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 $(H_{α,β,θ}f)(k) = f(k + 1) + f(k - 1) + β\cos(2παk + θ) f(k)$. We derive explicit estimates for the eigenvalues at the edge of the spectrum of the finite-dimensional almost Mathieu operator. We furthermore show that the (properly rescaled) $m$-th Hermite function $ϕ_m$ is an approximate eigenvector of this operator, and that it satisfies the same properties that characterize the true eigenvector associated to the $m$-th largest eigenvalue. Moreover, a properly translated and modulated version of $ϕ_m$ is also an approximate eigenvector of this operator, and it satisfies the properties that characterize the true eigenvector associated to 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 the infinite dimensional case. The ingredients for our proofs comprise Taylor expansions, basic time-frequency analysis, Sturm sequences, and perturbation theory for eigenvalues and eigenvectors. Numerical simulations demonstrate the tight fit of the theoretical estimates.