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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 ℓ 2 {\displaystyle \ell ^{2}}. Here α , ω ∈ T , λ > 0 {\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 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 λ = 1 {\displaystyle \lambda =1} , the almost Mathieu operator is sometimes called Harper's equation.