Spectral Function: Concluding Delta, Physical Interpretation, Imaginary Part

[aaaa202]

Attached is a line from my book about the spectral function for free electrons. How do they conclude that it is a delta function? I can see it from a handwaving argument since δ is infinitesimal but that does not explain the factor of 2pi. Rather I think that the equation really only make sense if set up as integral identity, but I don't see how exactly how. Also, what is the physical interpretation of the spectral function? My book relates it to how a particular energy can be excited but, I don't understand this. What does the imaginary part of the Greens function tell us?

 

[Simon Bridge]

Comes directly from the definition of the propagator ... the delta function is meaningless without an integration remember. So what's the imaginary part of the propagator?
What values does it take as you integrate over frequency?

 

[qbert]

There is a pretty standard representation of the delta-function here
$$\lim_{\eta \rightarrow 0} \frac{\eta}{x^2 + \eta^2} = \pi \delta(x).$$
See, for example, the wikipedia article on the delta function.

 

[aaaa202]

Does it follow if I try to integrate with the residue theorem?

I don't think my propagator is defined like yours. Mine is a thermal average of a commutator.