Green's function for resonant level

[aaaa202]

I would really like some help for exercise 2 in the attached pdf. I know it's a lot asking you to read through all the pages but maybe you can skim them and catch the main points leading to exercise 2.
What I don't understand is pretty basic. What is meant by the Green's function g(l,ikn)? In all of the examples we are working with green's function on the form g(vv',τ) so what does the other notation mean? Does it mean: g(ll,ikn)?
In any case isn't it simply found by identifying g^-1(l,τ) as stated in (2.23) and Fourier transforming? That's what I though but I don't think that yields the correct result with the sum over v-states.

 

[BigJDubs]

The Green's function g(l,ikn) is the Fourier transform of the retarded Green's function g(vv',τ) (defined in Equation (2.23)). It is defined as:g(l,ikn) = ∫∞−∞ dτ g(vv',τ)eiknτ To calculate it, one would need to use the formula for the Fourier transform of a function, which is given by:F[f(t)] = ∫∞−∞ dt f(t)e−iknt By substituting the definition of g(vv',τ) into the above equation, one would get:g(l,ikn) = ∫∞−∞ dτ ∑v,v' δ(v-v')g(v,τ)eiknτ The delta function can be used to simplify the above expression and we get:g(l,ikn) = ∑v ∫∞−∞ dτ g(v,τ)eiknτ Finally, this yields the desired result for the Green's function g(l,ikn).