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We revisit an inequality due to Onsager, which states that the (quantum)liquid structure factor has an upper bound of the form (const.) x |k|, for nottoo large modulus of the wave vector k. This inequality implies the validity ofthe Landau criterion in the theory of superfluidity with a definite, nonzerocritical velocity. We prove an auxiliary proposition for general Bose systems,together with which we arrive at a rigorous proof of the inequality for one ofthe very few soluble examples of an interacting Bose fluid, Girardeau's model.The latter proof demonstrates the importance of the thermodynamic limit of thestructure factor, which must be taken initially at k different from 0. It alsosubstantiates very well the heuristic density functional arguments, which arealso shown to hold exactly in the limit of large wave-lengths. We also brieflydiscuss which features of the proof may be present in higher dimensions, aswell as some open problems related to superfluidity of trapped gases.Comment: 28 pages, 2 figure, uses revtex

Onsager's inequality, the Landau–Feynman ansatz and superfluidity