On the regularization of the collision solutions of the one-center problem with weak forces

We study the possible regularization of collision solutions for one centre problems with a weak singularity. In the case of logarithmic singularities, we consider the method of regularization via smoothing of the potential. With this technique, we prove that the extended flow, where collision solutions are replaced with a transmission trajectory, is continuous, though not differentiable, with respect to the initial data.