Regularity of the Rotation Number for the One-Dimensional Time-Continuous Schrödinger Equation

Starting from results already obtained for quasi-periodic co-cycles in SL(2, R), we show that the rotation number of the one-dimensional time-continuous Schrödinger equation with Diophantine frequencies and a small analytic potential has the behavior of a 1/2-Hölder function. We give also a sub-exponential estimate of the length of the gaps which depends on its label given by the gap-labeling theorem.