Comment on “Generalized tanh-shaped hyperbolic potential: bound state solution of Schrödinger equation”

Abstract

Recently Ahmadov et al. obtained an analytical bound-state energy expression for the generalized tanh-shaped hyperbolic potential using the Nikiforov–Uvarov method together with a Pekeris approximation to the centrifugal term. In this Comment, we examine the numerical spectrum reported by Ahmadov et al. in Table 1 of their paper. Three issues are identified. First, sixteen of the tabulated energies lie below the minimum of the potential or of the corresponding effective potential obtained after the Pekeris approximation and therefore cannot represent bound-state eigenvalues. Second, direct substitution into the derived energy expression reproduces the reported \(s\)-state values but not the \(l>0\) entries. The quantum-number notation used by the authors in Table 1 is also inconsistent with their statement that \(n\) is the radial quantum number. The \(s\)-wave finite-state condition additionally excludes the \(5\text{s}\) value reported for the third parameter set. Third, comparison with a direct numerical integration of the original radial Schrödinger equation shows that the analytical expression is very accurate for small \(\alpha \), whereas the agreement becomes poorer as \(\alpha \) increases, particularly for states with larger \(l\).