Consistency of Local Density Approximations and Quantum Corrections for
Time-Dependent Quantum Systems
By: Joseph W. Jerome
Time dependent quantum systems are the subject of intense inquiry, in
mathematics, science, and engineering,
particularly at the atomic and molecular levels.
In 1984, Runge and Gross introduced time dependent density functional
theory (TDDFT), a non-interacting electron model, which predicts
charge exactly. An exchange-correlation potential is included in the
Hamiltonian to enforce this property.
We have previously investigated such systems
on bounded domains for Kohn-Sham potentials by use of evolution operators
and fixed point theorems.
In this article, motivated by usage in the physics community, we consider
local density approximations (LDA) for building
the exchange-correlation potential,
as part of a set of quantum corrections.
Existence and uniqueness of solutions are established separately
within a framework for general quantum corrections,
including time-history corrections and ionic Coulomb potentials,
in addition to LDA potentials.
In summary, we are able to demonstrate a unique weak solution,
on an arbitrary time
interval, for a general class of quantum corrections, including
those typically used in numerical simulations of the model.
This paper has appeared in the Journal of Applicable Analysis (corrected
version):
https://doi.org/10.1080/00036811.2020.1831163.