About a onedimensional stationary SchrödingerPoisson system with KohnSham potential
Abstract
The stationary SchrödingerPoisson system with a selfconsistent effective KohnSham potential is a system of PDEs for the electrostatic potential and the envelopes of wave functions defining the quantum mechanical carrier densities in a semiconductor nanostructure. We regard both Poisson's and Schrödinger's equation with mixed boundary conditions and discontinuous coefficients. Without an exchangecorrelation potential the SchrödingerPoisson 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 defining Schrödinger operator is antimonotone and boundedly Lipschitz continuous. The solution of the SchrödingerPoisson system without an exchangecorrelation potential depends boundedly Lipschitz continuous on the reference potential in Schrödinger's operator. By means of this relation a fixed point mapping for the vector of quantum mechanical carrier densities is set up which meets the conditions in Schauder's fixed point theorem. Hence, the KohnSham system has at least one solution. If the exchangecorrelation potential is boundedly Lipschitz continuous and the local Lipschitz constant is sufficiently small, then the solution of the KohnSham 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 onedimensional case requires special treatment, because in general the physically relevant exchangecorrelation potentials are not Lipschitz continuous mappings from the space L^{1} into L^{2}, but into L^{1}.
 Publication:

Zeitschrift Angewandte Mathematik und Physik
 Pub Date:
 1999
 DOI:
 10.1007/PL00001496
 Bibcode:
 1999ZaMP...50..423K