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

Any l-state solutions of the Woods-Saxon potential in arbitrary dimensions within the new improved quantization rule