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 understanding of the statistical properties of quantal energy spectra of strongly chaotic systems in terms of periodic orbits. It allows the semiclassical computation of those statistics that are bilinear in the spectral density d(E), like the spectral rigidity Delta ₃(L) and the number variance Sigma ²(L). Since Berry's (1985, 1991) 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 factor universally obeys K( tau )=1 for tau >>1. We show that for a wide class of strongly chaotic systems, including billiards with Neumann boundary conditions and the motion on some Riemann surfaces, the asymptotic behaviour of the semiclassical spectral form factor K( tau ) depends very sensitively on the averaging employed. A Gaussian averaging is preferable from a theoretical as well as from a numerical point of view to, for example, a rectangular averaging. However, we show in this paper that the Gaussian averaging leads in some cases to an asymptotic behaviour like K( tau ) approximately e^{c tau}. Where c>0 depends only on the energy E at which the statistic is considered.