In mathematical challenges from theoretical / computational chemistry by Committee on Mathematical Challenges from Computational Chemistry, "The N-and V-Representability Problems" is listed.
http://www.nap.edu/openbook.php?record_id=4886&page=48
> Unfortunately, only an incomplete set of necessary conditions are known, but these are already so complex that further work in this area has been abandoned by chemists.
For N-representability:
I think this is solved with negative result: computational complexity is QMA-complete.
For V-representability:
If I remembered correctly Ayers et al. have already published proof.
