The cylindrical δ-potential and the Dirac equation

In this paper, we discuss the Dirac equation in the presence of an attractive cylindrical δ-shell potential V(ρ) = −aδ(ρ − ρ0), where ρ is the radial coordinate and a > 0. We present a detailed discussion on the boundary conditions the wavefunction has to satisfy when crossing the support of the potential, proceeding then to explore the dependence of the ground state on the parameter a, analyzing the occurrence of supercritical effects. We also apply the Foldy–Wouthuysen transformation, discussing the non-relativistic limit of this problem.