Almost Mathieu operator

In mathematical physics, the almost Mathieu operator, named for its similarity to the Mathieu operator^[1] introduced by Émile Léonard Mathieu,^[2] 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.^[3] 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.