### Seemingly contradicting spectral functions in momentum- and position representation (partly solved)

When a single impurity (without interactions!) hybridizes with a band of conduction electrons, its spectral function broadens due to tunnelling processes in and out of that band. If we have a lattice of "impurities", or localized states, however, translational invariance is restored and each localized state with a defined quasi momentum /(k/) can only hybridize with conduction electrons of the same momentum /(k/), so that the spectral function is a finite sum of delta peaks (usually two).//

In this case, it is still possible to work in the position representation and calculate the spectral function of a localized state that has a defined position, but not a defined momentum. This spectral function turns out to be broadened, which is okay, but it also has spectral weight inside the region of the hybridization gap, which is a clear contradiction.//

This contradiction comes about from using a wrong ansatz for the self-energy, that takes only tunneling between a localized state at one fixed position and the conduction band into account, but not into localized states at different locations as intermediate states.

An exemplary process that is not included in the above diagram is