Foundations of time-dependent density functional theory

Time-dependent density functional theory is a reformulation of quantum mechanics based upon the one-particle density as a fundamental functional variable instead of the wave-function. The density can be generated either by solving the interacting multi-particle Schrödinger equation or by the solution of a numerically more favourable non-interacting so-called Kohn-Sham equation. Most of the information about the many-body system is then found in the associated effective potential. At the heart of the application of time-dependent density functional theory is an accurate approximation of this effective potential. These approximations are applied in various fields of physics, chemistry and biology. This project aims to clarify selected topics of the mathematical and theoretical foundations of time-dependent density functional theory. The insights gained will improve the reliability of this many-particle technique. On the mathematical side, a rigorous proof of existence of time-dependent effective potentials with as few restrictions as possible will be pursued. Further, the applicant will aim for a generalisation of the extended Runge-Gross theorem. On the theoretical side, the problem of resonant interactions from a density-functional perspective will be investigated. A further objective is the construction of an approximation to the effective potential capable of describing density-dynamics of a multi-particle system resonantly interacting with an external field is. This will lead to the opportunity to reliably study problems of many-body systems resonantly interacting with an external field. Furthermore, the important properties of ``memory`` and initial state dependence of the effective potentials will be considered. An implementation of ``memory`` without integrals over all previous times will be pursued and the change of the effective potential due to different initial states will be investigated. The applicant will study the theory of non-equilibrium Green`s functions as a constructive approach to approximate the effective potential. From a many-body perturbation theory incorporating arbitrary initial correlations based on Keldysh Green`s functions, equations for the effective potential will be derived. A variational equation based on the Luttinger-Ward functional extended to arbitrary initial conditions poses one possible route. Finally, beginning at the latest with the returning phase, applications of time-dependent density functional theory and related approaches to ultra-cold quantum gases and quantum optical problems will be pursued.