approximate statistical results which are close to those for the exact resonances in strongly localized regime [@GG:00]. Parametric resonances depend on a chosen parameter, for instance let $\tilde{k}=k_{0}$ and we fix it in the middle of the energy band, $k_{0}=\pi/2$. Writing explicitly $$\lambda(\tilde{k})=
\begin{cases}
\exp(i \tilde{k}), & \text {for exact resonances,}
\\
i, & \text {for parametric resonances.}
\end{cases}$$

From now on we set the energy scale by taking $t=1$, denoting the complex variable $\tilde{\mathcal{E}}/t$ by $\tilde{\mathcal{Z}}$. We denote the Hamiltonian matrix representing the closed-clean chain by $H$. It differs from $\mathcal{H}$ only at the diagonal as, for the clean chain case, all the site energies are zero. Calculation of the eigenvalues of $H$ is a standard exercise where one derives $z_{\alpha}=-2 \cos[\alpha\pi/(N+1)]$ for $\alpha=1,...,N$.

Before going into a detail treatment to the problem, we should first sketch the outline of our approach. We are interested in a weak disorder regime. Since our approach rely on perturbation theory, we need complex energies of open-clean chain, i.e., the $\tilde{z}_{\alpha}$s. So we will begin with calculating the resonances for open-clean chain of finite length. Then we will do the perturbation series expansion up to the second order of strength of the disorder. 