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

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.