About a 1 D Stationary Schrödinger Poisson System

The stationary SchrrdingerrPoisson system with a selffconsistent eeective KohnnSham potential is a system of PDEs for the electrostatic potential and the envelopes of wave functions deening the quantum mechanical carrier densities in a semiconductor nanostructure. We regard both Poisson's and Schrrdinger's equation with mixed boundary conditions and discontinu-ous coeecients. Without an exchangeecorrelation potential the Schrrdingerr Poisson system is a nonlinear Poisson equation in the dual of a Sobolev space which is determined by the boundary conditions imposed on the electrostatic potential. The nonlinear Poisson operator involved is strongly monotone and boundedly Lipschitz continuous, hence the operator equation has a unique solution. The proof rests upon the following property: the quantum mechanical carrier density operator depending on the potential of the deen-ing Schrrdinger operator is antiimonotone and boundedly Lipschitz continuous. The solution of the SchrrdingerrPoisson system without an exchangee correlation potential depends boundedly Lipschitz continuous on the reference potential in Schrrdinger's operator. By means of this relation a xed point mapping for the vector of quantum mechanical carrier densities is set up which meets the conditions in Schauder's xed point theorem. Hence, the KohnnSham system has at least one solution. If the exchangeecorrelation potential is suuciently small, then the solution of the KohnnSham system is unique. Moreover, properties of the solution as bounds for its values and its oscillation can be expressed in terms of the data of the problem. The onee dimensional case requires special treatment, because in general the physically relevant exchangeecorrelation potentials are not Lipschitz continuous map-pings from the space L