Higher-order

In the periodic orbit quantization of physical systems, usually only the leading-order $\hbar$ contribution to the density of states is considered. Therefore, by construction, the eigenvalues following from semiclassical trace formulae generally agree with the exact quantum ones only to lowest order of $\hbar$. In different theoretical work the trace formulae have been extended to higher orders of $\hbar$. The problem remains, however, how to actually calculate eigenvalues from the extended trace formulae since, even with $\hbar$ corrections included, the periodic orbit sums still do not converge in the physical domain. For lowest-order semiclassical trace formulae the convergence problem can be elegantly, and universally, circumvented by application of the technique of harmonic inversion. In this paper we show how, for general scaling chaotic systems, also higher-order $\hbar$ corrections to the Gutzwiller formula can be included in the harmonic inversion scheme, and demonstrate that corrected semiclassical eigenvalues can be calculated despite the convergence problem. The method is applied to the open three-disk scattering system, as a prototype of a chaotic system