Type I energies with positive Lyapunov exponent and gap-labelling condition bound open spectral gaps for irrational frequencies and trig-polynomial potentials, making the all-gaps-open property robust for perturbed almost-Mathieu operators.
The absolutely continuous spectrum of the almost Mathieu operator
3 Pith papers cite this work. Polarity classification is still indexing.
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abstract
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 Schr\"odinger operator problems for the twenty-first century.
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UNVERDICTED 3representative citing papers
ℓ¹→ℓ^∞ dispersive decay of order t^{-1/3} holds for the discrete Klein-Gordon equation on Z with small analytic quasi-periodic potentials, yielding Strichartz estimates and small-data global existence for the nonlinear problem.
The spectrum E = R²(e^p + e^{-p}) + (e^x + e^{-x}) from local P¹ × P¹ is identified with the almost Mathieu operator, yielding three spectral phases separated by transitions at R² = 1 and R² = e^β.
citing papers explorer
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Monotonicity, global symplectification and the stability of Dry Ten Martini Problem
Type I energies with positive Lyapunov exponent and gap-labelling condition bound open spectral gaps for irrational frequencies and trig-polynomial potentials, making the all-gaps-open property robust for perturbed almost-Mathieu operators.
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Dispersive estimates for discrete Klein-Gordon equations on one-dimensional lattice with quasi-periodic potentials
ℓ¹→ℓ^∞ dispersive decay of order t^{-1/3} holds for the discrete Klein-Gordon equation on Z with small analytic quasi-periodic potentials, yielding Strichartz estimates and small-data global existence for the nonlinear problem.
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Geometric Engineering and Almost Mathieu Operator
The spectrum E = R²(e^p + e^{-p}) + (e^x + e^{-x}) from local P¹ × P¹ is identified with the almost Mathieu operator, yielding three spectral phases separated by transitions at R² = 1 and R² = e^β.