The method of calculating electron spectral properties by substituting perturbation theoretic results for the self-energy into the Dyson equation is investigated for a model of a deep electron level in which exact results are known. The method gives wrong results in the most important range and also predicts spurious structure. On the other hand, if a perturbation expansion is made directly for the spectral function after extracting the appropriate energy shifts no such difficulty arises.
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Blomberg et al. (1972) studied this question.