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

TL;DR

This paper analyzes the complex solutions of the Schrödinger equation with exponential potentials, revealing how eigenenergies behave and explaining previous inaccuracies in resonance energy calculations using the Riccati-Padé method.

Contribution

It provides analytical insights into the eigenenergies of exponential potentials and clarifies the convergence issues of the Riccati-Padé method for these problems.

Findings

Eigenenergies tend to those of the exponential wall potential or to rational numbers as  

Explains inaccuracies in previous Riccati-Pade9 method applications

Studies convergence properties of the Riccati-Pade9 method

Abstract

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.