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 opera tors a and (A) over cap and (B) over cap, i.e., rho((A,B)) (E) = Sigma n[n\ (A) over cap\n][n\(B) over cap\n]delta(E-E-n), cannot, in general, be writt en as a trace formula, and therefore no simple extension of semiclassical t race formulas is known for this case. However, from the high resolution ana lysis of quantum spectra in the semiclassical regime we find strong evidenc e that weighting the delta functions in the quantum mechanical density of s tates with the product of diagonal matrix elements, [n\(A) over cap\n][n\(B ) over cap\n], is equivalent to weighting the periodic orbit contributions in the semiclassical periodic orbit sum with the product of the periodic or bit 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 an d near-integrable regime, and for the circle billiard. [S1063-651X(99)08008 -3].