Semiclassical spectra weighted with products of diagonal matrix elements of operators A(alpha), i.e., g(alphaalpha')(E)= summation operator(n)/(E-E(n)), are obtained by harmonic inversion of a cross-correlation signal constructed of classical periodic orbits. The method provides highly resolved semiclassical spectra even in situations of nearly degenerate states, and opens the way to reducing the required signal lengths to shorter than the Heisenberg time. This implies a significant reduction of the number of orbits required for periodic orbit quantization by harmonic inversion.

Semiclassical spectra beyond the Gutzwiller and Berry-Tabor approximation for chaotic and regular systems, respectively, are obtained by harmonic inversion of the ប expansion of the periodic orbit signal. The method is illustrated for the circle billiard, where the semiclassical error is reduced by one to several orders of magnitude with respect to the lowest order approximation used previously.

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Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: Firstly, the periodic orbit quantization will be extended to include higher order corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained.PACS. 03.65.Sq Semiclassical theories and applications

A method for the semiclassical quantization of chaotic maps is proposed, which is based on harmonic inversion. The power of the technique is demonstrated for the baker's map as a prototype example of a chaotic map.The harmonic inversion method for signal processing [1,2] has proven to be a powerful tool for the semiclassical quantization of chaotic as well as integrable dynamical systems [3][4][5]. Starting from Gutzwiller's trace formula for chaotic systems, or the Berry-Tabor formula for integrable systems [6], the harmonic inversion method is able to circumvent the convergence problems of the periodic orbit sums and to directly extract the semiclassical eigenvalues from a relatively small number of periodic orbits. The technique has successfully been applied to a large variety of Hamiltonian systems [4,5]. It has been shown that the method is universal in the sense that it does not depend on any special properties of the dynamical system.In this Letter we demonstrate that the range of application of the harmonic inversion method extends beyond Hamiltonian systems also to quantum maps. Starting from the analogue of Gutzwiller's trace formula for chaotic maps, we show that the semiclassical eigenvalues of chaotic maps can be determined by a procedure very similar to the one for flows. As an example system we consider the well known baker's map. For this map we can take advantage of the fact that the periodic orbit parameters can be determined analytically.We briefly review the basics of quantum maps that are relevant to what follows (for a detailed account of quantum maps see, e.g. Ref. [7]). We consider quantum maps, acting on a finite dimensional Hilbert space of dimension N, which possess a well-defined classical limit for N → ∞. The quantum dynamics is