Hydrogen atom in space with a compactified extra dimension and potential   defined by Gauss' law
  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. 
Martin Bure\v{s} and Petr Siegl
quant-ph hep-th math-ph math.MP