Renormalization of theelectron-phonon interaction: a reformulation of the BCS-gapequation

A recently developed renormalization approach is used tostudy the electron-phonon coupling in many-electron systems. Bystarting from an Hamiltonian which includes a small gaugesymmetry breaking field, we directly derive a BCS-like equationfor the energy gap from the renormalization approach. Theeffective electron-electron interaction for Cooper pairs doesnot contain any singularities. Furthermore, it is found thatphonon-induced particle-hole excitations only contribute to theattractive electron-electron interaction if their energydifference is smaller than the phonon energy.