Liouville-Arnold integrability for scattering under cone potentials

TL;DR

This paper demonstrates the Liouville-Arnold integrability of a broad class of scattering systems with cone potentials, including long-range interactions, and shows their stability under small perturbations.

Contribution

It extends integrability results to new systems with cone potentials, including long-range interactions, and establishes their robustness to perturbations.

Findings

Proved integrability for systems with cone potentials.

Included systems with very-long-range interactions.

Showed stability of integrability under small potential perturbations.

Abstract

The problem of scattering of particles on the line with repulsive interactions, gives rise to some well-known integrable Hamiltonian systems, for example, the nonperiodic Toda lattice or Calogero's system. The aim of this note is to outline our researches which proved the integrability of a much larger class of systems, including some that had never been considered, such as the scattering with very-long-range interaction potential. The integrability of all these systems survives any small enough perturbation of the potential in an arbitrary compact set. Our framework is based on the concept of cone potentials, as defined below, which include the scattering on the line as a particular case.