Existence and nonlinear stability of stationary states for the semi-relativistic Schr\"odinger-Poisson system

TL;DR

This paper investigates the existence and nonlinear stability of stationary states in the semi-relativistic Schr"odinger-Poisson system, extending previous results to higher regularity spaces using the energy-Casimir method.

Contribution

It establishes the existence and nonlinear stability of stationary states and generalizes global well-posedness to higher regularity spaces.

Findings

Proved existence of stationary states.

Demonstrated nonlinear stability of these states.

Extended well-posedness results to higher regularity spaces.

Abstract

We study the stationary states of the semi-relativistic Schr\"odinger-Poisson system in the repulsive (plasma physics) Coulomb case. In particular, we establish the existence and the nonlinear stability of a wide class of stationary states by means of the energy-Casimir method. We generalize the global well-posedness result of our previous work for the semi-relativistic Schr\"odinger-Poisson system to spaces with higher regularity.