  We introduce a semiclassical quantization method which is based on a
stroboscopic description of the classical and the quantum flows. We show that
this approach emerges naturally when one is interested in extracting the energy
spectrum within a prescribed and finite energy interval. The resulting
semiclassical expression involves a finite number of periodic orbits whose
energies are in the considered interval. Higher order corrections which reflect
the sharp restriction of the spectrum to an interval are explicitely given. The
relation to Fourier methods for extracting semiclassical spectra, such 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.

Chaotic Dynamics