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Distributions of the $S$-matrix poles in Woods-Saxon and cut-off Woods-Saxon potentials

The positions of the $l=0$ $S$-matrix poles are calculated in generalized Woods-Saxon (GWS) potential and in cut-off generalized Woods-Saxon (CGWS) potential. The solutions of the radial equations are calculated numerically for the CGWS potential and analytically for GWS using the formalism of Gy. Bencze \cite{[Be66]}. We calculate CGWS and GWS cases at small non-zero values of the diffuseness in order to approach the square well potential and to be able to separate effects of the radius parameter and the cut-off radius parameter. In the case of the GWS potential the wave functions are reflected at the nuclear radius therefore the distances of the resonant poles depend on the radius parameter of the potential. In CGWS potential the wave function can be reflected at larger distance where the potential is cut to zero and the derivative of the potential does not exist. The positions of most of the resonant poles do depend strongly on the cut-off radius of the potential, which is an unphysical parameter. Only the positions of the few narrow resonances in potentials with barrier are not sensitive to the cut-off distance. For the broad resonances the effect of the cut-off can not be corrected by using a suggested analytical form of the first order perturbation correction.