A theory of many-particle systems using the memory function method is formulated by expressing a response function in terms of that of a reference system and an effective potential. An exact formal expression for the effective potential is thereby obtained in terms of an interaction-part of a reduced second order memory function (IRSMF) and the second and fourth frequency moments of a spectral function of a canonical self-correlation function. Two regimes, statistical and collisionless, are introduced for the IRSMF to discuss the approximate properties of the effective potential. As an example of tractable schemes to determine the effective potential in a self-consistent manner, a standard short-time or high-frequency approximation is employed to take the IRSMF to be Gaussian. The formal theory is illustrated by developing a general theory of density fluctuations in classical and quantum liquids from a unified and systematic point of view. A spectral function of a density response function is expressed in terms of various characteristic frequencies describing single-particle and collective excitations, kinematical or Landau-type and dynamical or collisional dampings, time-decaying of the IRSMF, etc. A brief study is made on collective modes in liquid He4 and anharmonic solids with particular attention paid to the properties of the effective potential.
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Takeno et al. (1978) studied this question.