On the one-dimensional Coulomb problem

We analyse the one-dimensional Coulomb problem (1DCP) pointing out some mistaken beliefs on it. We show that no eigenstates of even or odd parity can represent states of the system. The 1DCP exhibits a sort of spontaneous breaking of parity. We also show that a superselection rule operates in the system. Such rule explains some of its peculiarities. We build the superpotential associated to the 1DCP.