  While the 1D Bose gas appears to exhibit superfluid response under certain
conditions, it fails the Landau criterion according to the elementary
excitation spectrum calculated by Lieb. The apparent riddle is solved by
calculating the dynamic structure factor of the Lieb-Liniger 1D Bose gas. A
pseudopotential Hamiltonian in the fermionic representation is used to derive a
Hartree-Fock operator, which turns out to be well-behaved and local. The
Random-Phase approximation for the dynamic structure factor based on this
derivation is calculated analytically and is expected to be valid at least up
to first order in $1/\gamma$, where $\gamma$ is the dimensionless interaction
strength of the model. The dynamic structure factor in this approximation
clearly indicates a crossover behavior from the non-superfluid Tonks to the
superfluid weakly-interacting regime, which should be observable by Bragg
scattering in current experiments.
