Long-Time Behaviour of Time-Dependent Density Functional Theory

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.