Analytical properties of the mean field in many fermion systems

The importance of the dispersion relation linking the real and the imaginary parts of the mean field in many fermion systems is pointed out. General models are built for the imaginary part. The properties of the real part close to the Fermi level are discussed in relation with these models. Its analytical behaviour is determined by the analytical behaviour of the imaginary part in the same domain, but the numerical values can be strongly influenced by the large energy behaviour of the same quantity. The sensitivity upon the behaviour in the intermediate domain is described semiquantitatively. The nuclear matter case is briefly discussed. The generality of our results is pointed out. (orig.)