Theory of the coexistence of superconductivity and magnetism: The superconducting transition temperature

The general strong-coupling equations are constructed for self-energy and gap for conduction electrons in a periodic lattice of localized magnetic moments. Both electron-phonon and electron-magnon interactions are taken into account. In the case of intermediate electron-phonon coupling, the superconducting transition temperature $T_c$ is obtained analytically for two limits: $T_c>T_m$ and $T_m\gg T_c$. The strong suppression of $T_c$ by electron-magnon interaction is investigated numerically and illustrated for various values of the parameters. The numerical results agree well with analytical solutions.