A method of self-consistent perturbation theory with respect to the energy overlap integral in the Hubbard model

It is shown that there exists a regular representation for the self-energy part in the form of a series in powers of the energy overlap integral $t(f-f')$. The Green's function that arises in the first order in $t(f-f')$ satisfies four exact relations for the spectral moments corresponding to the linear canonical Kalashnikov–Fradkin transformation and is the best single-particle approximation of the problem on the class of solutions with two delta functions. The self-consistent second order of the theory in $t(f-f')$ gives a basis for the following conclusions: Convergence of the theory is ensured in the regions of concentrations corresponding to a ferromagnetic metal, $n>n_c+\Delta n_f$, and a paramagnetic metal, $n<n_c-\Delta n_p$; in the region $n_c+\Delta n_f>n>n_c-\Delta n_p$ the results of the theory must be interpreted as interpolation results ($n_c$ is the critical concentration of the ferromagnetic–paramagnetic transition, $0<\Delta n_f<\Delta n_p<n_c$).