What is the current state of research into $v$-representability? In their proof, Hohenberg and Kohn (Phys Rev 136 (1964) B864) established that the ground state density, $\rho_\text{gs}$, uniquely determines the Hamiltonian.  This had the effect of establishing an implicit relationship between $\rho_\text{gs}$ and the external potential (e.g. external magnetic field, crystal field, etc.), $V$, as the form of the kinetic energy and particle-particle interaction energy functionals are universal since they are only functions of the density.  This implicit relationship defines a set of densities which are called $v$-representable. What is surprising is that there are "a number of 'reasonable' looking densities that have been shown to be impossible to be the ground state density for any $V$." (Martin, p. 130) On the surface, this restriction looks like it would reduce the usefulness of density functional theory, but, in practice, that is not the case. (See the proof by Levy - PNAS 76 (1979) 6062, in particular.) However, research continues into the properties of the $v$-representable densities, and I was wondering if someone could provide a summary of that work.
 A: There are three places to address the $v$-representability:
1) The Hohenberg-Kohn theorem for the one-to-one mapping from ground-state electronic density to potential $v$
2) The Hohenberg-Kohn variational theorem
3) The Kohn-Sham scheme
As far as I know,
1) The $v$-representability is automatically guaranteed. Since the HK theorem starts from the exact ground-state density, which comes from the eigenfunction with $v$
2) The domain of $v$-representable density is still unknown. It can be avoided by Levy's constrained search (as you quoted paper). In the constrained search, one only needs $N$-representable density, the domain is known.
3) The problem is non-interacting $v$-representability (associated with the potential of non-interacting system). If one restricts to integer occupation of the orbitals/single Slater determinant, the domain is unknown.  If one relaxes the integer occupation, the domain is known by Lieb's work. 
One recent summary can be found from Density Functional Theory: An Advanced Course (Springer, 2011 edition) by Eberhard Engel and Reiner M. Dreizler. The book includes references to Lieb's paper.
