Obaya R., Paramio M.
Directional Differentiability of the Rotation Number for the
Almost Periodic Schr\"{o}dinger Equation
(87K, LaTeX)

ABSTRACT.  Let us consider the one-dimensional Schr\"{o}dinger equation
\begin{equation}\label{prim}
        L_{E}(x)=-x^{''}+g_0(t)x-Ex=0
\end{equation}
 with almost periodic potential $g_0$. The set $A$ of energies where  the Lyapuo
v
exponent
vanishes is also known to be the essential support of the absolutely continuous
part of the spectral measure. This paper deals with the variation of the
rotation number on $A$, in particular with the differentiability and Lipschitz
character of this map.