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Cantor spectrum for the almost Mathieu equation. (English) Zbl 0516.47018
For a dense $$G_\delta$$ of pairs $$(\lambda, \alpha)$$ in $$\mathbb R^2$$, the authors prove that the operator $(Hu)(n)=u(n+1)+u(n-1)+\lambda\,\cos(2\pi\alpha n+\theta)u(n)$ has a nowhere dense spectrum. Additional results are proved for rational $$\alpha$$. For example, if $$\alpha=(2p+1)/2q$$ and $$\theta=0$$ [resp. $$\pi/2q$$] if $$q$$ is even [resp. odd] then one spectral gap is closed for all $$\lambda$$. The paper concludes with a conjecture about how the main theorem might be strengthened using $$K$$-theory or homotopy.
Reviewer: A. L. Andrew

MSC:
 47B39 Linear difference operators 47A10 Spectrum, resolvent
Keywords:
spectral gap; $$K$$-theory; homotopy
Full Text:
References:
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