Van Leeuwen's extended Runge–Gross conjecture for interaction replacement

Let a system with initial wave function $\psi_0$ evolve under interaction potential $w$ and external potential $v$. For a different interaction potential $w'$, there should be a unique external potential $v'$ and initial wave function $\psi'_0$ such that the new system reproduces exactly the same density $n$. Assume that $n$ is time-analytic and sufficiently smooth in space. The extended Runge–Gross conjecture. Such a choice of $v'$ and $\psi'_0$ is possible and unique. This is the proposed affirmative answer to non-interacting $v$-representability and underlies the Kohn–Sham construction; the source notes that its proof is formal and that further mathematical work on the relevant domains was needed.