In Phys. Rev. B, 66, 165421 (2002) by Nieminen
they state that rho = - 1/pi Im(G°), without explanation and in
Phys. Rev. B 60, Number 4 (1999) also by Nieminen:
delta rho = -2/pi I am (int (delta G dE))
There must be a way to derive the density of state by Green's functions.