Bethe -Salpeter Equation and Isotope Effect in Superconductivity

We evaluate the temperature of superconducting phase transition in terms of the Bethe-Salpeter equation for the many-body system. We consider the case when both electron-phonon interaction and strong Coulomb interaction coexist. To remove the artificial cutoff at high energy we solve the Bethe-Salpeter equation in terms of the two-particle vacuum scattering amplitude. We show that such an approach leads to the conclusion that the so-called Coulomb pseudopotential appears only due to artificial cutoff and does not exist at all in terms of the Bethe-Salpeter equation.