size $N\times M$, the elements $V_n^c$ of this matrix couple the $n$-th level to the $c$-th scattering channel. The poles of $S$ are given by the eigenvalues of ${\mathcal{H}_{\mathrm{eff}}}$. Assuming an independence of the coupling elements from the energy and neglecting direct processes [@Verb85], the effective Hamiltonian of the open systems is represented as follows: $$\label{heff}
  {\mathcal{H}_{\mathrm{eff}}}= H - \frac{i}{2}VV^{\dag} ,$$ where $H$ is the Hamiltonian of the closed system and the anti-Hermitian part $\frac{i}{2}VV^{\dag}$ describes coupling to the channels. In the case of the systems with preserved TRS considered below, $H$ is a real symmetric matrix of size $N\times N$ and $V$ is also real. As usual, the limit $N\rightarrow \infty$ is to be finally taken.

Since ${\mathcal{H}_{\mathrm{eff}}}$ is a non-Hermitian operator, the eigenvalue problems ${\mathcal{H}_{\mathrm{eff}}}{|  \psi_n \rangle}= \mathcal{E}_n{| \psi_n \rangle}$ and ${ \langle \tilde{\psi}_n |}{\mathcal{H}_{\mathrm{eff}}}= \mathcal{E}_n { \langle \tilde{\psi}_n |}$ define two sets of a priori independent eigenvectors, called right $\{{| \psi_n \rangle}\}$ and left $\{{ \langle \tilde{\psi}_n |}\}$ eigenvectors associated to the same set of eigenvalues $\{\mathcal{E}_n\}$. These eigenvectors form a bi-orthogonal set which satisfies conditions of orthogonality, ${ \langle \tilde{\psi}_n |}\psi_m\rangle = \delta_{nm}$, and completeness, $\sum_n{| \psi_n \rangle}{ \langle \tilde{\psi}_n |}=1$. Making use of the right 