Two-dimensional Boson-Fermion model

We studied the influence of the electron-boson interaction (hybridization) on the bosonic spectrum in the case of a two dimensional (2D) electronic sy stem. The boson dispersion is affected by this interaction and we showed th at for a constant density of states the wavevector dependence is quadratic for qv(F)/4T much less than 1, and the imaginary part is small for qv(F)/4T much greater than 1. However, the 2D electronic system presents a van Hove density of states, and we considered this energy dependence in the calcula tion of the bosonic energy spectrum. The real part becomes logarithmic depe ndent on the wavevector and the imaginary part can be neglected if qv(F) si milar to E-F or 2 E-c similar to z, for qv(F)/4T much greater than 1.