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An approach for particle-hole correlation functions, based on the so-calledSCRPA, is developed. This leads to a fully self-consistent RPA-like theorywhich satisfies the $f$-sum rule and several other theorems. As a first step, asimpler self-consistent approach, the renormalized RPA, is solved numericallyin the one-dimensional Hubbard model. The charge and the longitudinal spinsusceptibility, the momentum distribution and several ground state propertiesare calculated and compared with the exact results. Especially at half filling,our approach provides quite promising results and matches the exact behaviourapart from a general prefactor. The strong coupling limit of our approach canbe described analytically.Comment: 35 pages, 18 Figures, Feynman diagrams as 10 additional eps-files,  revised and enhanced version, accepted in Phys. Rev.

Dyson equation approach to many-body Green’s functions and self-consistent RPA: Application to the Hubbard model