Denote the pure system as system 1, with both continuum and discrete eigen energy. $G_0$ is its Green's function.
After introducing some impurities, we call the resultant system system 2 with new Green's function $G$, and $T$ is T matrix.
We have $G=G_0 + G_0 T G_0$
My question is, since the poles of Green's function are eigen energy of the system, and from the above equation, we find that all the poles of $G_0$ will also be the poles of $G$, does this mean the eigen energy of system 2 share the same eigen energy as its pure counterpart system 1? That is to say, both system 1's continuum and discrete eigen energy do not change in the presence of impurity?
Is there any possibility the $T$ matrix can cancel some of $G_0$'s poles?