A theoretical investigation of time-dependent Kohn–Sham equations: new proofs

ABSTRACT

In this paper, a new analysis for the existence, uniqueness, and regularity of solutions to a time-dependent Kohn–Sham equation is presented. The Kohn–Sham equation is a nonlinear integral Schrödinger equation that is of great importance in many applications in physics and computational chemistry. To deal with the time-dependent, nonlinear and non-local potentials of the Kohn–Sham equation, the analysis presented in this manuscript makes use of energy estimates, fixed-point arguments, regularization techniques, and direct estimates of the non-local potential terms. The assumptions considered for the time-dependent and nonlinear potentials make the obtained theoretical results suitable to be used also in an optimal control framework.

Keywords:

• Kohn–Sham model
• time-dependent density functional theory
• nonlinear Schrödinger equation

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35K55
• 81Q99

Acknowledgments

The first author wishes to thank Prof. Joseph W. Jerome for many fruitful discussions, for his encouragement, and for having read a first draft of this manuscript. G. Ciaramella thanks also to Dr. Maria Infusino and Stefan Hain for having read a first draft of this manuscript and provided several useful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.