Stroboscopic quantization of autonomous systems

We introduce a semiclassical quantization method which is based on a strobo scopic description of the classical and the quantum flows. We show that thi s approach emerges naturally when one is interested in extracting the energ y spectrum within a prescribed and finite energy interval. The resulting se miclassical expression involves a finite number of periodic orbits whose en ergies are in the considered interval. Higher order corrections which refle ct the sharp restriction of the spectrum to an interval are explicitly give n. The relation to Fourier methods for extracting semiclassical spectra, su ch as harmonic inversion, is worked out. The constraints due to the finite dimension of the Hilbert space and the unitarity of the restricted quantum evolution operator are important ingredients in this context.