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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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