Off-the-energy shell properties of the mass operator and spectral functions in nuclear matter

We investigate the off-the-energy-shell properties of the mass operator M(k; omega) = V(k; omega) + iW(k; omega), i.e., its dependence upon the nucleon momentum k and upon the nucleon frequency omega, separately. Particular attention is paid to the dispersion relation which connects its real and imaginary parts. We limit ourselves to the first two terms of the hole-line expansion of the mass operator, namely to the Brueckner-Hartree-Fock field M1(k; omega) and to the second-order "rearrangement" term M2(k; omega). Most previous works only dealt with "on-shell values" obtained by setting omega equal to the root e(k) of an energy-momentum relation, or equivalently by setting k equal to the root k(e) of this energy-momentum relation. We use as input a finite-rank representation of the realistic Argonne v14 nucleon-nucleon interaction. The Fermi momentum is set equal to 1.36 fm-1. For momenta larger than the Fermi momentum, the calculated k-dependence of the on-shell depth V1(k; e(k)) can be approximated by a gaussian. The corresponding nonlocality range is close to that assumed by Perey and Buck in their phenomenological analysis of scattering cross sections; it is somewhat smaller than that associated with the k-dependence of the off-shell potential V1(k; omega) for fixed omega. The calculated omega-dependence of V2(k; omega) is in excellent agreement with the dispersion relation which connects V2(k; omega) to the values of W2(k; omega') for all omega' < e(k(F)). The dispersion relation between V1(k; omega) and W1(k; omega') is also investigated; in that case, caution must be exercised because the values of W1(k; omega') for omega' larger than 500 MeV still play a sizeable role, and also because the dispersion relation involves a large omega-independent "background"; it is proved that the latter is equal to the Hartree-Fock potential. More generally, the dispersion relation between the real and imaginary parts of the exact mass operator involves an omega-independent background for which we derive a closed expression analogous to but different from the Hartree-Fock potential. The omega-dependence of the spectral function S(k; omega) is calculated for two typical values of k, namely 3/4k(F) and 5/4k(F). Its integral over all values of omega differs from unity by only a few percent, which provides an estimate of the reliability of our approximation scheme. We study the dependence upon E(A+1)* of the integrated "particle" strength located below the excitation energy E(A+1)* in the system formed by adding a nucleon with momentum k to the nuclear-matter ground state. We also calculated the dependence upon E(A-1)* of the integrated "hole" strength located below the excitation energy E(A-1)* in the system formed by taking out a nucleon with momentum k from the nuclear-matter ground state. In the limit E(A-1)* --> infinity, this integrated hole strength yields an approximation of the occupation probability of the momentum k in the ground state. We compare this result with estimates obtained from other approximate expressions which involve the partial derivative of V1(k; omega) with respect to omega. We also evaluate the "mean removal energies" and compare them to the "quasiparticle energies," i.e., to the energies at which the spectral function presents a maximum.