Hydrogen atom in space with a compactified extra dimension and potential defined by Gauss' law

TL;DR

This paper explores how an extra spatial dimension affects the hydrogen atom's stability and energy spectrum, revealing conditions under which bound states and unbounded energies occur, especially with compactification effects.

Contribution

It analyzes the hydrogen atom in a space with an extra dimension, considering both infinite and compactified cases, and identifies the conditions for bounded and unbounded energy spectra.

Findings

Energy spectrum bounded for charges below a critical value

Negative energy eigenstates emerge with compactification

Infinite bound states appear if the compactification radius is small

Abstract

We investigate the consequences of one extra spatial dimension for the stability and energy spectrum of the non-relativistic hydrogen atom with a potential defined by Gauss' law, i.e. proportional to $1/|x|^2$. The additional spatial dimension is considered to be either infinite or curled-up in a circle of radius $R$. In both cases, the energy spectrum is bounded from below for charges smaller than the same critical value and unbounded from below otherwise. As a consequence of compactification, negative energy eigenstates appear: if $R$ is smaller than a quarter of the Bohr radius, the corresponding Hamiltonian possesses an infinite number of bound states with minimal energy extending at least to the ground state of the hydrogen atom.