Displacement Correlations and Frequency Spectra for Mass-Disordered Lattices

The general form of the multiple-occupancy polynomials $P_n\mathrm{}\left(c\right)\mathrm{}$, shown by Langer to be necessary in the perturbation expansion of the Green's function of a mass-disordered lattice, is derived and its use is illustrated by the summation to all orders in the concentration of the self-energy diagrams involving one vertex. Different Green's functions must be used for the displacement correlations and for the frequency spectrum, and numerical calculations for a linear chain show that the two Green's functions can differ considerably in certain regions of frequency. Some problems of convergence are discussed.