  The mapping of time-dependent densities on potentials in quantum mechanics is
critically examined. The issue is of significance ever since Runge and Gross
(Phys. Rev. Lett. 52, 997 (1984)) established the uniqueness of the mapping,
forming a theoretical basis for time-dependent density functional theory. We
argue that besides existence (so called v-representability) and uniqueness
there is an important question of stability and chaos. Studying a 2-level
system we find innocent, almost constant densities that cannot be constructed
from any potential (non-existence). We further show via a Lyapunov analysis
that the mapping of densities on potentials has chaotic regions in this case.
In real space the situation is more subtle. V-representability is formally
assured but the mapping is often chaotic making the actual construction of the
potential almost impossible. The chaotic nature of the mapping, studied for the
first time here, has serious consequences regarding the possibility of using
TDDFT in real-time settings.
