Harmonic inversion is introduced as a powerful tool for both the analysis ofquantum spectra and semiclassical periodic orbit quantization. The methodallows to circumvent the uncertainty principle of the conventional Fouriertransform and to extract dynamical information from quantum spectra which hasbeen unattainable before, such as bifurcations of orbits, the uncovering ofhidden ghost orbits in complex phase space, and the direct observation ofsymmetry breaking effects. The method also solves the fundamental convergenceproblems in semiclassical periodic orbit theories - for both the Berry-Taborformula and Gutzwiller's trace formula - and can therefore be applied as anovel technique for periodic orbit quantization, i.e., to calculatesemiclassical eigenenergies from a finite set of classical periodic orbits. Theadvantage of periodic orbit quantization by harmonic inversion is theuniversality and wide applicability of the method, which will be demonstratedin this work for various open and bound systems with underlying regular,chaotic, and even mixed classical dynamics. The efficiency of the method isincreased, i.e., the number of orbits required for periodic orbit quantizationis reduced, when the harmonic inversion technique is generalized to theanalysis of cross-correlated periodic orbit sums. The method provides not onlythe eigenenergies and resonances of systems but also allows the semiclassicalcalculation of diagonal matrix elements and, e.g., for atoms in externalfields, individual non-diagonal transition strengths. Furthermore, it ispossible to include higher order terms of the hbar expanded periodic orbit sumto obtain semiclassical spectra beyond the Gutzwiller and Berry-Taborapproximation.
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