elastodynamics are available. In particular, in the case of vibrating plates, a complete knowledge of the eigenfunctions can be obtained through noninvasive measurements [@Xer09] even for moderate overlap of resonances. Indeed, the understanding of the statistics of eigenfunctions beyond the perturbative regime still remains an open problem. (We note that some relevant interesting numerical results for microwave billiards with large openings were recently reported [@Bul07].) Finally, one should also note that the complexness parameter may be considered as a sensitive probe of the crossover from localized to extended states in open disordered systems [@Van09].

We are grateful to H.-J. Sommers for his instructive advice and help with evaluating Eq. (\[p\_M(X)bessel\]). One of us (D.V.S.) acknowledges gratefully the generous hospitality of LPMC in Nice and the financial support of University of Nice during his stay there. The partial financial support by BRIEF grant (D.V.S.) is also acknowledged.

Derivation of Eqs. (\[p\_M(X)bessel\]) and (\[p\_M(X)odd\]) {#AppB}
===========================================================

We first note that the integrand of Eq. (\[p\_M(X)int\]) is a symmetric function in $z$ that allows us to restrict the $z$-integration to the positive axis. Then we deform the contour of integration over $\omega$ from the real to imaginary axis by putting $\Omega=i\omega$. Performing after 