Classical scattering at low energies

TL;DR

This paper develops a comprehensive classical scattering theory at low energies for certain negative, slowly decaying potentials, including Coulombic, by constructing a family of classical orbits and analyzing their properties.

Contribution

It introduces a complete classification of outgoing classical orbits at low energies for a broad class of potentials, including Coulombic, and shows these form smooth Lagrangian manifolds.

Findings

Constructed a continuous family of classical orbits parametrized by initial position, final direction, and energy.

Proved that the collection of orbits for fixed direction and energy forms a smooth Lagrangian manifold.

Provided a foundation for studying quantum scattering at low energies for these potentials.

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 $\omega\in S^{d-1}$ of escape (to infinity) and the energy $\lambda\geq 0$, 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 $\omega\in S^{d-1}$ and $\lambda\geq 0$ 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.