Quantization of the multifold Kepler system

Iwai, Toshihiro; Uwano, Yoshio; Katayama, Noriaki

doi:10.1063/1.531431

Cited by 36 publications

(50 citation statements)

References 28 publications

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“…• The so-called [5] which was the one used in [6] for this system, since it preserves the maximal superintegrability in a straightforward way due to the immediate quantum transcription of the (2N − 1) classical integrals of the motion. This property leads to a maximal degeneracy of the spectrum, which is exactly the same as in the quantum N D flat oscillator.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2011

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The full spectrum and eigenfunctions of the quantum version of a nonlinear oscillator defined on an N -dimensional space with nonconstant curvature are rigorously found. Since the underlying curved space generates a position-dependent kinetic energy, three different quantization prescriptions are worked out by imposing that the maximal superintegrability of the system has to be preserved after quantization. The relationships among these three Schrödinger problems are described in detail through appropriate similarity transformations. These three approaches are used to illustrate different features of the quantization problem on N -dimensional curved spaces or, alternatively, of position-dependent mass quantum Hamiltonians. This quantum oscillator is, to the best of our knowledge, the first example of a maximally superintegrable quantum system on an N -dimensional space with nonconstant curvature.

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Section: Introductionmentioning

confidence: 99%

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

et al. 2011

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“…Moreover H η can be naturally related to the Taub-NUT system [2,3,4,5,6,7,8,9,10,11,12,13] since M N can be regarded as the (Riemannian) ND Taub-NUT space [14]. It is also known that according to the Perlick classification [15,16,17,18] the system (1) pertains to the class II, and thus it has to be regarded as an "intrinsic oscillator".…”

mentioning

confidence: 99%

The classical Taub–Nut system: factorization, spectrum generating algebra and solution to the equations of motion

Latini¹,

Ragnisco²

2015

J. Phys. A: Math. Theor.

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The formalism of SUSYQM (SUperSYmmetric Quantum Mechanics) is properly modified in such a way to be suitable for the description and the solution of a classical maximally superintegrable Hamiltonian System, the so-called Taub-Nut system, associated with the Hamiltonian:In full agreement with the results recently derived by A. Ballesteros et al. for the quantum case, we show that the classical Taub-Nut system shares a number of essential features with the Kepler system, that is just its Euclidean version arising in the limit η → 0, and for which a "SUSYQM" approach has been recently introduced by S. Kuru and J. Negro. In particular, for positive η and negative energy the motion is always periodic; it turns out that the period depends upon η and goes to the Euclidean value as η → 0. Moreover, the maximal superintegrability is preserved by the η-deformation, due to the existence of a larger symmetry group related to an η-deformed Runge-Lenz vector, which ensures that in R 3 closed orbits are again ellipses. In this context, a deformed version of the third Kepler's law is also recovered. The closing section is devoted to a discussion of the η < 0 case, where new and partly unexpected features arise.We consider the classical Hamiltonian in R N given by:where k and η are real parameters, q = (q 1 , . . . , q N ), p = (p 1 , . . . , p N ) ∈ R N are conjugate coordinates and momenta, and q 2 ≡ |q| 2 = ∑ N i=1 q 2 i . We recall that H η has been proven to be a maximally superintegrable Hamiltonian by making use of symmetry techniques [1]. This means that H η is endowed with the maximum possible number (2N − 1) of functionally independent constants of motion (including H η itself). In fact, besides the integrals of motion provided by the so(N) symmetry, H η is endowed with an η−deformed ND Laplace-Runge-Lenz vector R implying the existence of N additional constants of motion coming from the components of R, which are given by:

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“…Therefore, we are dealing with a system defined on a conformally flat and spherically symmetric space M with metric It turns out that the mathematical and physical relevance of the Hamiltonian (1.1) relies on two important facts. On one hand, when N = 3 the Hamiltonian H is directly related to a reduction [1] of the geodesic motion on the Taub-NUT space [2,3,4,5,6,7,8,9,10,11,12]. On the other hand, it was shown in [13] that (1.1) defines a maximally superintegrable classical system, that is, H is endowed with the maximum possible number of (2N − 1) functionally independent integrals of motion, all of which are, in this case, quadratic in the momenta.…”

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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Quantization of the multifold Kepler system

Cited by 36 publications

References 28 publications

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

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

The classical Taub–Nut system: factorization, spectrum generating algebra and solution to the equations of motion

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

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