  We study states arising from fluctuations in the disorder potential in
systems with long-range hopping. Here, contrary to systems with short-range
hopping, the optimal fluctuations of disorder responsible for the formation of
the states in the gap, are not rendered shallow and long-range when $E$
approaches the band edge ($E\to 0$). Instead, they remain deep and short-range.
The corresponding electronic wave functions also remain short-range-localized
for all $E<0$. This behavior has striking implications for the structure of the
wave functions slightly above $E=0$. By a study of finite systems, we
demonstrate that the wave functions $\Psi_E$ transform from a localized to a
quasi-localized type upon crossing the $E=0$ level, forming resonances embedded
in the $E>0$ continuum. The quasi-localized $\Psi_{E>0}$ consists of a
short-range core that is essentially the same as $\Psi_{E=0}$ and a delocalized
tail extending to the boundaries of the system. The amplitude of the tail is
small, but it decreases with $r$ slowly. Its contribution to the norm of the
wave function dominates for sufficiently large system sizes, $L\gg L_c(E)$;
such states behave as delocalized ones. In contrast, in small systems, $L\ll
L_c(E)$, quasi-localized states are overwhelmingly dominated by the localized
cores and are effectively localized.
