Given a general, N-particle Hamiltonian operator, analogs of the Hohenberg-Kohn theorem are derived for functions that are more general than the particle density, including density matrices and the diagonal elements thereof. The generalization of Lieb's Legendre transform ansatz to the generalized Hohenberg-Kohn functional not only solves the upsilon-representability problem for these entities, but, more importantly, also solves the N-representability problem. Restricting the range of operators explored by the Legendre transform leads to a lower bound on the true functional. If all the operators of interest are incorporated in the restricted maximization, however, the variational principle dictates that exact results are obtained for the systems of interest. This might have important implications for practical work not only for density matrices but also for density functionals. A follow-up paper will present a useful alternative approach to the upsilon- and N-representability problems based on the constrained search formalism.
It is shown that if
n(r) is the discrete density on a lattice (enclosed in a finite box) associated with a nondegenerate ground state in an external potential
v(r) (i.e., is "
v-representable"), then the density $n(r)+\ensuremath{\mu}m(r)$, with
m(r) arbitrary (apart from trivial constraints) and $\ensuremath{\mu}$ small enough, is also associated with a nondegenerate ground state in an external potential ${v}^{\ensuremath{'}}(r)$ near
v(r); i.e., $n(r)+\ensuremath{\mu}m(r)$ is also
v-representable. Implications for the Hohenberg-Kohn variational principle and the Kohn-Sham equations are discussed.