Random Matrix Elements and Eigenfunctions in Chaotic Systems

The expected root-mean-square value of a matrix element $A_{\alpha \beta }$ in a classically chaotic system, where $A$ is a smooth, $\mathrm{\hslash }$-independent function of the coordinates and momenta, and $\alpha$ and $\beta$ label different energy eigenstates, has been evaluated in the literature in two different ways: by treating the energy eigenfunctions as gaussian random variables and averaging $\mathrm{\mid }{A}_{\alpha \beta }{\mathrm{\mid }}^2$ over them; and by relating $\mathrm{\mid }{A}_{\alpha \beta }{\mathrm{\mid }}^2$ to the classical time-correlation function of $A$. We show that these two methods give the same answer only if Berry’s formula for the spatial correlations in the energy eigenfunctions (which is based on a microcanonical density in phase space) is modified at large separations in a manner which we previously proposed.