Mapping from Densities to Potentials in Time-Dependent Density-Functional Theory

We show that a time-dependent particle density n(rt) obtained from a given many-particle system can, under mild restrictions on the initial state, always be reproduced by an external potential v^'(rt) in a many-particle system with different two-particle interactions. Given the initial state of this other many-particle system, the potential v^'(rt) is unique up to a purely time-dependent function. As a special case we obtain the well-known Runge-Gross theorem.