Spectrum of the linearized Vlasov--Poisson equation around steady states from galactic dynamics

TL;DR

This paper analyzes the spectral properties of the linearized Vlasov-Poisson equation in gravitational systems, identifying the spectrum and conditions for oscillations, including external potentials.

Contribution

It provides a detailed spectral analysis of the linearized Vlasov-Poisson equation around steady states, including criteria for oscillating modes and the impact of external potentials.

Findings

Identified the absolutely continuous spectrum of the linearized operator.

Established criteria for the existence of oscillating modes.

Estimated the number of such modes under various conditions.

Abstract

We study the linearized Vlasov-Poisson equation in the gravitational case around steady states that are decreasing and continuous functions of the energy. We identify the absolutely continuous spectrum and give criteria for the existence of oscillating modes and estimate their number. Our method allows us to take into account an attractive external potential.