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

Abstract

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.

Multicomplex Wave Functions for Linear And Nonlinear Schrödinger Equations

“…This would be one possible extension of the Schrödinger equation to manifold domains. For recent work along these lines, see [6,47,48] and references therein. Furthermore, the Laplacian on bicomplex numbers was recently studied in [26].…”