A semiclassical sum rule for matrix elements of classically chaotic systems

In the semiclassical limit, the sum S(E, Delta E)= Sigma nm mod Anm mod 2 delta (E-1/2(En+Em)) delta ( Delta E-(En-Em)) of matrix elements of an arbitrary operator -A can be related to the classical correlation function of the Weyl symbol A(q, p) of -A: CA(E, t)= integral d alpha delta (E-H( alpha ))A( alpha )A( alpha t). S(E, Delta E) is proportional to the Fourier transform of CA(E, t) over t, plus a set of correction terms associated with periodic trajectories in phase space. If the system has a chaotic classical limit, the matrix elements are independently Gaussian distributed with mean value zero, and S(E, Delta E) gives the variance of this distribution.