B. Grebert, T. Kappeler Estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system (113K, latex) ABSTRACT. We prove new estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system with complex periodic potentials. Our method allows us to characterize in a precise way the decreasing properties of the gap length sequence in terms of regularity of complex potentials in weighted Sobolev spaces.