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)
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.