Degenerate ground states and nonunique potentials: breakdown and   restoration of density functionals
  The Hohenberg-Kohn (HK) theorem is one of the most fundamental theorems of quantum mechanics, and constitutes the basis for the very successful density-functional approach to inhomogeneous interacting many-particle systems. Here we show that in formulations of density-functional theory (DFT) that employ more than one density variable, applied to systems with a degenerate ground state, there is a subtle loophole in the HK theorem, as all mappings between densities, wave functions and potentials can break down. Two weaker theorems which we prove here, the joint-degeneracy theorem and the internal-energy theorem, restore the internal, total and exchange-correlation energy functionals to the extent needed in applications of DFT to atomic, molecular and solid-state physics and quantum chemistry. The joint-degeneracy theorem constrains the nature of possible degeneracies in general many-body systems. 
K. Capelle, C. A. Ullrich and G. Vignale