Almost Eigenvalues and Eigenvectors of Almost Mathieu Operators

TL;DR

This paper provides explicit eigenvalue estimates and constructs approximate eigenvectors for the finite-dimensional almost Mathieu operator, extending results to the infinite case with numerical validation.

Contribution

It introduces explicit eigenvalue bounds and approximate eigenvectors for the almost Mathieu operator at the spectrum edge, applicable under mild conditions.

Findings

Explicit eigenvalue estimates at the spectrum edge

Construction of approximate eigenvectors using Hermite functions

Extension of results to the infinite-dimensional case

Abstract

The almost Mathieu operator is the discrete Schr\"odinger operator $H_{\alpha,\beta,\theta}$ 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. We furthermore show that the (properly rescaled) $m$-th Hermite function $\phi_m$ 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 $\phi_m$ 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 $\theta$ and under very mild conditions on $\alpha$ and $\beta$. 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.