The Non-Trapping Degree of Scattering

TL;DR

This paper studies classical potential scattering, introducing a topological degree for non-trapping energies, explicitly calculating it for all potentials, and exploring implications for the structure of Hill's Region and symbolic dynamics.

Contribution

It explicitly calculates the topological degree for all potentials and links non-trapping conditions to the topology of Hill's Region, extending understanding of multi-obstacle scattering.

Findings

The topological degree deg(E) is less than 2 for non-trapping energies.

For bounded potentials, the Hill's Region boundary is either empty or a sphere.

Decomposition of potentials allows embedding symbolic dynamics in multi-obstacle scattering.

Abstract

We consider classical potential scattering. If no orbit is trapped at energy E, the Hamiltonian dynamics defines an integer-valued topological degree. This can be calculated explicitly and be used for symbolic dynamics of multi-obstacle scattering. If the potential is bounded, then in the non-trapping case the boundary of Hill's Region is empty or homeomorphic to a sphere. We consider classical potential scattering. If at energy E no orbit is trapped, the Hamiltonian dynamics defines an integer-valued topological degree deg(E) < 2. This is calculated explicitly for all potentials, and exactly the integers < 2 are shown to occur for suitable potentials. The non-trapping condition is restrictive in the sense that for a bounded potential it is shown to imply that the boundary of Hill's Region in configuration space is either empty or homeomorphic to a sphere. However, in many situations one can decompose a potential into a sum of non-trapping potentials with non-trivial degree and embed symbolic dynamics of multi-obstacle scattering. This comprises a large number of earlier results, obtained by different authors on multi-obstacle scattering.