On the complex solution of the Schr\"odinger equation with exponential potentials

We study the analytical solutions of the Schr\"odinger 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 $\lambda \rightarrow \infty$. In the light of these results, we explain the wrong resonance energies obtained in a previous application of the Riccati-Pad\'e method to the Schr\"odinger equation with a repulsive exponential potential, and further study the convergence properties of this approach.