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.
