On F and E, in DFT
  Rigorous mathematical foundations of density functional theory are revisited, with some use of infinitesimal (nonstandard) methods. A thorough treatment is given of basic properties of internal energy and ground-state energy functionals along with several improvements and clarifications of known results.A simple metrizable topology is constructed on the space of densities using a hierarchy of spatial partitions. This topology is very weak, but supplemented by control of internal energy, it is, in a rough sense, essentially as strong as $L^1$. Consequently, the internal energy functional $F$ is lower semicontinuous with respect to it. With separation of positive and negative parts of external potentials, very badly behaved, even infinite, positive parts can be handled. Confining potentials are thereby incorporated directly into the density functional framework. 
Paul E. Lammert
cond-mat.other math-ph math.MP quant-ph