Semiclassical non-trace-type formulas for matrix-element fluctuations and weighted densities of states

Densities of states weighted with the diagonal matrix elements of two operators $Â$ and $\mathrm{B̂},$ i.e., ${\rho }^{\mathrm{}(A,B)\mathrm{}}\mathrm{}\left(E\right)={\sum }_n〈n|Â|n〉〈n|\mathrm{B̂}|n〉\delta \left(E-E_n\right),$ cannot, in general, be written as a trace formula, and therefore no simple extension of semiclassical trace formulas is known for this case. However, from the high resolution analysis of quantum spectra in the semiclassical regime we find strong evidence that weighting the δ functions in the quantum mechanical density of states with the product of diagonal matrix elements, $〈n|Â|n〉〈n|\mathrm{B̂}|n〉,$ is equivalent to weighting the periodic orbit contributions in the semiclassical periodic orbit sum with the product of the periodic orbit means, $〈A{〉}_p〈B{〉}_p,$ of the classical observables A and B. Results are presented for the hydrogen atom in a magnetic field for both the chaotic and near-integrable regime, and for the circle billiard.