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In this series of papers we present a detailed study of theparticle--particle collective excitations of the Hubbard model, and theircontribution to the density and spin excitation spectrum. In the first paper,we shall investigate the singlet particle--particle pair with momentum$(\pi,\pi)$, the $\eta$ particle, of the negative--$U$ Hubbard model. We reviewthree previously obtained theorems about the $\eta$ particle and develop aself-consistent linear response theory which takes into account itscontribution to the density excitation spectrum in the superconducting state.We show that this self--consistent theory agrees with the exact theorems aswell as the results of numerical Monte Carlo simulations.Comment: 30 pages, revtex 3.0 and epsf.sty (optional), 13 PostScript figures.  Submitted to Phys. Rev. B. Minor changes, typos corrected, references adde

A CLASS OF COLLECTIVE EXCITATIONS OF THE HUBBARD MODEL: η EXCITATION OF THE NEGATIVE-U MODEL