The Numerical Solution of the Imaginary-Axis Eliashberg Equations

In the paper, we solve the imaginary-axis Eliashberg equations. We calculate numerically self-consistently the superconducting order function, the wave function renormalization factor, and the energy shift function as a function of the Matsubara frequency. We consider different values of the average number of the electrons per lattice site. Additionally, we study the temperature dependence of the order function and the wave function renormalization factor. The possible extension of the Eliashberg theory to the case of the high-T$\text{}_{C}$ superconductors was also briefly discussed.