Elimination of divergences in quantum mechanics

The solution of the one-dimensional Schrödinger equation, written In the $S$-matrix form (ordered exponential function) is expanded in a series whose terms diverge. It is shown that the infinities arise because the phase of a matrix elenaent of the exact solution has an infinite value. By means of a separate expransion for the modulus and phase of the matrix element and comparison with the expansion for the $S$-matrix, it is possible to find a series for the square of the modulus that does not contain an infinity. The technique deycleped for calculating the modulus of the matrix element is illustrated for an example having an analytic solution.