Existence and nonlinear stability of steady states of the Schrödinger-Poisson system

2001/01/18 by Peter A. Markowich, Gerhard Rein, Gershon Wolansky
Physics and Astronomy · Mathematics · #math-ph #math.AP #math.MP #msc:35Q40 #msc:35Q55 #msc:35B35 #msc:82D10

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published as Journal of Statistical Physics, 106 (5-6), 1221-1239 (2002) · 16 pages

arxiv created 2001/01/18 · arxiv updated 2009/11/30

Abstract

We consider the Schrödinger-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 Schrödinger-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.

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