Current densities in density-functional theory

It is well known that any given density ρ(x) can be realized by a determinantal wave function for N particles. The question addressed here is whether any given density ρ(x) and current density j(x) can be simultaneously realized by a (finite kinetic energy) determinantal wave function. In case the velocity field v(x)=j(x)/ρ(x) is curlfree, we provide a solution for all N, and we provide an explicit upper bound for the energy. If the velocity field is not curl-free, there is a finite energy solution for all N≥4, but we do not provide an explicit energy bound in this case. For N=2 we provide an example of a non-curl-free velocity field for which there is a solution and an example for which there is no solution. The case N=3 with a non-curl-free velocity field is left open.