Semiclassical form factor for spectral and matrix element fluctuations of multidimensional chaotic systems

Abstract

We present a semiclassical calculation of the generalized form factor K ab(τ) which characterizes the fluctuations of matrix elements of the operators â and b̂ in the eigenbasis of the Hamiltonian of a chaotic system. Our approach is based on some recently developed techniques for the spectral form factor of systems with hyperbolic and ergodic underlying classical dynamics and f = 2 degrees of freedom, that allow us to go beyond the diagonal approximation. First we extend these techniques to systems with f>2. Then we use these results to calculate K ab(τ). We show that the dependence on the rescaled timer (time in units of the Heisenberg time) is universal for both the spectral and the generalized form factor. Furthermore, we derive a relation between K ab(τ) and the classical time-correlation function of the Weyl symbols of â and b̂. © 2005 The American Physical Society.