Strong coupling theory of impure superconductors : real space formalism

In the electron-phonon model, the influence of nonmagnetic impurities on the transition temperature of superconductors is revisited. Anderson's pairing condition between time-reversed eigenstate pairs is derived from the physical constraint of the Anomalous Green's function. After this pairing condition is incorporated into the self-consistency equation, one may find that the phonon-mediated interaction decreases exponentially by localization.