About a one-dimensional stationary Schrödinger-Poisson system with Kohn-Sham potential
- Kaiser, Hans-Christoph
- Rehberg, Joachim
2010 Mathematics Subject Classification
- 35B45 35D05 35D10 35J05 35J10 35J20 35J60 35J65 35Q40 47A55 47A60 47A75 47H05 47H10 47H15
- stationary Schrödinger-Poisson system, Kohn-Sham system, Schrödinger operator with mixed boundary conditions, quantum mechanical particle density operator, nonlinear Poisson equations, simulation of electronic devices, electron gas with reduced dimension, nanoelectronics
The stationary Schrödinger-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 nano-structure. We regard both Poisson's and Schrödinger's equation with mixed boundary conditions and discontinuous coefficients. Without an exchange-correlation potential the Schrödinger-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 Schrödinger operator is anti-monotone and boundedly Lipschitz continuous. The solution of the Schrödinger-Poisson system without an exchange-correlation 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 Kohn-Sham system has at least one solution. If the exchange-correlation potential 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 L1 into L2, but into L1.
- Z. Angew. Math. Phys., 50 (1999), No. 3, pp. 423-458.