completely rigid spectrum, which may be viewed as an approximation of the GOE spectrum where the fluctuations are neglected.

The picket-fence model {#sec:pf}
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In this model the eigenenergies of the closed system are equally spaced, i.e. $E_n-E_{n\pm k}=\pm k\Delta$, and the eigenvector components are random Gaussian variables. The complexness parameter is then given by $$\label{Xpf}
  X_n = \gamma_n \sum_{k\neq 0}\frac{Z_k}{4k^2}\,.$$ This expression does not have any divergence problems of the GOE case, thus statistics of (\[Xpf\]) can be also characterized by its moments. In particular, the average value is easily found to be exactly given by Eq. (\[Xvar\]), with the factor $f$ being $$\label{fpf}
 f = \sum_{k\ne 0}\frac{1}{4k^2} = \frac{\pi^2}{12}\,.$$ Figure 1(b) illustrates the dependence ${\left \langle X \right \rangle}=\frac{\pi^2}{24}\mathrm{var}(\gamma)$.

We now derive an exact expression for the probability distribution ${\mathcal{P}^{\mathrm{pf}}}_M(X)$ in the picket-fence case. First we substitute in the definition (\[PX\]) the Fourier representation of the delta function, $\delta(X-X_n)=\int\frac{d\omega}{2\pi}e^{i\omega(X-X_n)}$, where $X_n$ is given by Eq. (\[Xpf\]). Then the integration over the projections $Z_k$ with the help of Eq. (\[pZ\]) becomes trivial, yielding $$\label{pdem}
 {\mathcal{P}^{\mathrm{pf}}}_M(X) = \int\limits_{-\infty}^\infty \frac{d\omega}{2\pi}\;e^{i\omega X} \int \limits_{0}^\infty d\gamma \, P_M(\gamma) \prod_{k=1}^\infty \frac{1}{1+i\frac{\omega\gamma}{2k^2}} \,.$$ The infinite product here can be evaluated explicitly [@Abr65]. Making use of 