respectively. In both cases we have considered $N=100$, $\eta=0.81$ and $W=0.015$ so that $\xi(0)>>N$. As seen in these figures, complex energies of the disordered chain are scattered around the $\tilde{z}_{\alpha}$s.

We now calculate the corrections to $\tilde{z}_{\alpha}$ for such weak disorder case. It is suggestive here to deal with the self-energy. Let $\mathcal{S}_{1}(\epsilon_{2},...,\epsilon_{N};z)$ be the self-energy for the first site, defined via $$\begin{aligned}
\label{GS}
\mathcal{G}_{11}(z)=\dfrac{1}{z-\epsilon_{1}-\mathcal{S}_{1}(\{\epsilon\};z)}.\end{aligned}$$ Here $\{\epsilon\}$ denotes the set $\epsilon_{2},...,\epsilon_{N}$ and $\mathcal{G}_{11}$ is the $\{1,\,1\}$ element of the resolvent $\mathcal{G}(z)=(z-\mathcal{H})^{-1}$, defined for the Hermitian matrix $\mathcal{H}$. For the later convenience we write $$\begin{aligned}
\label{Hborn}
\mathcal{H}=H+\mathcal{W}, \end{aligned}$$ where $\mathcal{W}=\sum_{\ell=1}^{N}\epsilon_{\ell}P_{\ell}$ and $P_{\ell}=|\,\ell\,\rangle\langle\,\ell\,|$ is the projection for the $\ell$’th site.

In the rest of the paper we will work out results only for the exact resonances. For the parametric resonance theses results can be carried out following similar steps, so we skip all the intermediate steps merely by stating the result at the end.

As before in Eq. (\[charcpol\]), for disordered chain, resonances correspond to the roots of the secular equation $$\label{secular}
\mathcal{F}(z)=0=z-\epsilon_{1}-\mathcal{S}_{1}(\{\epsilon\};z)+\lambda\eta.$$ Preserving $\tilde{z}_{\alpha}$ as the roots of Eq. (\[charcpol\]), we define $\tilde{\mathcal{Z}}_{\alpha}$ as the roots of Eq. (\[secular\]). Now we expand the roots $\tilde{\mathcal{Z}}_{\alpha}=\tilde{z}_{\alpha}+(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})+(\delta_{2} \tilde{\mathcal{Z}}_{\alpha})$, assuming that $(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})$ are 