Regularity of the rotation number for the one-dimensional time-continuous Schrödinger equation (Q1928783)

In this paper, the author studies the behavior of the rotation number of the one-dimensional time-continuous Schrödinger equation with Diophantine frequencies and a small analytic potential. It is proved that the rotation number of this equation has the behavior of a \(\frac{1}{2}\)-Hölder function. The author also gives a sub-exponential estimate of the length of the gaps which depends on its label given by the gap-labeling theorem.