2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/27010Various sum rules accounting for the coupling between density and particle excitations and emphasizing in an explicit way the role of the Bose-Einstein condensation are discussed. Important consequences on the fluctuations of the particle operator as well as on the structure of elementary excitations are reviewed. These include a recent generalization of the Hohenberg-Mermin-Wagner theorem holding at zero temperature. (To appear in "Bose-Einstein Condensation", A.Griffin, D.Snoke and S.Stringari eds., Cambridge Univ. Press)16 pages, Plain TeX, no figures, UTF-308-10/93Condensed Mattertext