Semiclassical spectra and diagonal matrix elements by harmonic inversion of cross-correlated periodic orbit sums

Semiclassical spectra weighted with products of diagonal matrix elements of operators $A_{\alpha },$ i.e., $g_{\alpha {\alpha }^{\prime }}\mathrm{}\left(E\right)={\sum }_n〈n|A_{\alpha }|n〉〈n|A_{{\alpha }^{\prime }}|n〉/\left(E-E_n\right),$ are obtained by harmonic inversion of a cross-correlation signal constructed of classical periodic orbits. The method provides highly resolved semiclassical spectra even in situations of nearly degenerate states, and opens the way to reducing the required signal lengths to shorter than the Heisenberg time. This implies a significant reduction of the number of orbits required for periodic orbit quantization by harmonic inversion.