Improving the analytical determination of bound state energies and scattering lengths in molecular potentials - especially near threshold

Conventional WKB quantization can be improved substantially by including the appropriate reflection phases at the classical turning points. By application of the Effective-Range-theory we are able to calculate the reflection phase at the outer turning point in an attractive potential up to linear order in energy. For arbitrary energy we estimate the reflection phase by matching the low energy expansion with known formulas for high energies. This model, which includes only one free parameter is a significant improvement over the approximate eigenenergies obtained by other methods. The scattering length is completely determined by the knowledge of one of the highest bound energy levels (not necessarily by the highest one) and the asymptotic behavior of the potential