show that our result (\[resz\]) is close to the numerical solution. The agreement gets better as $\eta\rightarrow1$ (not shown here separately). However, the ansatz (\[ansatz\]) is not valid near the band edges. Moreover, the agreement fails for parametric resonances near the middle of the band as $\eta\rightarrow 1$; see Fig. \[ansatz-parametric\] for $\eta=0.99$.

The Weak disorder limit
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![Scatter plot for parametric resonances where $N=100$, $\eta=0.81$ and $W=0.015$, for $5000$ realizations. As in Fig.\[scatter\_exact\], here also dots represent the clean chain and points represent the disordered chain.[]{data-label="scatter_parametric"}](scatter-parametric.eps){width="75.00000%"}

In the next stage of the problem we switch on a very weak disorder in the chain. From a second order perturbation theory we know that for a disordered infinitely long chain the localization length, $\xi(E)$, is maximum at the middle of the band. For small $W$ it is given by [@krammac; @thou] $$\begin{aligned}
\label{xi}
\xi(E)=\dfrac{24(4t^2-E^2)}{W^2},\end{aligned}$$ implying thereby, $\xi(0)=\dfrac{96 t^2}{W^2}$. However, the exact result shows a small deviation at the band center due to the breakdown of the second-order perturbation theory [@KWegner:1981]. We consider a limiting situation when $\xi(0)/N >> 1$. For instance, in Fig. \[scatter\_exact\] and in Fig. \[scatter\_parametric\], we show the scatter plot ($\Re\{\tilde{\mathcal{Z}}_{\alpha}\}$ vs $\Im\{\tilde{\mathcal{Z}}_{\alpha}\}$) for exact and parametric resonances 