  This paper analyzes dynamical properties of small Fermi-Hubbard and
Bose-Hubbard systems, focusing on the structure of the underlying Hilbert
space. We evaluate time-dependent quantities such as the return probability to
the initial state and the spin imbalance of spin-1/2 fermions. The results are
compared with recent experimental observations in ultracold gases. For the
symmetric two-site Fermi-Hubbard model we find that the spin imbalance and the
return probability are controlled by two and three frequencies, respectively.
The spin imbalance and the return probability are identical for the asymmetric
Falicov-Kimball limit and controlled by only one frequency. In general, the
transition probabilities between the initial state and energy eigenstates
depend strongly on the particle-particle interaction. This is discussed for
"self trapping" of spinless bosons in a double-well potential. We observe that
the available Hilbert space is reduced significantly by strong interaction.
