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Corrected Analytical Solution of the Generalized Woods-Saxon Potential for Arbitrary $\ell$ States

The bound state solution of the radial Schrödinger equation with the generalized Woods-Saxon potential is carefully examined by using the Pekeris approximation for arbitrary $\ell$ states. The energy eigenvalues and the corresponding eigenfunctions are analytically obtained for different $n$ and $\ell$ quantum numbers. The obtained closed forms are applied to calculate the single particle energy levels of neutron orbiting around $^{56}$Fe nucleus in order to check consistency between the analytical and Gamow code results. The analytical results are in good agreement with the results obtained by Gamow code for $\ell=0$.