On a Vlasov-Schrödinger-Poisson model

Abstract Weak solutions of a Vlasov-Schrödinger-Poisson system are shown to exist in the stationary and time-dependent situations. This system models the transport and interaction of electrons in a bidimensional electron gas. The particles are assumed to have a wave behaviour in the confinement directions (z) and to behave like point particles in the directions parallel to the electron gas (x). For each fixed x and at each time t, the eigenfunctions and the eigen-energies of the Schrödinger operator in the z are computed. The occupation number of each eigenfunction is computed through the resolution of a Vlasov equation in the x direction, the force field being the gradient of the eigen-energy. The whole system is coupled to the Poisson equation for the electrostatic interaction. Existence of weak solutions is shown for boundary value problems in the stationary and time-dependent regimes. The proofs rely on the one hand on the study of quasistatic Schrödinger-Poisson systems and on the other hand on an energy estimate (for the time-dependent case) and on supersolution techniques (for the stationary case).