  The approximation of fixed points by numerical fixed points was presented in
the elegant monograph of Krasnosel'skii et al. (1972). The theory, both in its
formulation and implementation, requires a differential operator calculus, so
that its actual application has been selective. The writer and Kerkhoven
demonstrated this for the semiconductor drift-diffusion model in 1991. In this
article, we show that the theory can be applied to time dependent quantum
systems on bounded domains, via evolution operators. In addition to the kinetic
operator term, the Hamiltonian includes both an external time dependent
potential and the classical nonlinear Hartree potential. Our result can be
paraphrased as follows: For a sequence of Galerkin subspaces, and the
Hamiltonian just described, a uniquely defined sequence of Faedo-Galerkin
solutions exists; it converges in Sobolev space, uniformly in time, at the
maximal rate given by the projection operators.
