systems, namely, spectral rigidity, we investigate the case of a system whose closed limit is described by a pure picket-fence spectrum. An exact analytic prediction for the probability distribution of the complexness parameter is derived, depending on only two parameters: the number of open scattering channels and the mean resonance width. Finally, we consider the more realistic case of systems modeled by the Gaussian orthogonal ensemble (GOE). We derive an analytic expression for the probability distribution of the complexness parameter in this case and discuss the effect of spectral fluctuations.

Effective Hamiltonian formalism
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Scattering approach
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Open wave systems are commonly described using the so-called projection formalism [@Mah69; @Oko03]. The exterior coupling is modeled by $M$ scattering channels connected to $N$ levels of a closed system. The coupling to the environment turns modes, with a infinite life time, into resonances, with a finite life time. Being initially introduced in nuclear physics, this formalism has been later applied successfully to wave billiards [@Dit00] for which antennas and absorption can be described by scattering channels [@Bar05pre]. In this approach, the resonance part of the $S$-matrix is given by: $$S(E) = 1 - iV^{\dag}\frac{1}{E-{\mathcal{H}_{\mathrm{eff}}}}V ,$$ where $V$ is the coupling matrix of 