hyperbolic plane H 2 of the Euclidean anisotropic oscillator Hamiltonian with 'centrifugal' terms given byis presented. The resulting generalized Hamiltonian H κ depends explicitly on the constant Gaussian curvature κ of the underlying space, in such a way that all the results here presented hold simultaneously for S 2 (κ > 0), H 2 (κ < 0) and E 2 (κ = 0). Moreover, H κ is explicitly shown to be integrable for any values of the parameters δ, Ω, λ 1 and λ 2 . Therefore, H κ can also be interpreted as an anisotropic generalization of the curved Higgs oscillator, that is recovered as the isotropic limit Ω → 0 of H κ . Furthermore, numerical integration of some of the trajectories for H κ are worked out and the dynamical features arising from the introduction of a curved background are highlighted.The superintegrability issue for H κ is discussed by focusing on the value Ω = 3δ, which is one of the cases for which the Euclidean Hamiltonian H is known to be superintegrable (the 1:2 oscillator). We show numerically that for Ω = 3δ the curved Hamiltonian H κ presents nonperiodic bounded trajectories, which seems to indicate that H κ provides a non-superintegrable generalization of H even for values of Ω that lead to commensurate frequencies in the Euclidean case. We compare this result with a previously known superintegrable curved analogue H κ of the 1:2 Euclidean oscillator, which is described in detail, showing that the Ω = 3δ specialization of H κ does not coincide with H κ . Hence we conjecture that H κ would be an integrable (but not superintegrable) curved generalization of the anisotropic oscillator that exists for any value of Ω and has constants of the motion that are quadratic in the momenta. Thus each commensurate Euclidean oscillator could admit another specific superintegrable curved Hamiltonian which would be different from H κ and endowed with higher order integrals. Finally, the geometrical interpretation of the curved 'centrifugal' terms appearing in H κ is also discussed in detail.MSC: 37J35 70H06 14M17 22E60

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Reduction of superintegrable systems: The anisotropic harmonic oscillator

Cited by 76 publications

References 35 publications

Superintegrability and the dual −1 Hahn algebra in superconformal quantum mechanics

Superintegrability and the dual −1 Hahn algebra in superconformal quantum mechanics

The anisotropic oscillator on curved spaces: A new exactly solvable model

The anisotropic oscillator on the 2D sphere and the hyperbolic plane

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