# Asymptotic relation of Green's function for diverging self energy

I am considering the derivation on pages 64 to 66 of Zagoskin's Quantum Theory of Many-Body Systems. They consider a Green's function in the Lehmann representation:

$$G(p,\,\omega)=(2\pi)^3 \sum_s \left( \frac{A_s \delta(p-P_s)}{\omega-\epsilon_s^++\mu+i0} \pm \frac{B_s \delta(p+P_s)}{\omega-\epsilon_s^-+\mu-i0} \right)$$

where $\epsilon_s^+=E_s(N+1)-E_0(N)>\mu$ and $\epsilon_s^-=E_0(N)-E_s(N-1)<\mu$.

They they derive $G(\omega)\sim 1/\omega$ by stating that

in this limit we can neglect all other terms in the denominator...so that

$$G(p,\,\omega)\sim \frac{1}{\omega}(2\pi)^3 \sum_s \left(A_s\delta(p-P_s)\pm B_s \delta(p+P_s)\right)$$

The problem here is that, for some general interacting Green's function, I feel as if they are making an assumption about the self energy--i.e., that $\omega>\Sigma(k,\,\omega)\,\forall \omega$. If $\Sigma(\omega)\sim k,\,\omega^2$, for example, then the above analysis fails, although I can't find any explanation of such divergent behavior in my books.

What makes me concerned is the behavior of the real and imaginary parts of $G(k,\,\omega)$ as $|\omega|\rightarrow \infty$. I have seen authors use the above argument to state that the imaginary part of the Green's function disappears faster than the real part at diverging $\omega$. However, for some self energy that grows faster than $\omega$, does this argument break down?. If feel as if it would, but I can't find any references that talk about it. Any explanation or references (preferably original papers that can be easily accessed online) would be greatly appreciated.

EDIT 1: Am I correct in stating that the self energy is completely contained in $A_s$ and $B_s$, and therefore $G(p,\,\omega)\sim 1/\omega$ as $|\omega|\rightarrow \infty$ regardless of the self energy behavior?

EDIT 2: Is it simply because the limit of $\Sigma(\omega,\,k)\rightarrow \infty$ is unphysical for $\omega\rightarrow \pm \infty$ (even though $\frac{\partial \Sigma}{\partial \omega}\rightarrow \infty$ is legal in a non-Fermi liquid)?

$$\lim_{ik_n\rightarrow \infty}G(k,\,ik_n)=\lim_{ik_n\rightarrow \infty}\frac{1}{ik_n}$$ To get this correct behavior, the self energy cannot diverge at high frequency; i.e.,
$$\lim_{ik_n\rightarrow \infty}\Sigma(k,\,ik_n)=\textrm{constant}$$