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Qualitative aspects of classical potential scattering. (English) Zbl 0982.81054

Summary: 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 nontrivial 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 nontrivial degree. For \(d\geq 2\) 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.

MSC:

81U05 \(2\)-body potential quantum scattering theory
70F05 Two-body problems
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
81V35 Nuclear physics
81V45 Atomic physics
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