Global fixed point proof of time-dependent density-functional theory

TL;DR

This paper presents a new, more general proof of the core principles of time-dependent density-functional theory using a fixed point approach, avoiding previous restrictions and broadening its applicability.

Contribution

It reformulates the proofs of uniqueness and existence in TDDFT as a global fixed point problem, extending the theoretical foundation beyond earlier limitations.

Findings

Fixed point approach proves unique potential for a given density.

Existence of densities is established under bounded response function norms.

Avoids restrictions of time Taylor-expandability in traditional proofs.

Abstract

We reformulate and generalize the uniqueness and existence proofs of time-dependent density-functional theory. The central idea is to restate the fundamental one-to-one correspondence between densities and potentials as a global fixed point question for potentials on a given time-interval. We show that the unique fixed point, i.e. the unique potential generating a given density, is reached as the limiting point of an iterative procedure. The one-to-one correspondence between densities and potentials is a straightforward result provided that the response function of the divergence of the internal forces is bounded. The existence, i.e. the v-representability of a density, can be proven as well provided that the operator norms of the response functions of the members of the iterative sequence of potentials have an upper bound. The densities under consideration have second time-derivatives that are required to satisfy a condition slightly weaker than being square-integrable. This approach avoids the usual restrictions of Taylor-expandability in time of the uniqueness theorem by Runge and Gross [Phys.Rev.Lett.52, 997 (1984)] and of the existence theorem by van Leeuwen [Phys.Rev.Lett. 82, 3863 (1999)]. Owing to its generality, the proof not only answers basic questions in density-functional theory but also has potential implications in other fields of physics.