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We study the analytical solutions of the Schr\"odinger equation with arepulsive exponential potential $\lambda e^{- r}$, and that with an exponentialwall $\lambda e^r$, both with $\lambda > 0$. We show that the complexeigenenergies obtained for the latter tend either to those of the former, or toreal rational numbers as $\lambda \rightarrow \infty$. In the light of theseresults, we explain the wrong resonance energies obtained in a previousapplication of the Riccati-Pad\'e method to the Schr\"odinger equation with arepulsive exponential potential, and further study the convergence propertiesof this approach.

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