Random matrix analysis of the monopole strength distribution in ${}^{208}$Pb

We study statistical properties of the $0^{+}$ spectrum of ${}^{208}$Pb in the energy region $E_x\le 20$ MeV. We use the Skyrme interaction SLy4 as our model Hamiltonian to create a single-particle spectrum and to analyze excited states. The finite-rank separable approximation for the particle-hole interaction enables us to perform the calculations in large configuration spaces. We show that while the position of the monopole resonance centroid is determined by one phonon excitations of $0^{+}$, the phonon-phonon coupling is crucial for the description of a strength distribution of the $0^{+}$ spectrum. In fact, this coupling has an impact on the spectral rigidity ${\mathrm{\Delta }}_3(L)$ which is shifted towards the random matrix limit of the Gaussian orthogonal ensembles.