HARMONIC INVERSION AS A GENERAL-METHOD FOR PERIODIC ORBIT QUANTIZATION

In semiclassical theories for chaotic systems, such as Gutzwiller's pe riodic orbit theory, the energy eigenvalues and resonances are obtaine d as poles of a non-convergent series g(w) = Sigma(n) A(n) exp(is(n) w ). We present a general method for the analytic continuation of such a non-convergent series by harmonic inversion of the 'time' signal, whi ch is the Fourier transform of fi(w) We demonstrate the general applic ability and accuracy of the method on two different systems with compl etely different properties: the Riemann zeta function and the three-di sk scattering system. The Riemann zeta function serves as a mathematic al model for a bound system. We demonstrate that the method of harmoni c inversion by filter-diagonalization yields several thousand zeros of the zeta function to about 12 digit precision as eigenvalues of small matrices. However, the method is not restricted to bound and ergodic systems, and does not require the knowledge of the mean staircase func tion, i.e, the Weyl term in dynamical systems, which is a prerequisite in many semiclassical quantization conditions. It can therefore be ap plied to open systems as well. We demonstrate this on the three-disk s cattering system, as a physical example. The general applicability of the method is emphasized by the fact that one does not have to resort to a symbolic dynamics, which is, in rum, the basic requirement for th e application of cycle expansion techniques.