Bound states of the potentialV(r)=−Z(r+β)

We have studied the bound states of the potential $V\left(r\right)=-\frac{Z}{(r+\beta )}$. The $S$-wave energy level $E\left(\beta \right)$, as a function of the parameter $\beta$, has a logarithmic singularity at the origin. The imaginary part of $E\left(\beta \right)$ for $\beta \to 0$ goes as $Z^{4+2l}{\beta }^{2+2l}$, where $l$ is the angular momentum.