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Conference on "Mathematical properties of large quantum systems"
Organizers: Maria J. Esteban (CNRS), Mathieu Lewin (CNRS), Robert Seiringer (McGill, Montréal) & Jan Philip Solovej (Copenhagen)


Elliott H. Lieb (Princeton, USA): "Current Densities in Density Functional Theory"
Monday 17 June 2013, 09:30 - 10:15
Amphi Hermite


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 $Nge 4$, which uses the recent smooth extension of the Hobby-Rice theorem, but we do not provide an explicit energy bound in this case. For the case $N=1$ the curl freeness is necessary for a solution to exist. 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.)