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

Peter A. Markowich·Gerhard Rein·Gershon Wolansky
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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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Peter A. Markowich, Gerhard Rein, Gershon Wolansky. 2001-01-18. Existence and nonlinear stability of steady states of the Schrödinger-Poisson system. https://arxiv.org/abs/math-ph/0101020

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