On the symmetry of a one-dimensional hydrogen atom

Abstract: We touch upon a long-standing question of the "true" one-dimensional hydrogen atom solution. From a symmetry point of view, Kepler problem in $d\ge2$ dimension is characterized by geometrical rotational symmetry, $SO(d)$, as well as dynamical, "accidental" $SO(d+1)$ symmetry. Because of topology, these two symmetries are mutually exclusive in one dimension, regardless of the regularization employed, drawing one to a conclusion that the question of "true" hydrogen atom in one dimension doesn't have an answer because a single dimension can not support both of the symmetries of Kepler problem. We argue our findings using a novel method to recover and classify solutions appearing in the literature according to the symmetry they respect. In particular, curious features of some of the solutions - double degeneracy and particle confinement - are directly attributed to the dynamical symmetry behind them.