A semiclassical sum rule for matrix elements of classically chaotic systems
- 21 June 1987
- journal article
- Published by IOP Publishing in Journal of Physics A: General Physics
- Vol. 20 (9) , 2415-2423
- https://doi.org/10.1088/0305-4470/20/9/028
Abstract
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.Keywords
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