Geometric quantization of the multidimensional Kepler problem

Mladenov, Iväılo M.; Tsanov, Vasil

doi:10.1016/0393-0440(85)90016-6

Cited by 41 publications

(29 citation statements)

References 4 publications

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“…Note that such solutions correspond to the type of solution discussed in [21]. The solution for the next term, X [1] , is in general non-separable and gives the first order perturbation theory, which may be found in a manner similar to the methods discussed in Sect. 3.…”

Section: Discussionmentioning

confidence: 92%

“…Expanding X (r, − → ω ), the multiplicative factor of the wave function governed by the variables (r, − → ω ), in a γ -series as X = X [0] + γ X [1] + · · · we see that the Schrödinger operator (5.1) leads to a system of PDEs from which the X [k] 's may be obtained. The order zero term corresponds to a solution of…”

Section: Discussionmentioning

confidence: 99%

“…We should mention that some results for the higher dimensional Kepler problem do exist; see, e.g., Mladenov and Tsanov [1]. Naturally, the case of non-compact extra dimensions have been more amenable to mathematical analysis.…”

Section: Introductionmentioning

confidence: 99%

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Schrödinger equations on $${\mathbb{R}^3 \times \mathcal{M}}$$ with non-separable potential

Gorder

2012

J Math Chem

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We consider the problem of defining the Schrödinger equation for a hydrogen atom on R 3 × M where M denotes an m dimensional compact manifold. In the present study, we discuss a method of taking non-separable potentials into account, so that both the non-compact standard dimensions and the compact extra dimensions contribute to the potential energy analogously to the radial dependence in the case of only non-compact standard dimensions. While the hydrogen atom in a space of the form R 3 × M, where M may be a generalized manifold obeying certain properties, was studied by Van Gorder (J Math Phys 51:122104, 2010), that study was restricted to cases in which the potential taken permitted a clean separation between the variables over R 3 and M. Furthermore, though there have been studies on the Coulomb problems over various manifolds, such studies do not consider the case where some of the dimensions are non-compact and others are compact. In the presence of nonseparable potential energy, and unlike the case of completely separable potential, a complete knowledge of the former case does not imply a knowledge of the latter.

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Section: Discussionmentioning

confidence: 92%

Section: Discussionmentioning

confidence: 99%

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Schrödinger equations on $${\mathbb{R}^3 \times \mathcal{M}}$$ with non-separable potential

Gorder

2012

J Math Chem

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“…When applied to such reduced manifolds, the geometric quantization scheme produces the quantization of charge, spin, and energy levels of some physical systems [10,11].…”

Section: Geometric Quantizationmentioning

confidence: 99%

Quantization on curved surfaces

Mladenov

2002

Int J of Quantum Chemistry

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“…Such questions as Moser [26] and Kustaanheimo-Stiefel [15] regularization procedures as well as the relationship between them [14] are well known for celestial mechanics specialists. Also the questions concerning the quantization of the Kepler system and the MIC-Kepler system, which is its natural generalization, are the subject of many publications, see, e.g., [8,17,19,24,25,28,31]. There are other interesting generalizations of Kepler and MIC-Kepler problems, for example see [2,11,19,20,21,22].…”

Section: Introductionmentioning

confidence: 99%