Instability zones of periodic 1-dimensional Schrodinger and Dirac operators

The spectra of Schrodinger and Dirac operators with periodic potentials on the real line R have a band structure, that is, the intervals of continuous spectrum alternate with spectral gaps, or instability zones. The sizes of these zones decay, and the rate of decay depends on the smoothness of the potential. In the opposite direction, one can make conclusions about the smoothness of a potential based on the rate of decay of the instability zones. In the 1960s and 1970s this phenomenon was understood at the level of infinitely differentiable or analytic functions in the case of Schrodinger operators. However, only recently has the relationship between the smoothness of the potential and the rate of decay of the instability zones become completely understood and analyzed - for a broad range of classes of differentiable functions, - for Dirac operators and not just for Hill-Schrodinger operators, - in both the self-adjoint and non-self-adjoint cases. This paper is devoted to a survey of these results, mostly with complete proofs based on an approach developed by the authors.