Hi guys In some articles I've read, they all mention that the (local) density of states is related to the retarded Greens function for a non-interacting system by -(1/π)Im[G(r,ω)] = LDOS(r,ω), i.e. the imaginary part of the Greens function. The above relation holds because in k-space the retarded Greens function for translational invariant systems is given by Gretarded = 1/(ω-Ek+iΓ), where Γ is the self-energy (and Ek is the self-energy). In the limit where Γ goes to zero, we can take the imaginary part, and then we obtain the above relation - so far so good. But this is just for the case of a translational invariant systems - i.e. we have not assumed any impurities. Now lets say I place an impurity in my system. Now my retarded Greens function is not given as above. How do we know that the LDOS is still just proportional to the imaginary part of the retarded Greens function for the system?