Long-time divergence of semiclassical form-factors

Abstract

The quantum form-factor K( tau )-the Fourier transform of the spectral 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 tau . It is shown that whilst K itself tends to unity, the periodic-orbit sum typically grows exponentially as tau to infinity . This behaviour is related to the fact that leading-order semiclassical quantization methods yield complex eigenvalues with imaginary parts that are of higher order in Planck's constant. Divergence from the quantum limit begins when tau = tau *(h(cross)), which, for typical two-degrees-of-freedom systems and maps, is shown to be independent of h(cross) as h(cross) to 0. In the case of the baker's map, however, quantum diffraction from the classical discontinuity instead causes the analogue of tau * to tend to zero like N^-12/, Where N is the integer that corresponds to the inverse of Planck's constant. This is in agreement with recent numerical studies. Finally, we consider the implications of the semiclassical divergence studied here for the method developed by Argaman et al. (1993) of investigating correlations between the periodic orbits of chaotic systems.