The weakly coupled polaron
The Green's Function technique has been used to investigate the properties of weakly coupled longitudinal optical polarons. By taking into consideration the contribution of the multi-phonon and the phonon-cross-exchange diagrams in the self-energy, a vertex correction has been made to the lowest order diagram. The real part and the imaginary part of the self-energy are obtained. The dispersion relationship obtained by this approximation gives the ground state energy which is lower than the upper bound obtained from the variational method. Also, the spectral weight function for several momentum values has been calculated. It shows that the quasi-particle description is good for polarons with small momentum while for polarons with momentum values near the phonon emission threshold, the quasi-particle description is not adequate.