Abstract
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, 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.
| Original language | English (US) |
|---|---|
| Journal | Applicable Analysis |
| DOIs | |
| State | Published - Jan 1 2019 |
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Keywords
- 35Q41
- 81Q05
- Siegfried Carl
- Time-dependent quantum systems
- local density approximation
- quantum corrections
- smoothing
- time-history
ASJC Scopus subject areas
- Analysis
- Applied Mathematics
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Consistency of local density approximations and quantum corrections for time-dependent quantum systems. / Jerome, Joseph W.
In: Applicable Analysis, 01.01.2019.Research output: Contribution to journal › Article
TY - JOUR
T1 - Consistency of local density approximations and quantum corrections for time-dependent quantum systems
AU - Jerome, Joseph W.
PY - 2019/1/1
Y1 - 2019/1/1
N2 - 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, 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.
AB - 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, 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.
KW - 35Q41
KW - 81Q05
KW - Siegfried Carl
KW - Time-dependent quantum systems
KW - local density approximation
KW - quantum corrections
KW - smoothing
KW - time-history
UR - http://www.scopus.com/inward/record.url?scp=85060847041&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85060847041&partnerID=8YFLogxK
U2 - 10.1080/00036811.2019.1573988
DO - 10.1080/00036811.2019.1573988
M3 - Article
AN - SCOPUS:85060847041
JO - Applicable Analysis
JF - Applicable Analysis
SN - 0003-6811
ER -