Almost Mathieu operator

In mathematical physics, the almost Mathieu operator, named for its similarity to the Mathieu operator introduced by Émile Léonard Mathieu, arises in the study of the quantum Hall effect. It is given by

${\displaystyle [H_{\omega }^{\lambda ,\alpha }u](n)=u(n+1)+u(n-1)+2\lambda \cos(2\pi (\omega +n\alpha ))u(n),\,}$

acting as a self-adjoint operator on the Hilbert space ${\displaystyle \ell ^{2}(\mathbb {Z} )}$. Here ${\displaystyle \alpha ,\omega \in \mathbb {T} ,\lambda >0}$ 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. In physics, the almost Mathieu operators can be used to study metal to insulator transitions like in the Aubry–André model.

For ${\displaystyle \lambda =1}$, the almost Mathieu operator is sometimes called Harper's equation.