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)

TL;DR

This paper discusses open theoretical questions in density functional theory, focusing on the existence of derivatives of the Lieb functional and the non-interacting $v$-representability conjecture, providing reflections without conclusive answers.

Contribution

It offers a collection of thoughts and reminders on unresolved issues in DFT, aiming to stimulate further research and reflection.

Findings

No conclusive proof of derivative existence of Lieb functional

Insights on non-interacting $v$-representability conjecture

Provides a reflective overview without definitive results

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).