Dynamic structure function of a Bose gas atT=0

The dynamic structure function $\mathcal{S}(k, \omega )$, as a function of wave vector $k$ and frequency $\omega$, is calculated rigorously for a simple Bose gas at $T=0\mathrm{}$, The dielectric formulation is used to satisfy general symmetry requirements for a Bose system, and the calculation is carried out in the first approximation beyond Bogoliubov taking into account the three-phonon process. An analytic solution in terms of elliptic integrals is obtained for the width of $\mathcal{S}(k, \omega )$ valid for arbitrary $k$ and $\omega$. Qualitatively, $\mathcal{S}(k, \omega )$ at finite but small $k$ is found to have a square-root behavior near the threshold frequency for two-phonon production, a peak at the elementary excitation frequency, and a long power-law tail at high frequencies. Illustrative numerical results are presented for the width of $\mathcal{S}(k, \omega )$, the imaginary part of the phonon spectrum, and $\mathcal{S}(k, \omega )$ itself. Finally, the impulse approximation and the extraction of the condensate density $n_0$ from $\mathcal{S}(k, \omega )$ in the large-$k$ limit are discussed.