LONG-TIME DIVERGENCE OF SEMICLASSICAL FORM-FACTORS

The quantum form-factor K(tau)-the Fourier transform of the spectral a uto-correlation function-may be represented semiclassically in terms o f a sum over classical periodic orbits. We consider the problem of how this approximation behaves in the limit of long (scaled) time tau. It is shown that whilst K itself tends to unity, the periodic-orbit sum typically grows exponentially as tau --> infinity. This behaviour is r elated to the fact that leading-order semiclassical quantization metho ds yield complex eigenvalues with imaginary parts that are of higher o rder in Planck's constant. Divergence from the quantum limit begins wh en tau = tau(hBAR), which, for typical two-degrees-of-freedom systems and maps, is shown to be independent of hBAR as hBAR --> 0. In the ca se of the baker's map, however, quantum diffraction from the classical discontinuity instead causes the analogue of tau to tend to zero lik e N-1/2, where N is the integer that corresponds to the inverse of Pla nck's constant. This is in agreement with recent numerical studies. Fi nally, we consider the implications of the semiclassical divergence st udied here for the method developed by Argaman et al (1993) of investi gating correlations between the periodic orbits of chaotic systems.