The Quantum Faedo-Galerkin Equation: Evolution Operator and Time
Discretization
By: Joseph W. Jerome
Time dependent quantum systems have become indispensable in science and
nanotechnology. Disciplines including chemical physics and electrical
engineering have used approximate evolution operators to solve these
systems for targeted physical quantities. Here, we discuss the
approximation of closed time dependent quantum systems on bounded domains
via evolution operators. The work builds upon the use of weak solutions,
which includes a framework for the evolution operator based upon dual
spaces. We are able to derive the corresponding Faedo-Galerkin equation as
well as its time discretization, yielding a fully discrete theory. We
obtain corresponding approximation estimates. These estimates make no
regularity assumptions on the weak solutions other than their inherent
properties. Of necessity, the estimates are in the dual norm, which is
natural for weak solutions. This appears to be a novel aspect of this
approach.
This paper has appeared
in the Journal of
Numerical Functional Analysis and Optimization (vol. 38 (2017), 590-601)
and has been published online:
http://dx.doi.org/10.1080/01630563.2016.1252393.
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