Periodic orbit quantization: How to make semiclassical trace formulae convergent

Periodic orbit quantization requires an analytic continuation of non-conver gent semiclassical trace formulae. We propose two different methods for sem iclassical quantization. The first method is based upon the harmonic invers ion of semiclassical recurrence functions. A band-limited periodic orbit si gnal is obtained by analytical frequency windowing of the periodic orbit su m. The frequencies of the periodic orbit signal are the semiclassical eigen values, and are determined by either linear predictor, Pade approximant, or signal diagonalization. The second method is based upon the direct applica tion of the Pade approximant to the periodic orbit sum. The Pade approximan t allows the resummation of the, typically exponentially, divergent dynamic s, and can be applied to bound as well as to open systems. Numerical result s are presented for two different systems with chaotic and regular classica l dynamics, viz. the three-disk scattering system and the circle billiard.