M. Wilkinson et Pn. Walker, A BROWNIAN-MOTION MODEL FOR THE PARAMETER DEPENDENCE OF MATRIX-ELEMENTS, Journal of physics. A, mathematical and general, 28(21), 1995, pp. 6143-6160
We introduce a Brownian motion model for the parametric evolution of e
igenstates of a complex quantum system, modelled by a random matrix. T
he model is analogous to Dyson's model for the evolution of the eigenv
alues. We use this approach to analyse correlation functions describin
g the parameter dependence of diagonal and off-diagonal matrix element
s of a generic operator. In the case of diagonal matrix elements, we c
ompare our results with a semiclassical approach, which relates sums o
f matrix elements to periodic classical orbits. For systems with a cha
otic classical limit, the semiclassical correlation function agrees ex
actly with the random matrix theory.