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GLOBAL THEORY OF ONEFREQUENCY SCHRÖDINGER OPERATORS I: STRATIFIED ANALYTICITY OF THE LYAPUNOV EXPONENT AND THE BOUNDARY OF NONUNIFORM HYPERBOLICITY
, 2009
"... We study Schrödinger operators with a onefrequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials. From the dynamical point of view, the transition signals the emergence of nonuniform hyperbolicity, s ..."
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Cited by 21 (3 self)
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We study Schrödinger operators with a onefrequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials. From the dynamical point of view, the transition signals the emergence of nonuniform hyperbolicity, so the dependence of the Lyapunov exponent with respect to parameters plays a central role in the analysis. Though often illbehaved by conventional measures, we show that the Lyapunov exponent is in fact remarkably regular in a “stratified sense ” which we define: the irregularity comes from the matching of nice (analytic or smooth) functions along sets with complicated geometry. This result allows us to stablish that the “critical set ” for the transition has at most codimension one, so for a typical potential the set of critical energies is at most countable, hence typically not seen by spectral measures. Key to our approach are two results about the dependence of the Lyapunov exponent of onefrequency SL(2, C) cocycles with respect to perturbations in the imaginary direction: on one hand there is a severe “quantization” restriction, and on the other hand “regularity” of the dependence characterizes uniform hyperbolicity when the Lyapunov exponent is positive. Our method is independent of arithmetic conditions on the frequency.
Absolutely continuous spectrum for the almost Mathieu operator
, 2008
"... We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twentyfirst century. ..."
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Cited by 15 (5 self)
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We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twentyfirst century.
Discussion of reference (I
"... mercury attenuator. This irradiator facilitates the irradiation of small animals with dose rate patterns relevant to internal radionuclides, thereby making it possible to investigate the biological effects of timevarying dose rates and to calibrate biological dosimeters. ACKNOWLEDGMENTS ..."
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Cited by 11 (1 self)
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mercury attenuator. This irradiator facilitates the irradiation of small animals with dose rate patterns relevant to internal radionuclides, thereby making it possible to investigate the biological effects of timevarying dose rates and to calibrate biological dosimeters. ACKNOWLEDGMENTS
Regularity and convergence rates for the Lyapunov exponents of linear cocycles, preprint
, 2012
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Opening gaps in the spectrum of strictly ergodic Schrödinger operators
, 2009
"... We consider Schrödinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in t ..."
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Cited by 6 (2 self)
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We consider Schrödinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap, the sampling function may be continuously deformed so that the gap immediately opens. As a corollary, we conclude that for generic sampling functions, all gaps are open. The proof is based on the analysis of continuous SL(2, R) cocycles, for which we obtain
KotaniLast problem and Hardy spaces on surfaces of Widom type
 Department of Mathematics, Rice University, Houston TX 77005, U.S.A. Email address: damanik@rice.edu Department of Mathematics, Rice University, Houston TX 77005, U.S.A. Email address: milivoje.lukic@rice.edu Department of Mathematics, University of Tor
, 1210
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HÖLDER CONTINUITY OF ABSOLUTELY CONTINUOUS SPECTRAL MEASURES FOR ONEFREQUENCY SCHRÖDINGER OPERATORS
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Hölder Continuity of the Spectral Measures for OneDimensional Schrödinger Operator in Exponential Regime
, 2013
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