Generating coefficients of lattice Green's functions

Generating coefficients for time-dependent Green's functions $G\left(\tau \right)$, which contain the whole vibrational properties of the crystal, are defined. It is shown that they are the Fourier-expansion coefficients of the time-independent or frequency-dependent Green's function $G\left(\Omega \right)$, where $\Omega$ is a frequency variable. Their properties are studied in cases where the perturbation, applied to the crystal, does not preserve the translationary symmetry as well as in the case when the symmetry is preserved. As the knowledge of these coefficients allows the calculation of the real and imaginary parts of $G\left(\Omega \right)$ it is shown that the vibrational properties of the perturbed crystal can be extracted from their numerical values. A calculation method is given and applied to the case of a face-centered-cubic crystal with (001) face. Marked differences appear in comparing the surface-atom and bulk-atom "dispersion relation."