  The dynamic structure factor of a trapped Bose-Einstein condensed gas is
investigated at zero temperature in the framework of Bogoliubov theory.
Different values of momentum transfer are considered, ranging from the phonon
to the single-particle regime. Various approximated schemes are discussed,
including the local density approximation, where the system is locally
described as a uniform gas, and the impulse approximation, where the response
is fixed by the momentum distribution of the condensate. A comprehensive
approach, based on the eikonal expansion, is presented. The predictions of
theory are successfully compared with the results of recent two-photon Bragg
scattering experiments, both at low and high momentum transfer. Relevant
features of the dynamic structure factor are also discussed using the formalism
of sum rules and the concept of scaling. Particular emphasis is given to the
regime of high momentum transfer, where the dynamic structure factor is
sensitive to the behaviour of the order parameter in momentum space, and some
instructive examples showing the consequence of long-range coherence are
presented.
