A distributional approach for the one-dimensional hydrogen atom

Abstract

We consider the one-dimensional Hydrogen atom, with the Coulomb interaction V(x)=γ|x| (γ < 0), and use Schwartz's theory of distributions to address the non-integrable singularity at the origin. This singularity renders the interaction term V(x)ψ(x) in the Schrödinger's equation, where ψ(x) is the wave function, an ill-defined product in the ordinary sense. We replace this ill-defined product by a well-defined interaction distribution, S[ψ, V](x), and by imposing that it should satisfy some fundamental mathematical and physical requirements, we show that this distribution is defined up to a 4-parameter family of contact interactions, in agreement with the method of self-adjoint extensions. By requiring that the interaction distribution be invariant under parity, we further restrict the 4-parameter family of interactions to the subfamily of all the parity invariant Coulomb interactions. Finally, we present a systematic study of the bound states within this subfamily, addressing the frequently debated issues of the multiplicity and parity of the bound states, and the boundedness of the ground state energy.