Dynamic and static density-density correlations in the one-dimensional Bose gas: exact results and approximations

We discuss approximate formulas for the dynamic structure factor of the one-dimensional Bose gas in the Lieb-Liniger model that appear to be applicable over a wide range of the relevant parameters such as the interaction strength, frequency, and wavenumber. The suggested approximations are consistent with the exact results known in limiting cases. In particular, we encompass exact edge exponents as well as Luttinger liquid and perturbation theoretic results. We further discuss derived approximations for the static structure factor and the pair distribution function g(x). The approximate expressions show excellent agreement with numerical results based on the algebraic Bethe ansatz.