We investigate theoretically excitonic effects on the optical properties of one-dimensional (1D) semiconductors. In particular, absorption spectra near a band edge are exactly calculated within the effective-mass approximation for the 1D system with a direct allowed or forbidden gap. We employ two kinds of interaction potentials between an electron and a hole describing a modified Coulomb interaction and a short-range interaction, both of which are free from the well-known divergence problem of the 1D Coulomb system. The Sommerfeld factor, which is the absorption intensity ratio of the unbound (continuum) exciton to the free-electron-hole pair above the band edge, is found to be smaller than unity for the direct allowed transition, in striking contrast to the 3D and 2D cases. This peculiar feature is interpreted in terms of the anomalously strong concentration of the oscillator strength on the lowest discrete exciton state. On the other hand, for the direct forbidden transition, the Sommerfeld factor in the 1D system is larger than unity and shows similar behavior to those in the 3D and 2D cases. These properties hold irrespective of the interaction range of the electron-hole attractive potential. The feasibility of the model potentials is examined, and the Coulomb potential having a cusp-type cutoff is found to be the most effective to describe the potential in an actual semiconductor wire. A dielectric effect in the wire structure is shown to enhance these peculiar features of the 1D system.
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