It is well known that any given density $\rho(x)$ can be realized by a finite energy determinantal wave function for $N$ particles. The question addressed here is whether any given density $\rho(x) $ and current density $j(x)$ can be simultaneously realized by a (finite kinetic energy) determinantal wave function. In case the velocity field $v(x) =j(x)/\rho(x)$ is curl free, 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 $N\ge 4$, which uses the recent smooth extension of the Hobby-Rice theorem, 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. (Joint work with Robert Schrader.)
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