Existence and nonlinear stability of steady states of the Schrödinger-Poisson system
نویسندگان
چکیده
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. Acknowledgement: This research was supported by the OEAD, the International Erwin Schrödinger Institute in Vienna, the Wittgenstein 2000 prize of P. A. M. funded by the Austrian FWF, and the EU-funded TMRnetwork.
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