Deep inelastic scattering on ultracold gases

I shall discuss the dynamic structure factor of both Bose and Fermi gases with strong short-range interactions, focussing on the deep inelastic regime of large wave vector transfer $q$. Here, the dynamic structure factor is dominated by a resonance at the free-particle energy $\hbar \omega = \varepsilon_{\bf q} = \hbar^2 q^2/2m$ and is described in terms of scaling functions. I will show that the high-momentum structure has a rich scaling behavior characterized by two separate scaling regions: first, for frequencies that differ from the single-particle energy by terms of order ${\cal O}(q)$ (i.e., small deviations compared to the single-particle energy), the dynamic structure factor is described by the impulse approximation (IA) of Hohenberg and Platzman. Second, deviations of order ${\cal O}(q^2)$ (i.e., of the same order or larger than the single-particle energy) are described by the operator product expansion (OPE), with a universal cross-over connecting both regimes. Furthermore, I present an exact expression for the shift of the single-particle peak at large momentum due to interactions, which extends an old result by S. T. Beliaev for the low-density Bose gas to arbitrary values of the scattering length