Dispersion Relation Approach to the Nuclear Mean Field

The mean field discussed here covers both large negative and large positive energies. It can be identified with the optical-model potential at positive energies and with the shell-model potential at negative energies. At negative as well as at positive energies, it contains an imaginary part which describes, in an average way, the coupling between single-particle and more complicated degrees of freedom. The real and imaginary parts of the mean field are connected by a dispersion relation. The latter predicts that the real part presents a characteristic energy dependence in the transition domain between the optical-model potential and the shell-model potential. It also provides a constraint which enables one to accurately extrapolate the mean field from positive towards negative energies. This is quite useful since much more empirical information is available at positive than at negative energies. The energy dependence of the resulting mean field can be related to properties of single-particle and of quasiparticle excitations. In particular, one can predict the single-particle energies, the spectroscopic factors and the distribution of single-particle strengths in the two nuclei obtained either by adding a nucleon to, or subtracting a nucleon from, a doubly-closed shell nucleus. Examples are given and compared with empirical data.