Representability problem of density functional theory for superconductors

Aiming at a unified treatment of correlation and inhomogeneity effects in superconductors, Oliveira, Gross, and Kohn proposed in 1988 a density functional theory for the superconducting state. This theory relies on the existence of a Kohn- Sham scheme, i.e., an auxiliary noninteracting system with the same electron and anomalous densities of the original superconducting system [Phys. Rev. Lett. 60, 2430 (1988)]. However, the question of noninteracting $v$-representability has never been properly addressed and the existence of the Kohn-Sham system has always been assumed without proof. Here, we show that indeed such a noninteracting system does not exist at zero temperature. In spite of this result, we also show that the theory is still able to yield good results, although in the limit of weakly correlated systems only.