On stationary Schrödinger-Poisson equations modelling an electron gas with reduced dimension

Abstract

We regard the Schrödinger–Poisson system arising from the modelling of an electron gas with reduced dimension in a bounded up to three-dimensional domain and establish the method of steepest descent. The electrostatic potentials of the iteration scheme will converge uniformly on the spatial domain. To get this result we investigate the Schrödinger operator, the Fermi level and the quantum mechanical electron density operator for square integrable electrostatic potentials. On bounded sets of potentials the Fermi level is continuous and bounded, and the electron density operator is monotone and Lipschitz continuous. As a tool we develop a Riesz–Dunford functional calculus for semibounded self-adjoint operators using paths of integration which enclose a real half-axis. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.