Solution of the Schrödinger equation over an infinite integration interval by perturbation methods, revisited

We consider the solution of the one-dimensional Schrödinger problem over an infinite integration interval. The infinite problem is regularized by truncating the integration interval and imposing the appropriate boundary conditions at the truncation points. The Schrödinger problem is then solved on the truncated integration interval using one of the piecewise perturbation methods developed for the regular Schrödinger problem.We select the truncation points using a procedure based on the WKB approximation. However for problems which behave as a Coulomb problem both around the origin and in the asymptotic range, a more accurate treatment of the numerical boundaries is possible. Taking into account the asymptotic form of the Coulomb equation, adapted boundary conditions can be constructed and as a consequence smaller truncation points can be chosen. To deal with the singularity of the Coulomb-like problem around the origin, a special perturbative algorithm is applied in a small interval around the origin.