A Coulomb-related family of potentials having zero-energy bound states

• Central potentials are proposed.
• Zero-energy bound states are found.
• Point transformation is discussed.
• Quantum–classical comparison is made.

We propose here a new large class of singular central potentials having zero-energy bound states for many values of angular momenta. The potentials are shown to be closely related to the standard attractive Coulomb interaction. Some of them admit the E≠0E≠0 bound states as well. A quantum–classical correspondence is also discussed.