  The density of states, even for a perfectly ordered tight-binding model, can
exhibit a tail-like feature at the top of the band, provided the hopping
integral falls off in space slowly enough. We apply the coherent potential
approximation to study the eigenstates of a tight-binding Hamiltonian with
uncorrelated diagonal disorder and long-range hopping, falling off as a power
$\mu$ of the intersite distance. For a certain interval of hopping range
exponent $\mu$, we show that the phase coherence length is infinite for the
outermost state of the tail, irrespectively of the strength of disorder. Such
anomalous feature can be explained by the smallness of the phase-space volume
for the disorder scattering from this state. As an application of the theory,
we mention that ballistic regime can be realized for Frenkel excitons in
two-dimensional molecular aggregates, affecting to a large extent the optical
response and energy transport.
