Existence, Uniqueness, and Construction of the Density-Potential Mapping in Time-Dependent Density-Functional Theory

TL;DR

This paper reviews the theoretical foundations and conditions for the density-potential mapping in time-dependent density-functional theory, including mathematical details, fixed-point procedures, and extensions to vector potentials and photons.

Contribution

It provides a comprehensive analysis of the existence, uniqueness, and construction methods for the density-potential mapping, including new conditions and extensions beyond previous work.

Findings

Derived conditions for the density-potential mapping in quantum mechanics.

Presented a fixed-point iterative procedure for constructing potentials from densities.

Extended the mapping framework to include vector potentials and photon interactions.

Abstract

In this work we review the mapping from densities to potentials in quantum mechanics, which is the basic building block of time-dependent density-functional theory and the Kohn-Sham construction. We first present detailed conditions such that a mapping from potentials to densities is defined by solving the time-dependent Schr\"odinger equation. We specifically discuss intricacies connected with the unboundedness of the Hamiltonian and derive the local-force equation. This equation is then used to set up an iterative sequence that determines a potential that generates a specified density via time propagation of an initial state. This fixed-point procedure needs the invertibility of a certain Sturm-Liouville problem, which we discuss for different situations. Based on these considerations we then present a discussion of the famous Runge-Gross theorem which provides a density-potential mapping for time-analytic potentials. Further we give conditions such that the general fixed-point approach is well-defined and converges under certain assumptions. Then the application of such a fixed-point procedure to lattice Hamiltonians is discussed and the numerical realization of the density-potential mapping is shown. We conclude by presenting an extension of the density-potential mapping to include vector-potentials and photons.