Invertibility of retarded response functions for Laplace transformable potentials: application to one-body reduced density matrix functional theory

TL;DR

This paper establishes a rigorous invertibility theorem for response functions in time-dependent and static regimes under Laplace transformability, providing foundational insights for one-body reduced density matrix functional theory and potential uniqueness.

Contribution

It introduces a novel invertibility theorem for response functions, enabling a complete classification of potential non-uniqueness in 1RDM functional theory.

Findings

The theorem applies to both time-dependent and static response functions.

It provides a foundation for density-functional-like theories in linear response.

A complete classification of potential non-uniqueness in 1RDM theory is achieved.

Abstract

A theorem for the invertibility of arbitrary response functions is presented under the following conditions: the time-dependence of the potentials should be Laplace transformable and the initial state should be a ground state, though it might be degenerate. This theorem provides a rigorous foundation for all density-functional-like theories in the time-dependent linear response regime. Especially for time-dependent one-body reduced density matrix (1RDM) functional theory this is an important step forward, since a solid foundation has currently been lacking. The theorem is equally valid for static response functions in the non-degenerate case, so can be used to characterize the uniqueness of the potential in the ground state version of the corresponding density-functional-like theory. Such a classification of the uniqueness of the non-local potential in ground state 1RDM functional theory has been lacking for decades. With the aid of presented invertibility theorem presented here, a complete classification of the non-uniqueness of the non-local potential in 1RDM functional theory can be given for the first time.