The cylindrical \delta-potential and the Dirac equation

TL;DR

This paper investigates the Dirac equation with an attractive cylindrical delta-shell potential, analyzing boundary conditions, ground state dependence on potential strength, supercritical effects, and the non-relativistic limit via Foldy-Wouthuysen transformation.

Contribution

It provides a detailed analysis of boundary conditions and supercritical phenomena for the Dirac equation with cylindrical delta potentials, including the non-relativistic limit.

Findings

Boundary conditions for wave functions crossing the delta-shell

Dependence of ground state on potential strength a

Occurrence of supercritical effects

Abstract

In this article we discuss the Dirac equation in the presence of an attractive cylindrical \delta-shell potential V(\rho)=-a\delta(\rho-\rho_0), where \rho is the radial coordinate and a>0. We present a detailed discussion on the boundary conditions the wave function 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.