UPSILON-REPRESENTABILITY FOR SYSTEMS WITH LOW DEGENERACY

We consider the upsilon-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(lambda), E2(lambda), and E3(lambda) are three energy levels associated with some local potential V-lambda (r) that is a continuous function of lambda = (lambda-1,lambda-2,lambda-3) over all possible points lambda, where lambda-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 lambda 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 upsilon-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 upsilon-representable for small N (N less-than-or-equal-to 14 and N less-than-or-equal-to 9 for three- and two-dimensional cases, respectively). The implications for density-functional theory are discussed.