Classical scattering at low energies

Abstract

For a class of negative slowly decaying potentials including the attractive Coulombic one we study the classical scattering theory in the low-energy regime. We construct a (continuous) family of classical orbits parametrized by initial position $x\in {\mathbb{R}}^d$, final direction $\omega \in S^{d-1}$ of escape (to infinity) and the energy $\lambda \ge 0$, yielding a complete classification of the set of outgoing scattering orbits. The construction is given in the outgoing part of phase-space (a similar construction may be done in the incoming part of phase-space). For fixed $\omega \in S^{d-1}$ and $\lambda \ge 0$ the collection of constructed orbits constitutes a smooth manifold that we show is Lagrangian. The family of those Lagrangians can be used to study the quantum mechanical scattering theory in the low-energy regime for the class of potentials considered here. We devote this study to a subsequent paper.

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• Quantum scattering at low energies

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• Quantum scattering at low energies