we focus on the weak coupling regime, the imaginary part may be viewed as a perturbation of the Hamiltonian of the closed system. The repulsion of the energy levels exhibited by the systems under investigation allows us to consider the eigenenergies of $H$ as nondegenerate. One can therefore apply first-order perturbation theory to obtain from (\[eig\]) the eigenvalues and the eigenvectors of ${\mathcal{H}_{\mathrm{eff}}}$ straightforwardly. The eigenvalues read $E_n-\frac{i}{2}\Gamma_n$, where the $E_n$’s are the eigenvalues of $H$ and the widths $\Gamma_n$ are given by: $$\label{gam}
  \quad \Gamma_n=\sum_{c=1}^M (V_n^c)^2\,.$$ The perturbed eigenvectors of ${\mathcal{H}_{\mathrm{eff}}}$ written in the eigenbasis $\{{| \phi_n \rangle}\}$ of $H$ are easily found as follows: $${| \psi_n \rangle}={| \phi_n \rangle}-i\sum_{p\neq n}\frac{{ \langle \phi_p |}VV^T{| \phi_n \rangle}}{2(E_n-E_p)}{| \phi_p \rangle}\,.$$ Splitting then the real and imaginary parts of ${| \psi_n \rangle}$, the complexness parameter (\[qdef\]) of a given eigenvector reads $$\label{q2def}
  q^2_n=\sum_{p\neq n}\frac{\Gamma_{np}^2}{4(E_n-E_p)^2}\,,$$ where we have introduced $\Gamma_{np}=\sum_{c=1}^MV_n^cV_p^c$. These quantities are responsible for the coupling and interference of the resonance states due to the common decay channels [@Sok89].

In what follows, we study the statistical properties of the complexness parameter (\[q2def\]) for $H$ being described by a picket-fence or belonging to GOE. It is worth noting here that expression (\[q2def\]) is 