Qualitative Aspects of Classical Potential Scattering

We derive criteria for the existence of trapped orbits (orbits which are scattering in the past and bounded in the future). Such orbits exist if the boundary of Hill's region is non-empty and not homeomorphic to a sphere. For non-trapping energies we introduce a topological degree which can be non-trivial for low energies, and for Coulombic and other singular potentials. A sum of non-trapping potentials of disjoint support is trapping iff at least two of them have non-trivial degree. For $d \geqslant 2$ dimensions the potential vanishes if for any energy above the non-trapping threshold the classical differential cross section is a continuous function of the asymptotic directions.