Time-dependent V-representability on lattice systems

We study the mapping between time-dependent densities and potentials on small lattices. As discovered recently by Baer (arXiv:0704.1787), there exist well-behaved time-dependent density functions on lattices which cannot be constructed from any real potential. However, one finds that such densities can always be reproduced by complex potentials. We analyze the breakdown of time-dependent V-representability on lattices and show that it is related to problems with the continuity equation which ultimately arise from discretization of the momentum operator. This imposes fundamental restrictions on practical numerical applications of TDDFT. In the continuum limit, time-dependent V-representability is restored.