Coarse-grained V-representability

The unsolved problem of determining which densities are ground state densities of an interacting electron system in some external potential is important to the foundations of density functional theory. A coarse-grained version of this ensemble V-representability problem is shown to be thoroughly tractable. Averaging the density of an interacting electron system over the cells of a regular partition of space produces a coarse-grained density. It is proved that every strictly positive coarse-grained density is coarse-grained ensemble V-representable: there is a unique potential, constant over each cell of the partition, which has a ground state with the prescribed coarse-grained density. For a system confined to a box, the (coarse-grained) Lieb [Intl. J. Quantum Chem. 24, 243 (1983)] functional is also shown to be Gateaux differentiable. All results extend to open systems.