A Nonperturbative Eliasson's Reducibility Theorem

Abstract

This paper is concerned with discrete, one-dimensional Schrödinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasi-periodic Bloch wave if the potential is smaller than a certain constant which does not depend on the precise Diophantine conditions. The associated first-order system, a quasi-periodic skew-product, is shown to be reducible for almost all values of the energy. This is a partial nonperturbative generalization of a reducibility theorem by Eliasson. We also extend nonperturbatively the genericity of Cantor spectrum for these Schrödinger operators. Finally we prove that in our setting, Cantor spectrum implies the existence of a $G_{\delta }$-set of energies whose Schrödinger cocycle is not reducible to constant coefficients.

References in corpus (4)

• Reducibility or non-uniform hyperbolicity for quasiperiodic Schrodinger cocycles
• Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on $\mathbf{T}timesSL(2,\mathbf{R})$
• Nonperturbative localization
• Persistence under Weak Disorder of AC Spectra of Quasi-Periodic Schroedinger operators on Trees Graphs

Cited by in corpus (27)

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• Strange Nonchaotic Attractors in Harper Maps
• Asymptotics of spectral gaps of quasi-periodic Schrödinger operators
• Holder continuity of absolutely continuous spectral measures for one-frequency Schrodinger operators
• Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies
• Spectral Gaps of Almost Mathieu Operator in Exponential Regime
• Anderson localization for the completely resonant phases
• Expansions in the delay of quasi-periodic solutions for state dependent delay equations
• Hölder Continuity of the Spectral Measures for One-Dimensional Schrödinger Operator in Exponential Regime
• On the abominable properties of the almost Mathieu operator with well approximated frequencies
• Reducibility of finitely differentiable quasi-periodic cocycles and its spectral applications
• The dynamics of a class of quasi-periodic Schrödinger cocycles
• Almost Mathieu operators with completely resonant phases
• The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations
• Stability of the non-critical spectral properties I: arithmetic absolute continuity of the integrated density of states
• Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation
• Almost localization and almost reducibility
• Exponential decay of the size of spectral gaps for quasiperiodic Schrödinger operators
• Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles
• Hölder regularity of the integrated density of states for quasi-periodic long-range operators on ${\ell }^2({\mathbb{Z}}^d)$
• Exponential Decay of the lengths of Spectral Gaps for Extended Harper's Model with Liouvillean Frequency
• Holder Continuity of Absolutely Continuous Spectral Measure for Multi-frequency Schrodinger Operators
• Quantitative almost reducibility and Möbius disjointness for analytic quasiperiodic Schrodinger cocycles
• Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications