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S-Gatekeeper (talk | contribs) →The structure of the spectrum: Corrected links (they were leading to a wrong paper) |
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: <math> [H^{\lambda,\alpha}_\omega u](n) = u(n+1) + u(n-1) + 2 \lambda \cos(2\pi (\omega + n\alpha)) u(n), \, </math>
acting as a [[self-adjoint operator]] on the [[Hilbert_space#Sequence_spaces|Hilbert space <math>\ell^2(\mathbb{Z})</math>]]. Here <math>\alpha,\omega \in\mathbb{T}, \lambda > 0</math> are parameters. In [[pure mathematics]], its importance comes from the fact of being one of the best-understood examples of an [[ergodic]] [[Schrödinger operator]]. For example, three problems (now all solved) of [[Barry Simon]]'s fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator.<ref>{{cite book |first=Barry |last=Simon |chapter=Schrödinger operators in the twenty-first century |title=Mathematical Physics 2000 |pages=283–288 |publisher=Imp. Coll. Press |___location=London |year=2000 |isbn=978-1860942303 }}</ref> In physics, the almost Mathieu operators can be used to study metal to insulator transitions like in the [[Aubry–André model]].
For <math>\lambda = 1</math>, the almost Mathieu operator is sometimes called '''Harper's equation'''.
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