Confined excitons in semiconductors: Correlation between binding energy and spectral absorption shape

We calculate excitonic absorption spectra in several types of semiconductor nanostructures by using an original formalism involving fractional-dimensional spaces. This compact, analytical formulation predicts a surprising one-to-one mapping between the enhanced exciton Rydberg and the shape of the spectrum, whatever the physical origin of the confinement. By all-numerical calculation of the optical susceptibility we check that different systems with identical exciton Rydbergs exhibit nearly identical spectra, strikingly close to those given by the fractional-dimensional approach.