Long-time divergence of semiclassical form-factors

Abstract. The quantum form-factor K (r)-the Fourier transform of the spectnl auto-correlation function-may be represented semiclassically in terms of a sum over classical periodic orbits. We consider the problem of how this approximation behaves in the limit of long (scaled) time r. It is shown that whilst K itself tends to unity. the periodic-orbit sum typically grows exponentially as z + 00. This behaviour is related to the f a t that leadingorder semiclassical quantimion methods yield complex eigenvalues with imaginary pans that are of higher order in Planck's constant. Divergence from the quantum limil begins when I = r'p), which, for typical two-degreesof-freedom systems and maps, is shown to be independent of h as f i + 0. In the case of the baker's map. however. quantum diffraction from the classical discontinuity instead c a w s the analogue of r ' to tend to zero like N- ' D, where N is the integer that corresponds to the inverse of Planck's constant. This is in agreement with recent numerical sludies. Finally. we consider the implications of the semiclassical divergence studied here for the method developed by & g a m er 01 (1993) of investigating correlations between lhe periodic orbits of chaotic systems. 1