Existence and nonlinear stability of steady states of the Schr\"odinger-Poisson system

TL;DR

This paper proves the nonlinear stability of certain steady states in the Schrödinger-Poisson system using a dual energy-Casimir functional approach, with steady states parametrized by quantum state occupation probabilities.

Contribution

It introduces a novel dual functional method to construct and analyze steady states, establishing their nonlinear stability in the attractive Coulomb case.

Findings

Steady states are parametrized by a decreasing function of energy levels.

Existence of a unique maximizer of the dual functional as a steady state.

Nonlinear stability proven using an energy-Casimir Lyapunov functional.

Abstract

We consider the Schr\"odinger-Poisson system in the attractive (plasma physics) Coulomb case. Given a steady state from a certain class we prove its nonlinear stability, using an appropriately defined energy-Casimir functional as Lyapunov function. To obtain such steady states we start with a given Casimir functional and construct a new functional which is in some sense dual to the corresponding energy-Casimir functional. This dual functional has a unique maximizer which is a steady state of the Schr\"odinger-Poisson system and lies in the stability class. The steady states are parametrized by the equation of state, giving the occupation probabilities of the quantum states as a strictly decreasing function of their energy levels.