Gordon A., Jitomirskaya S., Last Y., Simon B.
Duality and Singular Continuous Spectrum in the
Almost Mathieu Equation
(38K, AMSTeX)
ABSTRACT. We study the almost Mathieu operator
$(h_{\lambda,\alpha,\theta}u)(n)=u(n+1)+u(n-1)+
\lambda\cos (\pi\alpha n+\theta)u(n)$
on $\ell^2(\Bbb Z)$, and prove that the dual of point spectrum
is absolutely continuous spectrum. We use this to show that for
$\lambda = 2$ it has purely singular continuous spectrum for
a.e.~pairs $(\alpha, \theta)$. The $\alpha$'s for which we prove
this are explicit.