AN EXPONENTIALLY INCREASING SEMICLASSICAL SPECTRAL FORM-FACTOR FOR A CLASS OF CHAOTIC SYSTEMS

The spectral form factor K(tau) plays a crucial role in the understand ing of the statistical properties of quantal energy spectra of strongl y chaotic systems in terms of periodic orbits. It allows the semiclass ical computation of those statistics that are bilinear in the spectral density d(E), like the spectral rigidity DELTA3(L) and the number var iance SIGMA2(L). Since Berry's work on the semiclassical approximation of the spectral rigidity in terms of periodic orbits, it is generally assumed that the periodic-orbit expression for the spectral form fact or universally obeys K(tau) = 1 for tau much greater than 1. Here we s how that for a wide class of strongly chaotic systems, including billi ards with Neumann boundary conditions and the motion on some Riemann s urfaces, the asymptotic behaviour of the semiclassical spectral form f actor K(tau) depends very sensitively on the averaging employed. A Gau ssian averaging is preferable from a theoretical as well as from a num erical point of view to, for example, a rectangular averaging. However , we show in this paper that the Gaussian averaging leads in some case s to an asymptotic behaviour like K(tau) approximately e(ctau), where c > 0 depends only on the energy E at which the statistic is considere d.