Resolvent technique projection operators

In order to become acquainted with the resolvent technique, and the use of projection operators, we re-examine the problem treated in 1.1. In doing so, we employ the cyclic crystal, which lends itself well to modelling a noninteracting (or unperturbed) substrate (Davison and Styslicka 1996). 

Clearly, the incorporation of the cut and stick tailoring feature of projection operators into the resolvent-technique formulation, makes it a particularly adaptable modelling tool. Moreover, it enables the atomic structure of the geometric space to be reflected in the so-called site representation of the GF. 

The best description of perturbation techniques in terms of projection operators and resolvents is given by Kato [28]. 

The desired matrix representation of the superoperator resolvent is then obtained in two steps by the inner projection technique (Pickup and Goscinski, 1973), where the superoperator resolvent is projected in the space of the complete set of excitation and de-excitation operators hn - First, we insert the resolution of the superoperator identity twice in Eq. (3.149) leading to... 

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