Investigating of Singularity of Central Shape Invariant Potentials

In this research, the singularity of the central shape-invariant potentials, which have a singularity of the inverse-square power , has been investigated. It has been shown that in quantum mechanics, for , the eigenvalue problem is well-defined and, as a result, the energy spectrum can be determined. In the transition region, for , both regular and irregular wave functions are square integrable and therefore acceptable, but the boundary conditions for determining the eigenvalues and eigenfunctions are not sufficient and there is no a specific predetermined mechanism for choosing a linear combination of wave functions. For , the particle is drawn to the singularity, and therefore, there is no any ground state with finite energy. It has also been shown using supersymmetric quantum mechanics that the inverse-square potential is the result of the singular inverse superpotential . Supersymmetric quantum mechanics provides a mechanism that, without any additional constraints, the less singular wave function is chosen and the potential is placed in the transition region for .