97-426 Jitomirskaya S., Last Y.

Anderson Localization for the Almost Mathieu Equation, III. Semi-Uniform Localization, Continuity of Gaps, and Measure of the Spectrum (44K, LaTeX) Jul 30, 97

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Abstract. We show that the almost Mathieu operator, $(H_{\omega,\lambda,\theta}\Psi)(n)=\Psi(n+1) + \Psi(n-1) + \lambda\cos(\pi\omega n +\theta)\Psi(n)$, has semi-uniform (and thus dynamical) localization for $\lambda > 15$ and a.e. $\omega,\theta$. We also obtain a new estimate on gap continuity (in $\omega$) for this operator with $\lambda > 29$ (or $\lambda < 4/29$), and use it to prove that the measure of its spectrum is equal to $|4-2|\lambda||$ for $\lambda$ in this range and all irrational $\omega$'s.

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