Combinatorics of Feynman Diagrams for the Problems with Gaussian Random Field

Abstract

The algorithm to calculate the generating function for the number of ``skeleton'' diagrams for the irreducible self-energy and vertex parts is derived for the problems with Gaussian random fields. We find an exact recurrence relation determining the number of diagrams for any given order of perturbation theory, as well as its asymptotics for the large order limit. These results are applied to the analysis of the problem of an electron in the Gaussian random field with the ``white-noise'' correlation function. Assuming the equality of all ``skeleton'' diagrams for the self-energy part in the given order of perturbation theory, we construct the closed integral equation for the one-particle Green's function, with its kernel defined by the previously introduced generating function. Our analysis demonstrate that this approximation gives the qualitatively correct form of the localized states ``tail'' in the density of states in the region of negative energies and is apparently quite satisfactory in the most interesting region of strong scattering close to the former band-edge, where we can derive the asymptotics of the Green's function and density of states in the limit of very strong scattering.