Runge–Gross density-potential mapping conjecture

Let $\psi_0$ be the initial wave function of an $N$-particle system, with initial density $n_0$ on a domain $\Omega$. Assume that the external potential is time-analytic, that $n_0$ tends to zero at the boundary of $\Omega$ and satisfies $n_0>0$ almost everywhere, and that the potential and $\psi_0$ are sufficiently smooth in space. The Runge–Gross conjecture. There is a one-to-one mapping between densities and external potentials, up to addition of a function depending only on time. This is the foundational time-dependent density-potential mapping in TDDFT; the source presents it only as a conjectural formulation and does not establish the full stated result.