On the spectrum of the screened Coulomb potential $V(r)=-r^{-1}{e}^{-C/r}$

Abstract

We analyse recent contradictory results and conclusions about the spectrum of the screened Coulomb potential $V(r)=-r^{-1}{e}^{-C/r}$. The well known Hellmann-Feynman theorem shows that all the bound states of the Coulomb potential ($C=0$) remain bounded as $C$ increases. We derive a simple approximate analytical expression for the eigenvalues for sufficiently small values of the screening parameter $C$ and an approximate asymptotic expression for the asymptotic behaviour of the s-state eigenvalues when $C\to \infty$. Present results are expected to resolve the discrepancy about the spectrum of the quantum-mechanical model just mentioned.