show these results for 14 energy indices near the middle of the energy band but on a different scale.[]{data-label="ex_finitevsLarge"}](ex-finitevsLarge.eps){width="75.00000%"}

What follows next is the calculation of large-$N$ results for $\langle\,( \Re\{\Delta\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$. Since we need first to calculate $\langle\,(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}\rangle$ and $\langle\,[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}\,\rangle$, from Eq. (\[FPT2\]) we obtain after averaging $$\begin{aligned}
\label{dzsq}
 \langle\,(\Delta \tilde{\mathcal{Z}}_{\alpha})^{2}\,\rangle
=\sigma^{2}\dfrac{\sum_{n=1}^{N}4\sin^{4}(n\tilde{\theta}_{\alpha})}
{D^{2}},\end{aligned}$$ and $$\begin{aligned}
\label{cdzsq}
 \langle\,[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}\,\rangle
=\sigma^{2}\dfrac{\sum_{n=1}^{N}4\sin^{4}(n\tilde{\theta}^{*}_{\alpha})}
{(D^{*})^{2}}.\end{aligned}$$

![Shown on the same pattern of Fig. \[ex\_finitevsLarge\] but for the parametric resonances where $N=500$ and $\eta=0.81$. 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. The inset is shown for the indices near the middle of the band.[]{data-label="pa_finitevsLarge"}](pa-finitevsLarge.eps){width="75.00000%"}

For summation we use the formula [@GR] $$\begin{aligned}
\label{sumsin4}
\sum_{n=1}^{N} \sin^{4}(n\,\theta)
&=&
\dfrac{1}{8}
\Bigg[
3N-\dfrac{\sin(N\theta)}{\sin(\theta)}
\big(4\cos[(N+1)\theta]
\nonumber
\\
&-&
\dfrac{\cos[2(N+1)\theta]\,\cos(N\theta)}{\cos(\theta)}
\big)\,
\Bigg],
\nonumber
\\
\sum_{n=1}^{N} \sin^{4}(n\,\tilde{\theta}_{\alpha})&=&
\dfrac{1}{8}
\left[
3N-4U_{N-1}T_{N+1}+\dfrac{T_{2N+2}U_{2N-1}}{\tilde{z}_{\alpha}}
\right],\end{aligned}$$ where in the second equality the polynomials have argument $\tilde{z}_{\alpha}/2$ with $2\cos(\tilde{\theta}_{\alpha})=\tilde{z}_{\alpha}$. Similarly for the summation in Eq. (\[cdzsq\]) one gets the polynomials with argument $\tilde{z}^{*}_{\alpha}/2$. One can now use the equality (\[sumsin4\]) in Eqs. (\[cdzsq\], \[cdzsq\]) in order to derive finite-$N$ result for $\langle\,(\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$. For large-$N$ we make use of the ansatz (\[ansatz\]) and calculate 