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

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