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

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Abstract

We consider the problem of defining the Schrödinger equation for a hydrogen atom on \({\mathbb{R}^3 \times \mathcal{M}}\) where \({\mathcal{M}}\) 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 \({\mathbb{R}^3 \times \mathcal{M}}\) , where \({\mathcal{M}}\) 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 \({\mathbb{R}^3}\) and \({\mathcal{M}}\) . 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 non-separable 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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  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816-1364, USA

    Robert A. Van Gorder

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  1. Robert A. Van Gorder
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Correspondence to Robert A. Van Gorder.

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Van Gorder, R.A. Schrödinger equations on \({\mathbb{R}^3 \times \mathcal{M}}\) with non-separable potential. J Math Chem 50, 1420–1436 (2012). https://doi.org/10.1007/s10910-012-9981-1

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  • DOI: https://doi.org/10.1007/s10910-012-9981-1

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