I'm reading these lecture notes on Anderson localization, and I cannot understand how the resonant regions contribute to the divergence of the resolvent expansion (sections 3.1 and 3.2). The relevant Hamiltonian is
$$ H=H_0+gT$$
where $$H_0=\sum_{i}\epsilon_i |i\rangle\langle i|,\quad T=-\sum_{\langle i,j\rangle}(|i\rangle\langle j|+|i\rangle\langle j|)$$
$i$,$j$ are sites on a cubic lattice, $\langle i,j\rangle$ are nearest neighbor. The author defines the resolvent as
$$ G(E)=\frac{1}{E-H}, \quad E\notin \sigma(H)$$
where $\sigma(H)$ denotes the spectrum of $H$. Also, call $G_0(E)=\frac{1}{E-H_0}$. After some algebra one can arrive at the series
$$ G(E)=G_0(E)+\sum_{n=1}^\infty (G_0T)^nG_0 $$
In this basis we can express $G(E)$ as a sum over walks from the starting point to the ending point: each $G_0$ contributes with a term like $\frac{1}{E-e_k}$ and $T$ makes us "walk" around the lattice.
$$ \langle i|G(E)|j\rangle=\frac{1}{E-\epsilon_i}+\sum_{n=1}^\infty (-g)^n\sum_{\substack{\pi:i\to j\\|\pi|=n}}\prod_{s=1}^n \frac{1}{E-\epsilon_{\pi(s)}}$$
The author later (beginning of section 3.2, page 13) says that if there are neighboring sites such that $\frac{g}{\epsilon_i-\epsilon_j}\geq 1$, then the series diverges because it contains terms of the form $$ \left(\frac{g}{\epsilon_i-\epsilon_j}\right)^m $$
I cannot see any such terms. I see $ \left(\frac{g}{E-\epsilon_i}\frac{g}{E-\epsilon_j}\right) $, but this does not cause divergences. What am I missing? Where does the divergence come from?
