Semiclassical quantization by Padé approximant to periodic orbit sums

– Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose a method for semiclassical quantization based upon the Padé approximant to the periodic orbit sums. The Padé approximant allows the re-summation of the typically exponentially divergent periodic orbit terms. The technique does not depend on the existence of a symbolic dynamics and can be applied to both bound and open systems. Numerical results are presented for two different systems with chaotic and regular classical dynamics, viz. the three-disk scattering system and the circle billiard. Semiclassical theories link the spectrum of a quantum system to the dynamics of its classical counterpart and thereby play an important role in the deeper understanding of the relation between quantum and classical mechanics. Of particular interest are periodic orbit theories which describe the quantum-mechanical density of states in terms of contributions from the periodic orbits of the classical system. The semiclassical trace formulae have been derived by Gutzwiller for classically chaotic systems [1,2] and by Berry and Tabor for regular (integrable) systems [3]. A common feature of these formulae is that in most cases the periodic orbit sum does not converge in those regions where the semiclassical eigenenergies or resonances are located. These resonances are given as the poles of the periodic orbit sum,