Rigorous proof of isotope effect by Bardeen-Cooper-Schrieffer theory

In this article, we present a rigorous proof of the isotope effect within the theory of Bardeen-Cooper-Schrieffer for isotropic superconductors. We show that, when the interaction kernel is a positive constant, the isotope effect is exact; when the interaction kernel is a positive function depending on the energies of the pairing electrons, the isotope effect is no longer exact but lies within a sharp range determined by the varying kernel function. Moreover, we show that our method here may be extended to establish an existence and uniqueness theorem for the transition temperature in the Bogoliubov-Tolmachev-Shirkov model which allows separate phonon and Coulomb dominance in their respective energy regimes.