A Trace Formula for Products of Diagonal Matrix Elements in Chaotic Systems

We derive a trace formula for ${\sum }_n{A}_{nn}{B}_{nn}\mathrm{.}\mathrm{.}\mathrm{.}\delta (E-E_n)$, where $A_{nn}$ is the diagonal matrix element of the operator $A$ in the energy basis of a chaotic system. The result takes the form of a smooth term plus periodic-orbit corrections; each orbit is weighted by the usual Gutzwiller factor times $A_p{B}_p\mathrm{.}\mathrm{.}\mathrm{.}$, where $A_p$ is the average of the classical observable $A$ along the periodic orbit $p$. This structure for the orbit corrections was previously proposed by Main and Wunner (chao-dyn/9904040) on the basis of numerical evidence.