Approximation of Separable Forces in the Generalized Theory of Finite Fermi Systems

The approximation of separable forces is applied for the first time in the generalized theory of finite Fermi systems (TFFS), which was developed in [1–4] with the aim of consistently taking into account, within the Green’s function method, complex configurations involving phonons. Standard multipole–multipole forces are used; for their quadrupole case, two parameters are fitted to effective quadrupole polarization charges, whose values are well known from experiments and within the standard theory of finite Fermi systems. It is shown that, in this approximation, the equations of the standard theory of finite Fermi systems for the vertex and total scattering amplitude can readily be solved. A useful relation between effective quadrupole polarization charges and separable-force parameters that describe the total scattering amplitude in a separable form is obtained. As an example of the application of the approach developed here, an equation for the regular part \(\Gamma^{r}\) of the scattering amplitude is studied for the first time. Both free terms of this equation are estimated, and it is shown that an additional term containing the product of two phonon-creation amplitudes is several times as great as the other free term, which is the TFFS effective interaction taken in the separable approximation. The solutions of the equation for \(\Gamma^{r}\) that are obtained in the present study lead to the conclusion that it is illegitimate to disregard this quantity. Since, within the generalized theory of finite Fermi systems, \(\Gamma^{r}\) naturally appears in describing two-phonon interactions, this means that the theory of two-phonon excitations may become substantially more complicated.