##
A Tight Nonlinear Approximation Theory
for Time Dependent Closed Quantum
Systems

#### By: Joseph W. Jerome

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.

This paper has been published online, February 19, 2018, in the Journal of
Numerical Mathematics.
It can also be accessed at: arXiv: 1709.09063.