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

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

2001-01-18

arxiv.org/abs/math-ph/0101020v1

Journal of Statistical Physics, 106 (5-6), 1221-1239 (2002)

Physics

Mathematical Physics

16 pages

Scientific paper

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.

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