linear while $(\delta_{2} \tilde{\mathcal{Z}}_{\alpha})$ are quadratic in the $\epsilon_{j}$ , for $j=1,...,N$. Then for $\mathcal{S}_{1}(\{\epsilon\};\tilde{\mathcal{Z}}_{\alpha})$, up to $\mathcal{O}(\{\epsilon\}^{2})$, we get $$\begin{aligned}
\label{Selfenergy}
\mathcal{S}_{1}(\{\epsilon\};\tilde{\mathcal{Z}}_{\alpha})
&=&
S_{1}(\{0\};\tilde{z}_{\alpha})+
\sum_{n=2}^{N}\epsilon_{n}
\left(\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial\epsilon_{n}}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber
\\
&+&
(\delta_{1} \tilde{\mathcal{Z}}_{\alpha}+\delta_{2} \tilde{\mathcal{Z}}_{\alpha})\left(\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial z}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber
\\
&+&
\dfrac{1}{2}\sum_{n,m=2}^{N}\epsilon_{n}\epsilon_{m}
\left(\dfrac{\partial^{2} \mathcal{S}_{1}(\{\epsilon\};z)}{\partial\epsilon_{n}\partial\epsilon_{m}}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber
\\
&+&
\dfrac{1}{2}(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})^{2}\left(\dfrac{\partial^{2} \mathcal{S}_{1}(\{\epsilon\};z)}{\partial^{2} z}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}.\end{aligned}$$ We will use this expansion in Eq. (\[secular\]). Before that we evaluate $$\begin{aligned}
\label{SGdz}
1-\left(\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial z}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
=
\dfrac{\partial}{\partial z} \dfrac{1}{G_{11}(z)}\bigg|_{z=\tilde{z}_{\alpha}},\end{aligned}$$ and, $$\begin{aligned}
\label{dGdz}
\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial \epsilon_{n}}\bigg |_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
&=&
\dfrac{1}{\left(G_{11}\right)^2}\,
\dfrac{\partial \mathcal{G}_{11}}{\partial \epsilon_{n}}\bigg|_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}},
\nonumber
\\\end{aligned}$$ for $n\geq2$. These equalities come from Eq. (\[GS\]). Finally, we calculate derivatives of $\mathcal{G}_{11}$, at $\{\epsilon\}=0$ and $z=\tilde{z}_{\alpha}$ with respect to $\{\epsilon\}$ by using Eq. (\[Hborn\]) for the [*Born-series*]{} expansion of $\mathcal{G}(z)$. We find $$\begin{aligned}
\label{dGdeps}
\dfrac{\partial \mathcal{G}_{11}}{\partial \epsilon_{n}}\bigg|_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
&=& 
G_{1n}G_{n1}\bigg|_{z=\tilde{z}_{\alpha}}.\end{aligned}$$ Grouping all these, for the first order corrections, we obtain $$\begin{aligned}
\label{FPT1}
&&(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})
=
\dfrac{\epsilon_{1}+\sum_{n=2}^{N}\epsilon_{n}
\dfrac{G_{1n}G_{n1}}
{\left[G_{11}\right]^{2}}\Bigg|_{z=\tilde{z}_{\alpha}}}
{\dfrac{\partial}{\partial z} \dfrac{1}{G_{11}(z)}\bigg|_{z=\tilde{z}_{\alpha}}
+
\dfrac{i\eta\exp[i\tilde{k}_{\alpha}]}
{2\sin(\tilde{k}_{\alpha})}
}. \end{aligned}$$ Similarly for the second order corrections we get $$\begin{aligned}
\label{dz2}
(\delta_{2} \tilde{\mathcal{Z}}_{\alpha}) 
&=&
\Bigg[\sum_{n,m=2}^{N}\epsilon_{n} \epsilon_{m}
\left\lbrace
\dfrac{G_{1n}G_{nm}G_{m1}}{[G_{11}]^{2}}
-
\dfrac{[G_{1n}G_{1m}]^{2}}
{[G_{11}]^{3}}
\right\rbrace
\nonumber
\\
&-&
\dfrac{(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})^{2}}{2}
\left\lbrace\left(\dfrac{\partial^{2}}{\partial^{2} z}\dfrac{1}{G_{11}}\right)+\eta\left(\dfrac{d^{2}\exp(ik(z)}{d^{2}z}\right)\right\rbrace
\Bigg]_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber\\
&\times &
\Bigg[{\dfrac{\partial}{\partial z} \dfrac{1}{G_{11}(z)}\bigg|_{z=\tilde{z}_{\alpha}}
+
\dfrac{i\eta\exp[i\tilde{k}_{\alpha})}
{2\sin(\tilde{k}_{\alpha})}
}\Bigg]^{-1}. 
\nonumber\\\end{aligned}$$

Note that $(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})$ and $(\delta_{2} \tilde{\mathcal{Z}}_{\alpha}) $ have been 