\[ex\_numericsvsLarge\], we compare asymptotic results with simulation done for the total number of realizations $L=2500$, for exact resonances. In Fig. \[pa\_numericsvsLarge\] we compare numerical results obtained for parametric resonances, where $N=500$, $\eta=0.81$ and $L=5000$, with our theory for large-$N$. Though we have considered only the flat disorder yet our results are valid for the Gaussian or other symmetric distribution functions. These figures show that our asymptotic results are in fair agreement with the numerical results for almost all $\alpha$. For instance, near the middle of the energy band it describes reasonably well a dip and a peak, respectively in the $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$. These two opposite effects, however, cancel out in $\langle\,(|\{\Delta \tilde{\mathcal{Z}}_{\alpha}\}|)^{2}\,\rangle$.

![ Shown on the same pattern of Fig. \[ex\_numericsvsLarge\] but for the parametric resonances where $N=500$, $W=0.015$ and $\eta=0.81$. These numerical results are obtained from the diagonalization of $N$-dimensional matrices for $5000$ realizations. In this figure $\langle\,|\delta_{1}\tilde{\mathcal{Z}}_{\alpha}|^{2}\,\rangle/\sigma^{2}$, $\langle\,(\Re\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ and $\langle\,(\Im\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ are shown respectively by pluses, crosses and stars while open circles are the rescaled theories (\[abdz7\], \[LargeNdx\]) and (\[LargeNdy\]).[]{data-label="pa_numericsvsLarge"}](pa-numericsvsLarge.eps){width="75.00000%"}

Conclusion
==========

In conclusion, we have studied resonances in a one dimensional discrete tight-binding open chain in a weak disorder limit. In this study we have 