  The approximated energy eigenvalues and the corresponding eigenfunctions of
the spherical Woods-Saxon effective potential in $D$ dimensions are obtained
within the new improved quantization rule for all $l$-states. The Pekeris
approximation is used to deal with the centrifugal term in the effective
Woods-Saxon potential. The inter-dimensional degeneracies for various orbital
quantum number $l$ and dimensional space $D$ are studied. The solutions for the
Hulth\'{e}n potential, the three-dimensional (D=3), the $% s$-wave ($l=0$) and
the cases are briefly discussed.
