Notes on open questions within density functional theory (existence of a derivative of the Lieb functional in a restricted sense and non-interacting $v$-representability)

Abstract

Density functional theory together with the Kohn-Sham scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary ``non-interacting'' system (one-body with a local potential). However, theoretical questions remain open. An important one is related to the existence of a derivative of the Lieb functional in some restricted (weak) sense (as pointed out by Lammert). Then, a further one would be related to the validity of the ``non-interacting $v$-representability'' conjecture. We gather here elements on these questions, providing reminders and thoughts that are non-conclusive but hopefully contribute to the reflection. This document must be considered as notes (it does not represent an article).