  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.
