Almost Eigenvalues and Eigenvectors of Almost Mathieu Operators

The almost Mathieu operator is the discrete Schrodinger Hα,β,θ on \(\ell ^2(\mathbb {Z})\) defined via \((H_{\alpha ,\beta ,\theta }f)(k) = f(k + 1) - \beta \cos {}(2\pi \alpha k \theta ) f(k)\). We derive explicit estimates for eigenvalues at edge of spectrum finite-dimensional \({H^{(n)}_{\alpha }}\). furthermore show that (properly rescaled) m-th Hermite function ϕm an approximate eigenvector }}\), and it satisfies same properties characterize true associated with largest eigenvalue Moreover, a properly translated modulated version also (in modulus) negative eigenvalue. results hold spectrum, any choice θ under very mild conditions α β. give precise size “edge,” extend some our to Hα,β,θ. ingredients proofs comprise special recursion functions, Taylor expansions, time-frequency analysis, Sturm sequences, perturbation theory eigenvectors. Numerical simulations demonstrate tight fit theoretical estimates.