AUTHOR=Calçada Marcos , Lunardi José T. , Manzoni Luiz A. , Monteiro Wagner , Pereira Marciano 

TITLE=A Distributional Approach for the One-Dimensional Hydrogen Atom

JOURNAL=Frontiers in Physics

VOLUME=Volume 7 - 2019

YEAR=2019

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2019.00101

DOI=10.3389/fphy.2019.00101

ISSN=2296-424X

ABSTRACT=We consider the one-dimensional Hydrogen atom, with the Coulomb interaction $V(x)=\frac{\gamma}{|x|}$ ($\gamma < 0$), and use Schwartz's theory of distributions to address the non-integrable singularity at the origin. This singularity renders the interaction term $V(x)\psi(x)$ in the Schr\"odinger's equation, where $\psi (x)$ is the wave function, an ill-defined product in the ordinary sense.  We replace this ill-defined product by a well defined interaction \emph{distribution}, $S[\psi, V](x)$, 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 \emph{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.