A unified scheme of central symmetric shape-invariant potentials

Most physical systems, whether classical or quantum mechanical, exhibit spherical symmetry. Angular momentum, denoted as \(\ell \), is a conserved quantity that appears in the centrifugal potential when a particle moves under the influence of a central force. This study introduces a formalism in which \(\ell \) plays a unifying role, consolidating solvable central potentials into a superpotential. This framework illustrates that the Coulomb potential emerges as a direct consequence of a homogeneous (r-independent) isotropic superpotential. Conversely, an \(\ell \)-independent central superpotential results in a 3-dimensional harmonic oscillator (3-DHO) potential. Moreover, a local \(\ell \)-dependent central superpotential generates potentials applicable to finite-range interactions such as molecular or nucleonic systems. Additionally, we discuss generalisations to arbitrary D dimensions and investigate the properties of the superpotential to determine when supersymmetry is broken or unbroken. This scheme also shows that the free-particle wave function in three dimensions is obtained from the spontaneous breakdown of supersymmetry and clarifies how a positive 3-DHO potential, as an upside-down potential, can have a negative energy spectrum. We also present complex isospectral deformations of the central superpotential and superpartners, which can have interesting applications for open systems in dynamic equilibrium. Finally, as a practical application, we apply this formalism to specify a new effective potential for the deuteron.