eventually escapes to the outer world.

Now we write down the Schrödinger equation for the entire system, $$\begin{aligned}
\label{Sch1}
-t\psi_{n+1}-t\psi_{n-1}&=&\tilde{\mathcal{E}}\psi_{n},~~~~~~~~\text{for $n<0$},
\\
\label{Sch2}
-t\psi_{-1}-t'\psi_{1}&=&\tilde{\mathcal{E}}\psi_{0}, ~~~~~~~~\text{for $n=0$},
\\
\label{Sch3}
-t'\psi_{0}-t\psi_{2}+\epsilon_{1}\psi_{1}&=&\tilde{\mathcal{E}}\psi_{1},~~~~~~~~\text{for $n=1$},
\\
\label{Sch4}
-t\psi_{n-1}-t\psi_{n+1}+\epsilon_{n}\psi_{n}&=&\tilde{\mathcal{E}}\psi_{n},~~~~~~~~\text{for $2\leq n\leq N$.}\end{aligned}$$ In order to avoid cluttering of notations we always represent quantities corresponding to disordered system by [*script letters*]{} while quantities for the clean system are represented in usual math notations. Tilde is used to discriminate the open system case from the closed one. Equation (\[Sch1\]) is for the lead where $\epsilon_{n}=0$. Equations (\[Sch2\], \[Sch3\]) describe the lead-chain coupling and Eq. (\[Sch4\]) is for the chain. As in, [@KS:08] we solve Eqs. (\[Sch1\]-\[Sch4\]) with a boundary condition of an outgoing plane wave in the lead, i.e., $\psi_{n_{\leq 0}}\propto \exp(-i\tilde{k}n)$ where $0<\Re\{\tilde{k}\}<\pi$ and $\Im \{\tilde{k}\}<0$. The condition on $\Re\{\tilde{k}\}$ ensures that the outgoing wave propagates to left in the lead. The condition on $\Im \{\tilde{k}\}$ is considered so that the amplitude of the resonance wave function grows in the lead. It comes from Eq. (\[Sch1\]) that the complex energy $\tilde{\mathcal{E}}$ is related to the complex wave vector $\tilde{k}$ via the dispersion relation $\tilde{\mathcal{E}}=-2 t \cos(\tilde{k})$. Now we eliminate all $\psi_{n}$ for $n<1$ from Eqs. (\[Sch1\]-\[Sch4\]) and 