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Classical scattering at low energies

J. Derezinski·E. Skibsted
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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 $x\in \R^d$, final direction $ω\in S^{d-1}$ of escape (to infinity) and the energy $λ\geq 0$, 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 $ω\in S^{d-1}$ and $λ\geq 0$ 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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J. Derezinski, E. Skibsted. 2006-04-18. Classical scattering at low energies. https://arxiv.org/abs/math-ph/0604042

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