  We analyze the inverse problem of Density Functional Theory using a
regularized variational method. First, we show that given $k$ and a target
density $\rho$, there exist potentials having $k^{\text{th}}$ excited mixed
states which densities are arbitrarily close to $\rho$. The state can be chosen
pure in dimension $d=1$ and without interactions, and we provide numerical and
theoretical evidence consistently leading us to conjecture that the same pure
representability result holds for $d=2$, but that the set of
potential-representable pure state densities is not dense for $d=3$. Finally,
we present an inversion algorithm taking into account degeneracies.
