Modified attractive inverse-square potential in the induced electric dipole system

We examine the spatial distribution of electric charges within an extended, non-conductive cylinder featuring an inner radius denoted as $r_{0}$. Our investigation unveils the emergence of a distinct modified attractive-inverse square potential, arising from the intricate interplay between the electric field and the induced electric dipole moment of a neutral particle. This modified potential notably departs from the conventional inverse-square potential, showcasing an additional term proportional to $r^{-1}$. As a result, we present compelling evidence for the realization of a discrete energy spectrum within this intricate system.