obtain $$\label{reduced}
-t\psi_{n+1}-t\psi_{n-1}+\tilde{\epsilon}_{n}\psi_{n}=\tilde{\mathcal{E}}\psi_{n},$$ where $$\label{energy}
\tilde{\epsilon}_{n}=\epsilon_{n}-t\eta\, \exp(i\tilde{k})\delta_{n1},$$ for $n=1,\,2,...,\,N$. The parameter $\eta=(t'/t)^2$ measures the coupling strength to the outside world.

An effective Hamiltonian defined by the Eq. (\[reduced\]) is non-Hermitian. For instance, if $\mathcal{H}$ is the $N\times N$ tridiagonal Hermitian matrix which represents the Hamiltonian of the closed-disordered chain then one may write the effective Hamiltonian, $\tilde{\mathcal{H}}$, as $$\label{Hamil}
\tilde{\mathcal{H}}=\mathcal{H}-t\,\eta\,\lambda(\tilde{k})\, P.$$ Here $P=|1\rangle\langle1|$ is the projection for site $n=1$ and $\lambda=\exp(i\tilde{k})$. The above non-Hermitian effective Hamiltonian has been first obtained by Terraneo and Guarnery [@GG:00]. The underlying result here is that the same relation (\[Hamil\]) is valid for any Hermitian $\mathcal{H}$ representing a (closed) quantum system [@JF:10] which has $N$-dimensional state space. Resonances are characterized by the complex eigenvalues, $\tilde{\mathcal{E}}_{\alpha}$, of $\tilde{\mathcal{H}}$.

Note here dependency of $\tilde{\mathcal{H}}$ on the complex wave vector $\tilde{k}$ which is related to the complex energies via the dispersion relation mentioned above - this is not a standard eigenvalue problem. To standardize this problem “parametric resonances” are often used as an alternative. In this approach the dependence of $\lambda$ on $\tilde{k}$ is typically neglected, reducing thereby the problem of finding the eigenvalues of the effective Hamiltonian at chosen value of $\tilde{k}$. As expected, parametric resonances yield 