Almost Mathieu operator: Difference between revisions

Revision as of 04:23, 19 February 2024

Revision as of 18:03, 19 May 2024

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{{Short description|Self-adjoint operator that arises in physical transition problems}}

In [[mathematical physics]], the '''almost Mathieu operator''', named for its similarity to the [[Mathieu function|Mathieu operator]]<ref name=simon1982almost>{{cite journal |last1=Simon |first1=Barry |year=1982 |title=Almost periodic Schrodinger operators: a review |journal=Advances in Applied Mathematics |volume=3 |issue=4 |pages=463-490}}</ref> introduced by [[Émile Léonard Mathieu]],<ref>{{cite web |title=Mathieu equation |url=https://encyclopediaofmath.org/wiki/Mathieu_equation |website=Encyclopedia of Mathematics |publisher=Springer |access-date=February 9, 2024}}</ref> arises in the study of the [[quantum Hall effect]]. It is given by

: <math> [H^{\lambda,\alpha}_\omega u](n) = u(u_{n+1)} + u(u_{n-1)} + 2 \lambda \cos(2\pi (\omega + n\alpha)) u(n), \, </math>

acting as a [[self-adjoint operator]] on the 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]].