shift, $$\begin{aligned}
\label{abdz6}
\langle|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2\rangle
=
\sigma^{2}\dfrac{N+\dfrac{1}{2}-\dfrac{U_{2N}+U_{2N}^{*}}{2}
+
\dfrac{T^{*}_{2N+2}T_{2N}-T_{2N+2}T^{*}_{2N}}{2[T^{*}_{2}-T_{2}]}}
{|D|^{2}}.\end{aligned}$$

We now turn our attention to large-$N$ behavior of the result (\[abdz6\]). For this purpose we use the ansatz (\[ansatz\]) and result the (\[res\]) for $\tilde{k}_{\alpha}$. Large-N behavior for the Chebyshev polynomials, with argument $\tilde{z}_{\alpha}$, may be calculated as $$\begin{aligned}
T_{2N}(\tilde{z}_{\alpha}/2)&=&\dfrac{\exp[2iN\tilde{\theta}_{\alpha}]+\exp[-2iN\tilde{\theta}_{\alpha}]}{2} 
\nonumber
\\
&\approx&
\dfrac{\Omega({k}_{\alpha};\eta)\exp(-2i{k}_{\alpha})+\left[\Omega({k}_{\alpha};\eta)\right]^{-1}\exp(2i{k}_{\alpha})}{2},$$ $$\begin{aligned}
T_{2N+2}(\tilde{z}_{\alpha}/2)
=
\dfrac{\Omega(k_{\alpha};\eta)+\left[\Omega(k_{\alpha};\eta)\right]^{-1}}{2}+\mathcal{O}(N^{-1}),\end{aligned}$$ $$\begin{aligned}
U_{2N}(\tilde{z}_{\alpha}/2)&=&\dfrac{\exp[i(2N+1)\tilde{\theta}_{\alpha}]-\exp[-i(2N+1)\tilde{\theta}_{\alpha}]}
{\exp(i\tilde{\theta}_{\alpha})-\exp(-i\tilde{\theta}_{\alpha})} 
\nonumber
\\
&\approx&
\dfrac{\Omega({k}_{\alpha};\eta)\exp(-i{k}_{\alpha})-\left[\Omega({k}_{\alpha};\eta)\right]^{-1}\exp(i{k}_{\alpha})}{\exp(-i{k}_{\alpha})-\exp(i{k}_{\alpha})}.\end{aligned}$$ Finally, $$\begin{aligned}
 T_{2}(\tilde{z}^{*}_{\alpha}/2)- T_{2}(\tilde{z}_{\alpha}/2)
&=& -\dfrac{2i\,\Im\{\Phi_{\alpha}\}}{N}\,
{z}_{\alpha}+\mathcal{O}(N^{-2})
\nonumber
\\
&\approx&
\dfrac{4i}{N}\, \cos(k_{\alpha})\,\Im\{\Phi_{\alpha}\},\end{aligned}$$ where we have used the ansatz (\[ansatz\]) in the second order polynomial $T_{2}(z)=2z^{2}-1$ and $\Im\{\Phi_{\alpha}\}=-\sin({k}_{\alpha})\,\text{ln}(|\Omega|)$, as obtained from Eqs. (\[ansatz\], \[res\]).

We can now plug in these results in Eq. (\[abdz6\]). These asymptotic results gives the numerator as ($N+a1+a2/(a3/N)$) where $a1,\,a2/a3$ are $\mathcal{O}(N^{0})$. Similarly we obtain denominator as ($\,N^{2}+b1\,N+b2$) where $b1$ and $b2$ are $\mathcal{O}(N^{0})$; see Appendix A for details. Thus in the leading order, we obtain $$\begin{aligned}
\label{abdz7}
\langle|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2\rangle
&=&
\dfrac{\sigma^{2}}{N}\,
\left(
1+\dfrac{1}{8}\,
\dfrac{
\left(
|\Omega|^{2}-|\Omega|^{-2}
\right)}
{\text{ln}(|\Omega|)}
\right).\end{aligned}$$

![Asymptotic results for $\langle\,|\delta_{1}\tilde{\mathcal{Z}}_{\alpha}|^{2}\,\rangle/\sigma^{2}$, $\langle\,(\Re\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ and $\langle\,(\Im\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$, shown by filled circles, squares and diamonds, vs energy index $\alpha$. We have compared here the finite-$N$ results, shown by open circles, for the exact resonances where $N=100$ and $\eta=0.81$. In the set we 