Long-time behaviour of time-dependent density functional theory

TL;DR

This paper analyzes the long-time behavior of the time-dependent Kohn-Sham equations in density functional theory, proving global existence and scattering under certain conditions using novel mathematical techniques.

Contribution

It introduces new methods to establish long-time existence and scattering results for the time-dependent Kohn-Sham equations in DFT, assuming weak self-interactions.

Findings

Proves global existence of solutions in the short-range regime

Establishes scattering behavior for the time-dependent KS equations

Develops new mathematical tools compatible with DFT structure

Abstract

The density functional theory (DFT) is a remarkably successful theory of electronic structure of matter. At the foundation of this theory lies the Kohn-Sham (KS) equation. In this paper, we describe the long-time behaviour of the time-dependent KS equation. Assuming weak self-interactions, we prove global existence and scattering in (almost) the full "short-range" regime. This is achieved with new and simple techniques, naturally compatible with the structure of the DFT and involving commutator vector fields and non-abelian versions of Sobolev-Klainerman-type spaces and inequalities.