Finite temperature dynamic structure function of the free Bose gas

A detailed calculation of the finite temperature dynamic structure function of the free Bose gas is presented and discussed. After a short derivation of the expressions describing the exact response $S(q,\omega ;T)$ and the Impulse Approximation (IA) $S_{IA}(q,\omega ;t)$, their main properties and their evolution with q and T are analyzed. The lowest order energy weighted sum rules of both $S(q,\omega ;T)$ and $S_{IA}(q,\omega ;T)$ are also derived and commented. Finally, the $q\to \mathrm{\infty }$ asymptotic behavior is analyzed and discussed in terms of scaling laws.