The effects of local inhomogeneities on the phonon modulated DOS in Bi2212

Recent scanning tunneling microscopy experiments on Bi2212 have reveled microscopic inhomogeneities in the local density of states and anomalous signatures of coupling to a bosonic mode. Gap referenced estimates for the mode energy are negatively correlated with the local gap size and the distribution of the mode estimates shows a clear isotope shift upon $^{18}$O substitution. Motivated by the clear isotope effect we examine electron-phonon coupling to the 55 meV apical oxygen mode in Bi2212 within the framework of Migdal-Eliashberg theory. The interplay of this interaction with local inhomogeneous broadening effects are also considered. The effects of the local dopant atoms on the electron-phonon interaction strength are examined using the Ewald summation technique.