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The Ten Martini problem. (English) Zbl 1166.47031
From the Introduction: In this paper, we solve the Ten Martini Problem as stated in [B. Simon, in: Mathematical physics 2000. International congress, London, GB, 2000 (London: Imperial College Press), 283–288 (2000; Zbl 1074.81521)].
Theorem. The spectrum of the almost Mathieu operator is a Cantor set for all irrational $$\alpha$$ and for all $$\lambda \neq 0$$.
The almost Mathieu operator is the Schrödinger operator on $$\ell^2(\mathbb{Z})$$,
$(H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+2\lambda\cos 2\pi(\theta+n\alpha)u_n,$
where $$\lambda,\alpha,\theta \in \mathbb{R}$$ are parameters (called the coupling, frequency, and phase, respectively), and one assumes that $$\lambda\neq 0$$. The interest in this particular model is motivated both by its connections to physics and by a remarkable richness of the related spectral theory. This has made the latter a subject of intense research in the last three decades.
If $$\alpha=\frac{p}{q}$$ is rational, it is well-known that the spectrum consists of the union of $$q$$ intervals called bands, possibly touching at the endpoints. In the case of irrational $$\alpha$$, the spectrum $$\Sigma_{\lambda,\alpha}$$ (which in this case does not depend on $$\theta$$) has been conjectured for a long time to be a Cantor set. To prove this conjecture has been dubbed The Ten Martini Problem by B. Simon [op. cit.]. For a history of this problem, see [Y. Last, in: Sturm–Liouville theory. Past and present (Basel: Birkhäuser), 99–120 (2005; Zbl 1098.39011)].

##### MSC:
 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47A10 Spectrum, resolvent
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##### References:
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