Gaussian weighting in the quasiclassical trajectory method

In standard quasiclassical trajectory calculations, each trajectory has the same statistical weight. Alternately, the Gaussian weighting method is an ad hoc procedure which consists in weighting each trajectory by a Gaussian-like coefficient such that the closer the final actions to integer values, the larger the coefficient. The goal of the present note is to justify such a method in the framework of classical S matrix theory.