The variable phase method used to calculate and correct scattering lengths

It is shown that the scattering length can be obtained by solving a Riccati equation derived from variable phase theory. Two methods of solving it are presented. The equation is used to predict how long-range interactions influence the scattering length, and upper and lower bounds on the scattering length are determined. The predictions are compared with others and it is shown how they may be obtained from secular perturbation theory.