Scattering map for the Vlasov--Poisson system with a repulsive harmonic potential

TL;DR

This paper proves the scattering and wave operators for the Vlasov--Poisson system with a repulsive harmonic potential across multiple dimensions, introducing a lens transform and simplifying initial data assumptions.

Contribution

It introduces the construction of wave operators and the lens transform for the Vlasov--Poisson system, providing a new approach and relaxing initial data conditions.

Findings

Proved (modified) scattering of solutions in dimensions d≥2.

Constructed wave operators for the system.

Introduced a lens transform specific to the Vlasov--Poisson system.

Abstract

We consider the Vlasov--Poisson system with a repulsive harmonic potential and prove the (modified) scattering of solutions, as well as the existence of wave operators, in any spatial dimension $d\geq 2$. The main novelty of this work is the construction of the wave operators and the introduction of the lens transform for the Vlasov--Poisson system. In addition, we provide a new and simpler proof that relaxes the assumptions on the initial data compared with those in Bigorgne, Velozo Ruiz, and Velozo Ruiz (2025) and Velozo Ruiz, and Velozo Ruiz (2024).