  Densities of states weighted with the diagonal matrix elements of two
operators A and B, i.e., rho^(A,B)(E) = sum_n <n|A|n><n|B|n> delta(E-E_n)
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 delta-functions in the quantum
mechanical density of states with the product of diagonal matrix elements,
<n|A|n><n|B|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.
