I'm reading Fetter Walecka on many body theory (Chapter 7). We have established the from of the many body green function \begin{equation} G(xt,x't')=\langle T \psi(xt)\, \psi^{\dagger}(x't')\rangle \end{equation} in the Lehmann representation \begin{equation} G(k,\omega)=\int_0^{\infty}dE \, \frac{A(k,E)}{\omega -\mu-E+i0}+\frac{B(k,E)}{\omega -\mu+E-i0} \end{equation}
$A$ and $B$ are positive semidefinite spectral functions. In the section on the physical interpretation, F&W make "an elementary model of the interacting assembly and assume that $G^R(\omega,k)$ has a simple pole close the the real axis at $\omega = \epsilon_k - i \gamma_k$ with residue $a$" (note that $\gamma$ is finite).
A short calculation yields
\begin{equation} G(k,t) = -i a e^{-i \epsilon_k t}\, e^{-\gamma_k t}, \end{equation} which describes the propagation of a quasiparticle with energy $\epsilon_k$ and lifetime $1/\gamma_k$.
Question: What is the microscopical justification for the simple pole at $\omega = \epsilon_k - i \gamma_k$? It seems to contradict the Lehmann representation, because the latter shows that all poles are infinitesimally close to the real axis.
