Ten martini problem

In mathematical physics, the almost Mathieu operator arises in the study of the quantum Hall effect. It is given by

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.

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

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