  We investigate the dynamic structure factor of atomic Bose and Fermi gases in
one-dimensional optical lattices at zero temperature. The focus is on the
generic behaviour of S(k,omega) as function of filling and interaction strength
with the aim of identifying possible experimental signatures for the different
quantum phase transitions. We employ the Hubbard or Bose-Hubbard model and
solve the eigenvalue problem of the Hamiltonian exactly for moderate lattice
sizes. This allows us to determine the dynamic structure factor and other
observables directly in the phase transition regime, where approximation
schemes are generally not applicable. We discuss the characteristic signatures
of the various quantum phases appearing in the dynamic structure factor and
illustrate that the centroid of the strength distribution can be used to
estimate the relevant excitation gaps. Employing sum rules, these quantities
can be evaluated using ground state expectation values only. Important
differences between bosonic and fermionic systems are observed, e.g., regarding
the origin of the excitation gap in the Mott-insulator phase.

Statistical Mechanics