small compared to that of the levels [@Leh95a; @Leh95b]. The coupling amplitudes may be equivalently chosen as fixed [@Verb85] or random [@Sok89]. In order to preserve orthogonal invariance of ${\mathcal{H}_{\mathrm{eff}}}$ under (complex) orthogonal transformations [@Sok89], we consider the $V_n^c$’s as real Gaussian random variables with zero mean and $$\label{V}
  {\left \langle V_n^c V_{n'}^{c'} \right \rangle}= (2\kappa\Delta/\pi) \delta_{nn'} \delta^{cc'}
  \equiv \sigma^2 \delta_{nn'} \delta^{cc'} \,.$$ Henceforth, ${\left \langle \cdots \right \rangle}$ stands for the statistical averaging over the ensembles. The coupling constant $\kappa$ determines a transmission coefficient $T=1-|{\left \langle S \right \rangle}|^2=4\kappa/(1+\kappa)^2$ of the channels (assumed to be statistically equivalent). The cases of $T\ll 1$ or $T=1$ correspond, respectively, to weak or perfect coupling. In the weak coupling regime considered below, $\kappa\ll1$, all the resonances are almost isolated and ${\left \langle \Gamma \right \rangle} \ll \Delta$.

Perturbative approach
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\[sec:level1\]Complexness parameter in the weak coupling regime
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We now derive an expression for the complexness parameter of the eigenvectors for weakly overlapping resonances. The matrix representation of ${\mathcal{H}_{\mathrm{eff}}}$ in an arbitrary basis $\{{| n \rangle}\}$ of the Hilbert space spanned by eigenvectors of $H$ reads: $${\mathcal{H}_{\mathrm{eff}}}= \sum_{n,p=1}^N {| n \rangle}H_{np} { \langle p |}-\frac{i}{2}\sum_{n,p=1}^N \sum_{c=1}^M {| n \rangle}V_n^cV_p^c{ \langle p |}$$ As 