Time Dependent Closed Quantum Systems: Nonlinear Kohn-Sham Potential
Operators and Weak Solutions
By: Joseph W. Jerome
We discuss
time dependent quantum systems on bounded domains
from the perspective of nonlinear, time-dependent
potentials. The time dependence of the Kohn-Sham potentials distinguishes
this study from that of the so-called nonlinear Schr\"{o}dinger equation,
much studied in the mathematical community.
We are interested in establishing a framework for potentials
including the external potential, the Hartree potential
and the exchange correlation potential
as occur in
time dependent density functional theory (TDDFT).
As in previous work, we make use of the
time-ordered evolution
operator.
A departure from the previous work is the
use of weak solutions for the nonlinear model; this necessitates a new
framework for the evolution operator based upon dual spaces.
We are able to obtain unique global solutions.
The author thanks Eric Polizzi for discussions
leading to the incorporation of a version of the exchange
correlation potential in the model.
This paper has been published in the Journal of
Mathematical Analysis and Applications: vol. 429 (2015), pp. 995--1006.
This preprint clarifies definitions in the journal article; we
have removed the complex conjugation on the test functions involved in the
definition of A(t) in section 2.1 and in the definition of S in section
2.2. Also, we have added a brief appendix regarding the
smoothing hypothesis on the exchange-correlation potential.
It can be viewed in the following format: