Perturbation Theory for Large Coupling Constants Applied to the Gauss Potential

Abstract

As an example of a perturbation technique for large coupling constants g2, we investigate the solutions and eigenvalues of the Schrödinger equation for a Gauss potential. In particular, we obtain the regular solution, valid for r2 < 1/|g|, in terms of confluent hypergeometric functions by expanding the potential in the neighborhood of the origin. The Jost solution is obtained in an analogous manner in terms of a certain integral and is valid for r2 > 1/|4g|. Both solutions are eigensolutions belonging to the same eigenenergy E = k2. These eigenvalues are derived in the form of large-g asymptotic expansions which are useful and valid over a wide range of g. A noteworthy aspect of the investigation is the close analogy of the underlying mathematics with that of well-known periodic equations.