# A theoretical investigation of time-dependent Kohn-Sham equations

## TL;DR

This paper provides a rigorous theoretical analysis of the existence, uniqueness, and regularity of solutions to the time-dependent Kohn-Sham equations, fundamental in quantum physics and density functional theory.

## Contribution

It offers the first comprehensive mathematical investigation into the well-posedness of these nonlinear coupled equations, including control and inhomogeneity effects.

## Key findings

- Proved existence of solutions under certain conditions
- Established uniqueness of solutions
- Analyzed regularity properties of solutions

## Abstract

In this work, the existence, uniqueness and regularity of solutions to the time-dependent Kohn-Sham equations are investigated. The Kohn-Sham equations are a system of nonlinear coupled Schr\"odinger equations that describe multi-particle quantum systems in the framework of the time dependent density functional theory. In view of applications with control problems, the presence of a control function and of an inhomogeneity are also taken into account.