Development of a self-consistent approximation

A self-consistent approximation of a higher level than the standard self-consistent approximation, known in various fields of physics as the Migdal, Kraichnan or Born self-consistent approximation, is derived taking into account both the first and second terms of the series for the vertex function. In contrast to the standard approximation, the new self-consistent approximation is described by a system of two coupled nonlinear integral equations for the self-energy and the vertex function. In addition to all the diagrams with non-intersecting lines of correlation/interaction taken into account by the standard self-consistent approximation, the new approach takes into account in each term of the Green's function expansion a significant number of diagrams with intersections of these lines. Because of this, the shape, linewidth, and amplitude of the resonance peaks of the dynamic susceptibility calculated in this approximation are much closer to the exact values of these characteristics. The advantage of the new self-consistent approach is demonstrated by the example of calculation of the dynamic susceptibility of waves in an inhomogeneous medium.