On Translational Superfluidity and the Landau Criterion for Trapped Bose gases in the Gross-Pitaevski Limit

The two-fluid and Landau criteria for superfluidity are compared for trapped Bose gases. While the two-fluid criterion predicts translational superfluidity, it is suggested, on the basis of the homogeneous Gross-Pitaevski limit, that a necessary part of Landau's criterion, adequate for non-translationally invariant systems, d oes not hold for trapped Bose gases in the GP limit. As a consequence, if the compressibility is detected to be very large (infinite by experimental standards) the two-fluid criterion is seen to be the relevant one in case the system is a tranlational superfluid, while the Landau criterion is the relevant one if translational superfluidity is absent.