Joaquim Puig 2006 Nonlinearity 19 355 doi:10.1088/0951-7715/19/2/007
Joaquim Puig
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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 Cantor spectrum implies the existence of a Gδ-set of energies whose Schrödinger cocycle is not reducible to constant coefficients.
34L40 Particular operators (Dirac, one-dimensional Schrödinger, etc.)
47D08 Schrödinger and Feynman-Kac semigroups
37J40 Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
Issue 2 (February 2006)
Received 15 June 2005, in final form 27 September 2005
Published 9 December 2005
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