  We study theoretically the optical properties of an exciton in a
two-dimensional ring threaded by a magnetic flux. We model the quantum ring by
a confining potential that can be continuously tuned from strictly
one-dimensional to truly two-dimensional with finite radius-to-width ratio. We
present an analytic solution of the problem when the electron-hole interaction
is short-ranged. The oscillatory dependence of the oscillator strength as a
function of the magnetic flux is attributed to the Aharonov-Bohm effect. The
amplitude of the oscillations changes upon increasing the width of the quantum
ring. We find that the Aharonov-Bohm oscillations of the ground state of the
exciton decrease with increasing the width, but remarkably the amplitude
remains finite down to radius-to-width ratios less than unity. We attribute
this resilience of the excitonic oscillations to the non-simply connectedness
of our chosen confinement potential.
