On a Vlasov–Schrödinger–Poisson Model

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Abstract
A Vlasov–Schrödinger–Poisson system is studied, modeling the transport and interactions of electrons in a bidimensional electron gas. The particles are assumed to have a wave behaviour in the confinement direction (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 eigenenergies 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 eigenenergy. 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.