Non-tangential limit of the Weyl m-functions for the ergodic Schrödinger equation

with Rafael Obaya,
J. Dynam. Differential Equations 10 (1998), 209-257.

This paper deals with the spectral and qualitative problems associated with the
one-dimensional ergodic Schrödinger equation. Let A2 be the set of those energies
for which the real projective flow admits an invariant linear measure with square
integrable density function. On this set the directional derivative of the Floquet
coefficient is calculated and the existence of non-tangential limit of the Weyl
-functions in the L1-topology is proved. As a consequence, it turns out that the
known Deift-Simon inequality for the derivative of the rotation number obtained
from Kotani's theory is in fact an equality. In the bounded orbit case, the uniform
boundedness of the Weyl m-functions is deduced and necessary and sufficient
conditions to ensure their uniform convergence are obtained.