Equation of motion for the Green’s function in anharmonic solids

We discuss the double-time Green’s-function approach for the phonon-phonon interaction in anharmonic solids. Using a certain interpretation of the phonon operators, the crystal Hamiltonian may be rewritten in a compact form. The equation of motion for the Green’s function is tremendously simplified with this formalism. The self-energy can be obtained in a very simple way, to any desired perturbation order. Our results reproduce the corresponding terms of the available literature results, which are limited to the fourth order in the perturbation parameter.