Exact Solution of Schr\"odinger Equation in (Anti-)de Sitter Spaces for Hydrogen Atom

TL;DR

This paper derives exact solutions for the Schr"odinger equation with Coulomb potential in de Sitter and Anti-de Sitter spaces, revealing how spatial curvature affects hydrogen-like bound states.

Contribution

It provides exact energy eigenvalues and wave functions for hydrogen atom in curved spacetimes using the Extended Uncertainty Principle and Nikiforov-Uvarov method.

Findings

Exact energy levels for de Sitter and Anti-de Sitter spaces

Wave functions expressed in special polynomials

Spatial deformation influences bound state properties

Abstract

We write Schr\"odinger equation for the Coulomb potential in both de Sitter and Anti-de Sitter spaces using the Extended Uncertainty Principle formulation. We use the Nikiforov-Uvarov method to solve the equations. The energy eigenvalues for both systems are given in their exact forms and the corresponding radial wave functions are expressed in associated Jacobi polynomials for de Sitter space, while those of Anti-de Sitter space are given in terms of Romanovski polynomials. We have also studied the effect of the spatial deformation parameter on the bound states in the two cases.