A semiclassical sum rule for matrix elements of classically chaotic systems

Abstract

In the semiclassical limit, the sum S(E, Delta E)= Sigma _nm mod A_nm mod ² delta (E-1/2(E_n+E_m)) delta ( Delta E-(E_n-E_m)) 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: C_A(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 C_A(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.