Magnetic impurity effect on superconductivity in systems with comparable Fermi and Debye energies

We obtain the basic system of equations for a superconductivity theory of a system with a chaotically distributed paramagnetic substitution impurity in which the Migdal theorem is violated (the condition $\omega_{\operatorname D}\ll E_{\operatorname F}$ cannot be assumed). The electron-phonon and impurity diagrams and also the additional diagrams corresponding to intersections of the electron-phonon and electron-impurity lines are taken into account. In the weak electron-phonon coupling limit, we obtain an equation for the superconducting transition temperature $T_C$ that differs from the corresponding equation for the usual superconductors by renormalizations of $T_{C0}$ and of the impurity scattering parameter $\rho$. These quantities depend essentially on the Migdal parameter $m=\omega_{\operatorname D}/E_{\operatorname F}$ and on the transferred momentum $q_c$. We show that the decrease of $T_C$ with the increase of the impurity concentration is slowed, as compared with usual superconductors, to an extent determined by $m$ and $q_c$. We also evaluate the isotopic coefficient $\alpha$, whose behavior as a function of the impurity concentration depends on $m$ and $q_c$.