Isotope effect in the presence of a pseudogap

We investigate the question of whether the unusual doping dependence of the isotope exponent observed in underdoped high-$T_c$ superconductors might be related to another unusual phenomenon observed in these systems: the pseudogap phenomenon. Within different approximations we study the influence of a phenomenological pseudogap on the isotope exponent and find that it generally strongly increases the isotope exponent, in qualitative agreement with experiments on underdoped high-$T_c$ compounds. This result is stable against strong-coupling self-energy corrections and also holds for recently proposed spin-fluctuation exchange models, if a weak additional electron-phonon coupling is considered.