About a one-dimensional stationary Schrödinger-Poisson system with Kohn-Sham potential

Abstract

The stationary Schrodinger-Poisson system with a self-consistent effective Kohn-Sham 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 Schrodinger's equation with mixed boundary conditions and discontinuous coefficients. Without an exchange-correlation potential the Schrodinger-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 defining Schrodinger operator is anti-monotone and boundedly Lipschitz continuous. The solution of the Schrodinger-Poisson system without an exchange-correlation potential depends boundedly Lipschitz continuous on the reference potential in Schrodinger'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 Kohn-Sham system has at least one solution. If the exchange-correlation potential is boundedly Lipschitz continuous and the local Lipschitz constant is sufficiently small, then the solution of the Kohn-Sham 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 one-dimensional case requires special treatment, because in general the physically relevant exchange-correlation potentials are not Lipschitz continuous mappings from the space \(L^1\) into \(L^2\), but into \(L^1\).