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Almost localization and almost reducibility. (English) Zbl 1185.47028
This paper deals with the dynamics of Schrödinger cocycles associated to a non-perturbatively small analytic potential and Diophantine frequency. Previous results in this field are mostly based on the classical Aubry duality: a Fourier-type transform matches localized eigenfunctions to analytic Bloch waves for the eigenvalue equation of the operator. A drawback in such approaches is the inability to study the set of energies where localization fails and hence they could not lead to an understanding of the whole spectrum. The present paper addresses this issue by developing a quantitive version of Aubry duality that can handle all energies and providing fine estimates for the dynamics of the Schrödinger operator. The results are used to establish the full version of Eliasson’s reducibility theory in the non-perturbative setting. The dynamical estimates also yield several immediate corollaries, including 1/2-Hölder continuity of the integrated density of states and the non-collapse of spectral gaps for the almost Mathieu operator.

MSC:
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
37C55 Periodic and quasi-periodic flows and diffeomorphisms
39A70 Difference operators
47A10 Spectrum, resolvent
47B39 Linear difference operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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