BOLTZMANN DISTRIBUTED QUANTUM STEADY-STATES AND THEIR CLASSICAL LIMIT

We prove existence and uniqueness of Boltzmann distributed quantum ste ady states of an electron ensemble which moves under the actions of th e self-consistent Coulomb potential and of an external potential. The case of the particles confined to a three-dimensional bounded domain a nd the whole space case in R3 is analyzed. Also, we prove the existenc e of the classical limit. As the Planck constant tends to zero we obta in a Maxwellian eqilibrium solution of the Vlasov-Poisson problem.