  An analytical study of discrete and resonant impurity quasi-Coulomb states in
a narrow gaped armchair graphene nanoribbon (GNR) is performed. We employ the
adiabatic approximation assuming that the motions parallel ("slow") and
perpendicular ("fast") to the boundaries of the ribbon are separated
adiabatically. The energy spectrum comprises a sequence of series of
quasi-Rydberg levels relevant to the "slow" motion adjacent from the low
energies to the size-quantized levels associated with the "fast" motion. Only
the series attributed to the ground size-quantized sub-band is really discrete,
while others corresponding to the excited sub-bands consist of quasi-discrete
(Fano resonant) levels of non-zero energetic widths, caused by the coupling
with the states of the continuous spectrum branching from the low lying
sub-bands. In the two- and three-subband approximation the spectrum of the
complex energies of the impurity electron is derived in an explicit form.
Narrowing the GNR leads to an increase of the binding energy and the resonant
width both induced by the finite width of the ribbon. Displacing the impurity
centre from the mid-point of the GNR causes the binding energy to decrease
while the resonant width of the first excited Rydberg series increases. As for
the second excited series their widths become narrower with the shift of the
impurity. A successful comparison of our analytical results with those obtained
by other theoretical and experimental methods is presented. Estimates of the
binding energies and the resonant widths taken for the parameters of typical
GNRs show that not only the strictly discrete but also the some resonant states
are quite stable and could be studied experimentally in doped GNRs.
