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

• Received 25 November 1998

DOI:https://doi.org/10.1103/PhysRevLett.82.3863