This will be followed by the derivation of the statistical results. Finally, we will consider the large-$N$ limit of these results.

Open-clean Chain
================

We begin with defining the resolvent $\tilde{\mathcal{G}}(z)=(z-\tilde{\mathcal{H}})^{-1}$. Using Eq. (\[Hamil\]) we may also write $$\label{resolvent}
\tilde{\mathcal{G}}(z)=(z-\mathcal{H}+\eta\,\lambda\,P)^{-1}.$$ For the open-clean chain we define the resolvent $$\begin{aligned}
\tilde{G}(z)&=&(z-H+\eta\,\lambda\,P)^{-1}
\nonumber
\\
&=&
(1+\eta\,\lambda\,GP)^{-1}\,G,\end{aligned}$$ where we have introduced $G(z)=(z-H)^{-1}$ as the resolvent for the “unperturbed" closed-clean chain. Resonances correspond to the singularities of the matrix $\tilde{G}_{mn}(z)$, or to the roots of the secular equation $$\label{charcpol}
F(z)=0
=1+\eta\lambda G_{11}(z),$$ where $G_{11}$ is the $\{1,\,1\}$ element of the matrix $G$ in site representation. ($G_{nm}(z)=\langle n|(z-H)^{-1}|m\rangle$.)

![Comparison of the result (\[resz\]) (pluses) with the numerical solution of the polynomial equation (\[fincharpol\]) (circles, squares and diamonds) for the exact resonances where $\eta=0.5,\,0.81$ and $0.99$. We have considered $N=100$.[]{data-label="ansatz-exact"}](ansatz-exact.eps){width="75.00000%"}

To obtain $G_{11}$ for finite $N$, we use the ordinary difference equation (ODE), $$\label{ODE}
\psi_{n+1}+\psi_{n-1}+z\,\psi_{n}=0,$$ with the boundary conditions $\psi_{0}=\psi_{N+1}=0$. This equation is obtained from Eq. (\[reduced\]) by setting all $\epsilon_{n}=0$. Next we consider $u_{n}(z)$ and $v_{n}(z)$ to be the two linearly independent functions which satisfy the ODE $$\begin{aligned}
\label{un}
u_{n+1}+u_{n-1}+z\,u_{n}&=&0,
\\
\label{vn}
v_{n+1}+v_{n-1}+z\,v_{n}&=&0,\end{aligned}$$ where $u_{0}=v_{N+1}=0$. Since norm of $u_{n},\,v_{n}$ is arbitrary, we fix $u_{1}=v_{N}=1$. Further 