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

TL;DR

This paper provides a rigorous mathematical analysis of the time-dependent Kohn-Sham equations, establishing conditions for existence, uniqueness, and regularity of solutions, which are crucial for applications in physics and chemistry.

Contribution

It introduces new proofs for the well-posedness of the nonlinear, non-local time-dependent Kohn-Sham equations using advanced mathematical techniques.

Findings

Proved existence and uniqueness of solutions under broad conditions

Established regularity results for solutions

Applicable to optimal control frameworks

Abstract

In this paper, a new analysis for existence, uniqueness, and regularity of solutions to a time-dependent Kohn-Sham equation is presented. The Kohn-Sham equation is a nonlinear integral Schroedinger 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.