Current densities in density-functional theory

It is well known that any given density $\rho \left(\mathbf{x}\right)$ can be realized by a determinantal wave function for $N$ particles. The question addressed here is whether any given density $\rho \left(\mathbf{x}\right)$ and current density $\mathbf{j}\left(\mathbf{x}\right)$ can be simultaneously realized by a (finite kinetic energy) determinantal wave function. In case the velocity field $\mathbf{v}\left(\mathbf{x}\right)=\mathbf{j}\left(\mathbf{x}\right)/\rho \left(\mathbf{x}\right)$ 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 $N\ge 4$, 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.