hbar expansion for the 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 hbar 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.