obtained in terms of the resolvent of the closed-clean chain which we already know in terms of Chebyshev polynomials; see Eq. (\[G0uv\]) and the relation between $u_{n}$ and $v_{n}$ with Chebyshev polynomials.

Statistics of the Scattered Complex Energies
============================================

We are interested in the statistics of the scattered complex energies. For instance, using the first order result (\[FPT1\]) of the perturbation theory, we calculate average of square of absolute shift in complex energies defined as, $\langle|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2\rangle\equiv\langle|(\tilde{\mathcal{Z}}_{\alpha}-\tilde{z}_{\alpha})|^2\rangle$. The angular brackets are used here to represent the averaging over many realizations of set of all random site energies $\{\epsilon_{n}\}$. This quantity gives a statistical account for the scattered complex energies. We also calculate $\langle\,( \Re\{\Delta\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$, viz, average of square of the real and the imaginary part of the shift $(\tilde{\mathcal{Z}}_{\alpha}-\tilde{z}_{\alpha})$, respectively. To obtain the latter quantities we need first to calculate $\langle\,(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}\rangle$ and $\langle\,[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}\,\rangle$, since $$\begin{aligned}
 (\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}=\dfrac{(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}+[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}+2(|\Delta \tilde{\mathcal{Z}}_{\alpha})\,|^{2})}{4},
\nonumber
\\
\\
 (\Im\{\Delta\tilde{\mathcal{Z}}_{\alpha}\})^{2}=-\dfrac{(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}+[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}-2(|\Delta \tilde{\mathcal{Z}}_{\alpha})\,|^{2})}{4}.
\nonumber
\\\end{aligned}$$ Here we have used $\{^{*}\}$ to represent the complex conjugate (c.c.).

For all these three statistics we simplify $(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})$, given in Eq. (\[FPT1\]), in terms of Chebyshev polynomials as $$\begin{aligned}
\label{FPT2}
(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})
&=&
\Bigg[\dfrac{\tilde{z}_{\alpha}^2-4}{2}
\sum_{n=1}^{N}\epsilon_{n}
\left(U_{N-n}(\tilde{z}_{\alpha}/2)\right)^{2}
