Quantum mechanics on spaces of nonconstant curvature: The oscillator problem and superintegrability

Ballesteros, Ángel; Enciso, Alberto; Herranz, Francisco J.; Ragnisco, Orlando; Riglioni, Danilo

doi:10.1016/j.aop.2011.03.002

Cited by 66 publications

(118 citation statements)

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“…Hence we will have, for our choice of s.a. Remark: the spectral analysis developed here for H + would be exactly the same as for H 0 and refines the results obtained in [2]. It explains also the apparent degeneracy for H 0 of the eigenfunction with m = 0: it is related to the non-uniqueness of the s.a. extensions.…”

Section: Transforming the Ode In (93) One Obtainssupporting

confidence: 81%

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Global structure and geodesics for Koenigs superintegrable systems

Valent¹

2016

Regul. Chaot. Dyn.

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Starting from the framework defined by Matveev and Shevchishin we derive the local and the global structure for the four types of super-integrable Koenigs metrics. These dynamical systems are always defined on non-compact manifolds, namely R 2 and H 2 . The study of their geodesic flows is made easier using their linear and quadratic integrals. Using Carter (or minimal) quantization we show that the formal superintegrability is preserved at the quantum level and in two cases, for which all of the geodesics are closed, it is even possible to compute the discrete spectrum of the quantum hamiltonian.

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Section: Transforming the Ode In (93) One Obtainssupporting

confidence: 81%

“…Let us observe that in [2] the quantization is done in flat space while we have quantized in curved space. Remarkably enough both approaches lead to the same energies while, of course, the eigenfunctions are markedly different.…”

Section: Proofmentioning

confidence: 99%

Global structure and geodesics for Koenigs superintegrable systems

Valent¹

2016

Regul. Chaot. Dyn.

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“…Secondly, another interesting open problem would be the construction of the constant curvature analogues of some integrable Hénon-Heiles systems (see [26] and references therein) that can be written as particular cases of the multiparametric family Finally, the explicit solution of the Schrödinger equation associated to H κ and H κ should be addressed, as well as the analysis of the quantum nonlinear dynamics generated by these new class of integrable nonlinear quantum models by following, e.g., the approaches presented in [27,28,29]. Work on all these lines is in progress.…”

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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“…We shall see how this problem can be solved by following the quantization procedure proposed in [14,15] for another maximally superintegrable quantum system on a different N D curved space: the so-called Darboux III oscillator system, which is an exactly solvable deformation of the harmonic oscillator potential that is associated to the Darboux III space [16,17]. Therefore, new exactly solvable deformations of the oscillator and Coulomb problems can be obtained when certain curved spaces with prescribed integrability properties are considered.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3 we present a quantization for H that preserves the full symmetry algebra of the classical system and, therefore, its maximal superintegrability. Explicitly, we shall prove that this is achieved through the conformal Laplacian quantization [15], namelŷ…”

Section: Introductionmentioning

confidence: 99%

An exactly solvable deformation of the Coulomb problem associated with the Taub–NUT metric

Ballesteros¹,

Enciso²,

Herranz³

et al. 2014

Annals of Physics

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In this paper we quantize the N -dimensional classical Hamiltonian system H = |q| 2(η + |q|)that can be regarded as a deformation of the Coulomb problem with coupling constant k, that it is smoothly recovered in the limit η → 0. Moreover, the kinetic energy term in H is just the one corresponding to an N -dimensional Taub-NUT space, a fact that makes this system relevant from a geometric viewpoint. Since the Hamiltonian H is known to be maximally superintegrable, we propose a quantization prescription that preserves such superintegrability in the quantum mechanical setting. We show that, to this end, one must choose as the kinetic part of the Hamiltonian the conformal Laplacian of the underlying Riemannian manifold, which combines the usual Laplace-Beltrami operator on the Taub-NUT manifold and a multiple of its scalar curvature. As a consequence, we obtain a novel exactly solvable deformation of the quantum Coulomb problem, whose spectrum is computed in closed form for positive values of η and k, and showing that the well-known maximal degeneracy of the flat system is preserved in the deformed case. Several interesting algebraic and physical features of this new exactly solvable quantum system are analysed, and the quantization problem for negative values of η and/or k is also sketched.

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Quantum mechanics on spaces of nonconstant curvature: The oscillator problem and superintegrability

Cited by 66 publications

References 50 publications

Global structure and geodesics for Koenigs superintegrable systems

Global structure and geodesics for Koenigs superintegrable systems

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

An exactly solvable deformation of the Coulomb problem associated with the Taub–NUT metric

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