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For general quantum systems the power expansion of the Gibbs potential andconsequently the power expansion of the self energy is derived in terms of theinteraction strength. Employing a generalization of the projector technique acompact representation of the general terms of the expansion results. Thegeneral aspects of the approach are discussed with special emphasis on theeffects characteristic for quantum systems. The expansion is systematic andleads directly to contributions beyond mean-field of all thermodynamicquantities. These features are explicitly demonstrated and illustrated for twonon-trivial systems, the infinite range quantum spin glass and the weaklyinteracting Bose gas. The Onsager terms of both systems are calculated, whichrepresent the first beyond mean-field contributions. For the spin glass newTAP-like equations are presented and discussed in the paramagnetic region. Theinvestigation of the Bose gas leads to a beyond mean-field thermodynamicdescription. At the Bose-Einstein condensation temperature complete agreementis found with the results presented recently by alternative techniques.Comment: 17 pages, 0 figures; revised version accepted by Phys Rev

Expansion of the Gibbs potential for quantum many-body systems: General formalism with applications to the spin glass and the weakly nonideal Bose gas