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We derive a trace formula for $\sum_n A_{nn}B_{nn}...\delta(E-E_n)$, where$A_{nn}$ is the diagonal matrix element of the operator $A$ in the energy basisof a chaotic system. The result takes the form of a smooth term plusperiodic-orbit corrections; each orbit is weighted by the usual Gutzwillerfactor times $A_p B_p ...$, where $A_p$ is the average of the classicalobservable $A$ along the periodic orbit $p$. This structure for the orbitcorrections was previously proposed by Main and Wunner (chao-dyn/9904040) onthe basis of numerical evidence.Comment: 8 pages; analysis made more rigorous in the revised versio

Trace formula for products of diagonal matrix elements in chaotic systems