On the complex solution of the Schrödinger equation with exponential potentials

We study the analytical solutions of the Schrödinger equation with a repulsive exponential potential $\lambda e^{-r}$, and that with an exponential wall $\lambda e^r$, both with $\lambda >0$. We show that the complex eigenenergies obtained for the latter tend either to those of the former, or to real rational numbers as $\lambda \to \mathrm{\infty }$. In the light of these results, we explain the wrong resonance energies obtained in a previous application of the Riccati-Padé method to the Schrödinger equation with a repulsive exponential potential, and further study the convergence properties of this approach.