Diffuse versus square-well confining potentials in modelling $A$@C$_{60}$ atoms

Attention: this version-$2$ of the manuscript differs from its previously uploaded version-$1$ (arXiv:1112.6158v1) and subsequently published in 2012 J. Phys. B \textbf{45} 105102 only by a removed typo in Eq.(2) of version-$1$; there was the erroneous factor "2" in both terms in the right-hand-side of the Eq.(2) of version-$1$. Now that the typo is removed, Eq.(2) is correct. A perceived advantage for the replacement of a discontinuous square-well pseudo-potential, which is often used by various researchers as an approximation to the actual C$_{60}$ cage potential in calculations of endohedral atoms $A$@C$_{60}$, by a more realistic diffuse potential is explored. The photoionization of endohedral H@C$_{60}$ and Xe@C$_{60}$ is chosen as the case study. The diffuse potential is modelled by a combination of two Woods-Saxon potentials. It is demonstrated that photoionization spectra of $A$@C$_{60}$ atoms are largely insensitive to the degree $\eta$ of diffuseness of the potential borders, in a reasonably broad range of $\eta$'s. Alternatively, these spectra are found to be insensitive to discontinuity of the square-well potential either. Both potentials result in practically identical calculated spectra. New numerical values for the set of square-well parameters, which lead to a better agreement between experimental and theoretical data for $A$@C$_{60}$ spectra, are recommended for future studies.