Classical transport effects on chaotic levels

We show that the quantum spectra of simple systems whose classical counterpart is of a mixed nature, i.e., partly regular with widespread chaos, manifest the effects of classical transport through imperfect barriers. The partial barriers are characterized by the flux crossing them. We derive the relationship between this flux and quantum Hamiltonian matrix elements. This in turn predicts new statistical fluctuations properties for the spectrum and partial localization of the wave functions. The example of two coupled quartic oscillators is given in detail.