  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.
