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 E₁({λ}), E₂({λ}), and E₃({λ}) are three energy levels associated with some local potential V_λ(r) that is a continuous function of {λ}=(λ₁,λ₂,λ₃) over all possible points {λ}, where λᵢ is the occupation number of the ith state and M=λ₁+λ₂+λ₃ 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.