  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.
