Almost Eigenvalues and Eigenvectors of Almost Mathieu Operators

Abstract

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

Dedicated to John Benedetto on the occasion of his eightieth birthday. In John’s rich scientific oeuvre, the topics of spectrum, sampling, discretization, Fourier transform and time-frequency analysis play a central role. This book chapter is partly inspired by John’s work and draws from all these topics.