Nontangential Limit of the Weyl m-Functions for the Ergodic Schrödinger Equation

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Abstract

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 we calculate the directional derivative of the Floquet coefficient and prove the existence of a nontangential limit of the Weyl m-functions in the L1-topology. In particular, we verify that the known Dieft–Simon inequality for the derivative of the rotation number obtained from Kotani's theory is in fact an equality. In the bounded orbit case we deduce the uniform boundedness of the Weyl m-functions and obtain necessary and sufficient conditions to assure their uniform convergence.

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