  Harper's equation (aka the "almost Mathieu" equation) famously describes the
quantum dynamics of an electron on a one dimensional lattice in the presence of
an incommensurate potential with magnitude $V$ and wave number $Q$. It has been
proven that all states are delocalized if $V$ is less than a critical value
$V_c=2t$ and localized if $V> V_c$. Here, we show that this result (while
correct) is highly misleading, at least in the small $Q$ limit. In particular,
for $V<V_c$ there is an abrupt crossover akin to a mobility edge at an energy
$E_c$; states with energy $|E|<E_c$ are robustly delocalized, but those in the
tails of the density of states, with $|E|>E_c$, form a set of narrow bands with
exponentially small bandwidths $ \sim t\ \exp[-(2\pi\alpha/Q)]$ (where $\alpha$
is an energy dependent number of order 1) separated by band-gaps $ \sim t Q$.
Thus, the states with $|E|> E_c$ are "almost localized" in that they have an
exponentially large effective mass and are easily localized by small
perturbations. We establish this both using exact numerical solution of the
problem, and by exploiting the well known fact that the same eigenvalue problem
arises in the Hofstadter problem of an electron moving on a 2D lattice in the
presence of a magnetic field, $B=Q/2\pi$. From the 2D perspective, the almost
localized states are simply the Landau levels associated with semiclassical
precession around closed contours of constant quasiparticle energy; that they
are not truly localized reflects an extremely subtle form of magnetic
breakdown.
