Reducing scattering problems under cone potentials to normal form by global canonical transformations

TL;DR

This paper introduces a method to transform a class of Hamiltonian scattering systems with cone potentials into a normal form using a global canonical transformation, facilitating analysis of their asymptotic behavior.

Contribution

The authors develop a framework for reducing Hamiltonian systems with cone potentials to a normal form via a global canonical transformation based on asymptotic trajectory properties.

Findings

Systems can be transformed to normal form under certain geometric conditions.

The transformation is robust under small perturbations of the potential.

Applicable to systems with diverse asymptotic behaviors.

Abstract

We introduce a class of Hamiltonian scattering systems which can be reduced to the "normal form" $\dot P=0$, $\dot Q=P$, by means of a global canonical transformation $ (P,Q)=A(p,q), p,q\in R^n$, defined through asymptotic properties of the trajectories. These systems are obtained requiring certain geometrical conditions on $\dot p=-\nabla V(q)$, $\dot q=p$, where $V$ is a bounded below "cone potential", i.e., the force $-\nabla V(q)$ always belongs to a closed convex cone which contains no straight lines. We can deal with very different asymptotic behaviours of the potential and the potential can undergo small perturbations in any arbitrary compact set without losing the existence and the properties of $A$.