  We study the total photodetachment cross section of H$^-$ near a repulsive
center. An analytical formula for the photodetachment cross section is obtained
using the standard closed-orbit theory and extending it to the energy range
below the zero-field threshold. The formula is found to be accurate by
comparing with an exact quantum calculation. A comparison with the
photodetachment cross section in an effective homogeneous electric field is
made, and we discuss the similarities and differences of the two systems.
