Nonperturbative approach to screened Coulomb potentials

The energy levels of two simple examples of screened Coulomb potentials have been analyzed using nonperturbative methods. The analysis indicates that the energy levels as a function of the perturbation parameter $\lambda$ have a branch cut along the negative real axis, starting from the origin. Furthermore, there are singularities on the second sheet, along $\left|\lambda \right|e^{\pm \frac{3\pi i}{2}}$ for $\left|\lambda \right|\to 0$. As a consequence of these singularities, the energy levels have an asymptotic series in $\lambda$, which means that one cannot use a power series in $\lambda$ to describe the energy levels to an arbitrary accuracy. The approximate but nonperturbative expression for the energy levels, which has been obtained by using dispersion relations, predicts energy levels which are in good agreement with those obtained from variational calculations.