  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. For a simple Einstein model we
calculate the renormalization of the electronic energies and the critical
temperature of superconductivity.
