Differential equation for the Uehling potential

The second-order differential equation for the Uehling potential is derived explicitly. The right side of this differential equation is a linear combination of the two Macdonald’s functions $K_0(br)$ and $K_1(br)$. This central potential is of great interest in many QED problems, since it describes the lowest-order correction for vacuum polarization in few- and many-electron atoms, ions, muonic and bi-muonic atoms/ions as well as in other similar systems.