Global theory of one-frequency Schrodinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity

Abstract

We study Schrodinger operators with a one-frequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials. From the dynamical point of view, the transition signals the emergence of nonuniform hyperbolicity, so the dependence of the Lyapunov exponent with respect to parameters plays a central role in the analysis. Though often ill-behaved by conventional measures, we show that the Lyapunov exponent is in fact remarkably regular in a ``stratified sense'' which we define: the irregularity comes from the matching of nice (analytic or smooth) functions along sets with complicated geometry. This result allows us to stablish that the ``critical set'' for the transition has at most codimension one, so for a typical potential the set of critical energies is at most countable, hence typically not seen by spectral measures. Key to our approach are two results about the dependence of the Lyapunov exponent of one-frequency $\SL(2,\C)$ cocycles with respect to perturbations in the imaginary direction: on one hand there is a severe ``quantization'' restriction, and on the other hand ``regularity'' of the dependence characterizes uniform hyperbolicity when the Lyapunov exponent is positive. Our method is independent of arithmetic conditions on the frequency.