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

摘要

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.