Reflections on the almost Mathieu operator
The author considers the almost Mathieu operator \(\phi_{n+1} + [\lambda\cos 2\pi (n \theta - \beta)] \phi_n + \phi_{n-1} = E\phi_n, \) where \(\{ \phi_n \}_{n \in Z}\) is a square summable sequence, \(\beta, \lambda\) are real constants and the rotation parameter \(\theta\) is a rational number \(\theta = p/q\). An explicit reduction of the spectral problem to the computation of the eigenvalues for a certain tridiagonal matrix is given. This reduction is valid for both odd and even values of \(q\). The tridiagonal form is useful for numerical eigenvalue computations.
scientific article; zbMATH DE number 1113532
scientific article