v-representability for systems with low degeneracy

We consider the v-representability of the particle density for a noninteracting system of spinless fermions by introducing the idea of proper order of a set of energy levels. It is shown that if ${\mathit{E}}_1$(λ), ${\mathit{E}}_2$(λ), and ${\mathit{E}}_3$(λ) are three energy levels associated with some local potential ${\mathit{V}}_{\lambda }$(r) that is a continuous function of λ=(${\lambda }_1$,${\lambda }_2$,${\lambda }_3$) over all possible points λ, where ${\lambda }_{\mathit{i}}$ is the occupation number of the ith state and M=${\lambda }_1$+${\lambda }_2$+${\lambda }_3$ is the total number of particles distributed over the three levels, then there must be at least one λ for which the three levels are in so-called proper order, in which the levels below the highest occupied level are filled. This result provides a basis for the proof of ensemble v-representability of some N-particle density for which the ground-state degeneracy of the system is no more than three. As examples, three- and two-dimensional central systems are examined, and an N-particle central density is shown to be ensemble v-representable for small N (N≤14 and N≤9 for three- and two-dimensional cases, respectively). The implications for density-functional theory are discussed.