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 [ H ω λ , α u ] ( n ) = u ( n + 1 ) + u ( n − 1 ) + 2 λ cos ( 2 π ( ω + n α ) ) u ( n ) , {\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 ℓ 2 ( Z ) {\displaystyle \ell ^{2}(\mathbb {Z} )} .