  We develop an analytical approach to the exciton optical absorption for
narrow gap armchair graphene nanoribbons (AGNR). We focus on the regime of
dominant size quantization in combination with the attractive electron-hole
interaction. An adiabatic separation of slow and fast motions leads via the
two-body Dirac equation to the isolated and coupled subband approximations.
Discrete and continuous exciton states are in general coupled and form quasi
Rydberg series of purely discrete and resonance type character. Corresponding
oscillator strengths and widths are derived. We show that the exciton peaks are
blue-shifted, become broader and increase in magnitude upon narrowing the
ribbon. At the edge of a subband the singularity related to the 1D density of
states is transformed into finite absorption via the presence of the exciton.
Our analytical results are in good agreement with those obtained by other
methods including numerical approaches. Estimates of the expected experimental
values are provided for realistic AGNRs.
