96-26 Gordon A., Jitomirskaya S., Last Y., Simon B.

Duality and Singular Continuous Spectrum in the Almost Mathieu Equation (38K, AMSTeX) Jan 30, 96

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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.

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