I am trying to understand better the properties of the resolvent and Green function of a bounded self-adjoint operator $H$ at $z=E+i\eta$ when $\eta \to 0^+$. The resolvent is the operator $R(H;z)=(H-zI)^{-1}$, which is defined when $z$ is is not in the spectrum. If I am not mistaken, the Green function is defined as $G:\mathcal{H}\times \mathcal{H\times \mathbb{C}}$ by $G(\psi,\phi; z):= \langle R(H,z)\phi,\psi \rangle$, and is defined when $z$ is not in the spectrum.
My question is what happens when $E$ is in the spectrum of $H$ to $\Vert R(H;E+i\eta)\Vert$ or $G(\psi, \phi;E+i\eta)$ when $\eta\to 0^+?$ It seems that when $E$ is an approximate eigenvalue, that $\Vert R(H;E+i\eta)\Vert\to \infty$ by Weyl's criterion for the spectrum. On the other hand, I don't see what should happen to the Green function when $\eta \to 0^+$. Should it tend to $\infty$? And is there an importance to when the Green function vanishes?
I am motivated by the case where $H= \Delta+D$ on $\ell^2(\mathbb{Z})$, with $\Delta$ being the discrete Laplacian and $D$ being a diagonal operator with finitely many diagonal values. I am trying to consider the Green function using the Random walk expansion for the Green function given in chapter 6 of Random operators by Aizenman and Warzel for such operators.