Similarity renormalization of the electron–phonon coupling

We study the problem of the phonon–induced electron–electron interaction in a solid. Starting with a Hamiltonian that contains an electron–phonon interaction, we perform a similarity renormalization transformation to calculate an effective Hamiltonian. Using this transformation singularities due to degeneracies are avoided explicitely. The effective interactions are calculated to second order in the electron–phonon coupling. It is shown that the effective interaction between two electrons forming a Cooper pair is attractive in the whole parameter space. The final result is compared with effective interactions obtained using other approaches.