Relationship between Feshbach′s and Green′s Function Theories of the Nucleon-Nucleus Mean Field

Abstract

We clarify the relationship and difference between theories of the optical-model potential which had previously been developed in the framework of Feshbach′s projection operator approach to nuclear reactions and of Green′s function theory, respectively. For definiteness, we consider the nucleon-nucleus system but all results can readily be adapted to the atomic case. The effects of antisymmetrization are properly taken into account. It is shown that one can develop along closely parallel lines the theories of "hole" and "particle" mean fields. The "hole" one-body Hamiltonians describe the single-particle properties of the system formed when one nucleon is taken away from the target ground state, for instance in knockout or pickup processes. The particle one-body Hamiltonians are associated with the system formed when one nucleon is elastically scattered from the ground state, or is added to it by means of stripping reactions. An infinite number of particle, as well as of hole, Hamiltonians are constructed which all yield exactly the same single-particle wave functions. Many "equivalent" one-body Hamiltonians can coexist because these operators have a complicated structure: they are nonlocal, complex, and energy-dependent. They do not have the same analytic properties in the complex energy plane. Their real and imaginary parts fulfill dispersion relations which may be different. It is shown that hole and particle Hamiltonians can also be constructed by decomposing any vector of the Hilbert space into two parts which are not orthogonal to one another, in contrast to Feshbach′s original theory; one interest of this procedure is that the construction and properties of the corresponding hole Hamiltonian can be justified in a mathematically rigorous way. We exhibit the relationship between the hole and particle Hamiltonians and the "mass operator." The latter is associated to the time-ordered Green′s function, rather than to its advanced and retarded parts separately as the hole and particle Hamiltonians. Similarities and differences between the hole and particle Hamiltonians and the mass operator are exhibited by constructing their explicit expressions in the case of nuclear matter, in the framework of second-order perturbation theory. Particular attention is paid to the connection of the mass operator and the various hole and particle Hamiltonians with observables which can be extracted from stripping, pickup and knockout reactions, in particular the spectroscopic factors and the spectral function. Since many different one-body Hamiltonians exist which all yield the same single-particle wave functions, their relative merits and drawbacks need to be discussed, with particular attention to their relationship to empirical shell- and optical-model potentials and to the possibility of developing practical approximation schemes.